AEPAHKME BMKCAHFK[ J@ BAUMWFAD[
Bfkcfnah Umkchjdjnfkad ]hfvmrsfty
MS_AHEFHN ME]KAUFJHAD CJWF\JH[, DDK
Cjunctjh, Bfkcfnah baecuvagdm.jrn 7?1=
Kjpyrfncts K 7?1= Mxpahefhn Meukatfjhad Cjrfzjhs, DDK baecuvagdm.jrn
Add rfncts rmsmrvme. Hj part j` tcfs pugdfkatfjh bay gm rmprjeukme, stjrme fh a rmtrfmvad systmb, jr trahsbfttme, fh ahy `jrb jr gy ahy bmahs, mdmktrjhfk, bmkcahfkad, pcjtjkjpyfhn, rmkjrefhn, jr jtcmrwfsm, wftcjut prfjr pmrbfssfjh j` Mxpahefhn Meukatfjhad Cjrfzjhs.
Kjhemhsme vmrsfjh j` F[GH< 6>3-?-6617550-5-=
Uj by nuru _rj`mssjr Eavfe D. [fiarsifm 16;>-1666
KJHUMHU[ _WM@AKM
FS
AKIHJ^DMENBMHU[
FS
KCA_UMW 1
[UWM[[ AHE [UWAFH
1
1.1
[trmss jh a sur`akm
1
1.7
[trmss at a pjfht [fnh kjhvmhtfjh `jr strmss [trmss mdmbmhts Kjhstruktfjh j` a strmss mdmbmht _dahm strmss
7 ; 5 5 =
1.;
[trmss trahs`jrbatfjh fh t wj efbmhsfjhs
=
1.;.1
Batrfx bmtcje fh twj efbmhsfjhs
0
1.;.7
_rfhkfpad strmssms
>
1.5.5
Baxfbub scmar strmss
6
1.5.1 1.5.7 1.5.; 1.5.5 1.5.=
[trmss trahs`jrbatfjh fh tcrmm efbmhsfjhs _rfhkfpad strmssms _rfhkfpad strmss kjhvmhtfjh Kcaraktmrfstfk mquatfjh ahe strmss fhvarfahts Jktacmerad strmssms Emvfatjrfk strmss
3 3 6 6 1? 11
1.=.1
Avmranm strafhs ]hfts j` avmranm strafhs
11 17
1.0.1 1.0.7
[trafh at a pjfht _dahm strafh _jdar kjjrefhatms
17 15 15
1.>
[trafh trahs`jrbatfjh
1=
1.>.1
_rfhkfpad strafhs
10
1.3
Batmrfad bjemd Dfhmar batmrfad bjemd Fsjtrjpfk batmrfad _dahm strmss ahe pdahm strafh Hjhdfhmar batmrfad bjemds Mdastfk‛pmr`mktdy pdastfk batmrfad bjemd Dfhmar strafh-caremhfhn batmrfad bjemd _jwmr daw batmrfad bjemd
10 10 1> 13 16 16 7? 7?
1.6
M``mkts j` tmbpmraturm
71
1.1?
@afdurm tcmjrfms Baxfbub scmar strmss tcmjry Baxfbub jktacmerad scmar strmss tcmjry Baxfbub hjrbad strmss tcmjry Bjef`fme Bjcr ‖s tcmjry
77 77 7; 7; 75
1.11
Kdjsurm
7=
1.17
[yhjpsfs j` mquatfjhs
70
KCA_UMW 7
JHM-EFBMH[FJHAD [UW]KU]WAD BMBGMW[
7>
7.1
Fhtmrhad `jrkms ahe bjbmhts Emkjupdfhn axfad, gmhefhn, ahe tjrsfjh ahadysfs
7> 73
1.7.1 1.7.7 1.7.; 1.7.5
1.5
1.= 1.0
1.3.1 1.3.7 1.3.; 1.3.5 1.3.= 1.3.0 1.3.>
1.1?.1 1.1?.7 1.1?.; 1.1?.5
7.1.1
KJHUMHU[ 7.7
Djnfk fh bmkcahfks j` batmrfads
76
7.; 7.;.1 7.;.7 7.;.5
Kdassfkad tcmjrfms j` 1-E strukturad bmbgmrs Dfbftatfjhs Em`fhftfjh j` varfagdms ahe sfnh kjhvmhtfjhs [cmar strmss fh tcfh sybbmtrfk gmabs
;? ;? ;1 ;=
7.5.1 7.5.7 7.5.;
Gjuheary vadum prjgdmbs Axfad efspdakmbmht Ujrsfjhad rjtatfjh Gmab em`dmktfjh
;> ;> ;3 ;3
7.=.1 7.=.7 7.=.; 7.=.5
Efskjhtfhufty `uhktfjhs Em`fhftfjhs Axfad efspdakmbmht Ujrsfjhad rjtatfjh Gmab em`dmktfjh
5? 5? 51 57 57
7.0
Kdjsurm
55
7.>
[yhjpsfs j` mquatfjhs
55
KCA_UMW ;
F H @ D ] M H KM @ ] H K U F J H [
5=
;.1 ;.1.1 ;.1.7 ;.1.; ;.1.5 ;.1.=
Gasfk kjhkmpts Em`fhftfjhs @jrkm (sjurkm) sfhnudarfty fh`dumhkm `uhktfjhs fh gmabs Bjbmht (ejugdmt) sfhnudarfty fh`dumhkm `uhktfjhs fh gmabs Hubmrfkad fhtmnratfjh Hjh-efbmhsfjhad varfagdms
5= 5= 5= 50 5> 5>
;.7.1
Fh`dumhkm `uhktfjhs `jr kdassfkad gmabs Fh`dumhkm `uhktfjhs `jr sfbpdy suppjrtme gmab
53 53
Gmabs jh mdastfk `juheatfjhs
=?
;.5.1 ;.5.1
@uheabmhtad sjdutfjhs `jr gmabs jh mdastfk `juheatfjhs [jbm [jbm prjp prjpmr mrtfm tfmss j` `uhe `uheab abmh mhtad tad sjdu sjdutfj tfjhs hs assj assjkf kfat atme me wftc wftc `jrkm `jrkm sfhn sfhnuda udarft rfty y
=1 =;
;.=.1 ;.=.7 ;.=.;
@fhftm gmabs [ybbmtrfk djaefhn Asybbmtrfk djaefhn ahe gjuheary kjheftfjhs Nmhmrad djaefhn ahe gjuheary kjheftfjhs
== =0 =0 =0
;.0
Kdjsurm
=>
;.>
[yhjpsfs j` mquatfjhs
=6
;.3
Batcmbatfkad emtafds j` tcm emrfvatfjh j` `uheabmhtad sjdutfjhs
0?
KCA_UMW 5
[UAGFDFUV
07
5.1
Mudmr gukidfhn Gukidfhn bjems M``mkts j` mhe kjheftfjhs Kdassf`fkatfjh j` kjdubhs
07 0; 05 05
5.7
Fbpmr`mkt kjdubhs
00
5.7
Fbpmr`mkt kjdubhs
00
5.;
Kdjsurm
03
5.5
[yhjpsfs j` mquatfjhs
03
KCA_UMW =
_DAUM[
06
=.1
Dfbftatfjhs jh pdatm tcmjry.
06
7.5
7.=
;.7 ;.; ;.5 ;.=
5.1.1 5.1.7 5.1.;
v
=.7
Ucmjry Ifhmbatfks Batmrfad bjemd [tatfk mqufvadmhky Djkatfjh j` hmutrad sur`akm [trmss `jrbudas Mqufdfgrfub Ef``mrmhtfad mquatfjh Gjuheary kjheftfjhs Kjbpdmxfty bap
>? >? >7 >7 >; >; >5 >5 >= >0
=.;
Havfmr ‖s sjdutfjh `jr rmktahnudar pdatms
>3
=.5
Haeaf-Dmvy sjdutfjh j` rmktahnudar pdatms Cjbjnmhmjus sjdutfjh _artfkudar sjdutfjh
3? 3? 31
Kfrkudar pdatms
37
Axfsybbmtrfk pdatms ]hf`jrb djae [jdfe kfrkudar pdatms Ahhudar pdatms Hjh-efbmhsfjhad mquatfjhs fh ahhudar pdatms
35 3= 3= 3= 3=
=.>
Kdjsurm
3>
=.3
[yhjpsfs j` mquatfjhs
33
KCA_UMW 0
FHUWJE]KUFJH UJ MDA[UFKFUV
33
0.1 0.1.1 0.1.7 0.1.; 0.1.5 0.1.=
Mdastfkfty mquatfjhs [trafh-efspdakmbmht mquatfjhs Kjbpatfgfdfty mquatfjhs _dahm strmss ahe pdahm strafh Mqufdfgrfub mquatfjhs Gjuheary kjheftfjhs
33 33 33 36 36 61
0.7.1 0.7.7
Axfsybbmtrfk prjgdmbs Axfsybbmtrfk pdahm strafh Axfsybbmtrfk pdahm strmss
61 61 67
0.;
Wjtatfhn efsis
67
0.5
Afry strmss `uhktfjh
67
0.= 0.=.1 0.=.7 0.=.;
[jdutfjh gy pjdyhjbfads Quaeratfk pjdyhjbfads Kugfk pjdyhjbfads @jurtc-jremr pjdyhjbfads
6; 6; 6; 65
0.0.1
Efspdakmbmhts `rjb strafhs fh 7-E Wfnfe gjey bjtfjh
6= 6=
0.>.1 0.>.7 0.>.;
Ujrsfjh j` hjh-kfrkudar sca`ts [afht-Pmhaht‖s bmtcje _rahetd‖s bmtcje _rjkmeurm `jr sjdvfhn prjgdmbs j` tjrsfjh j` hjh-kfrkudar sca`ts
6= 60 6> 63
0.3
Bmbgrahm ahadjny `jr tjrsfjh j` hjh-kfrkudar sca`ts
63
0.3.1
Bmbgrahm ahadjny `jr krjss smktfjh wftc cjdms
1??
0.6
Ujrsfjh j` tcfh-waddme jpmh smktfjh
1??
0.6.1
Mhe m``mkts jh rmktahnudar krjss smktfjhs
1?1
0.1?
Kdjsurm
1?1
0.11
[yhjpsfs j` mquatfjhs
1?;
=.7.1 =.7.7 =.7.; =.7.5 =.7.= =.7.0 =.7.> =.7.3 =.7.6
=.5.1 =.5.7 =.= =.0 =.0.1 =.0.7 =.0.; =.0.5
0.7
0.0 0.>
KJHUMHU[
vff
KCA_UMW >
PAWFAUFJHAD AHE MHMWNV BMUCJE[
10 >
>.1
Gasfk kjhkmpts fh varfatfjhad kadkudus Mxtrmbub ahe statfjhary Padums @uhktfjhads
10> 103 106
^jri
106
[trafh mhmrny [trafh mhmrny fh sybbmtrfk gmhefhn j` gmabs [trafh mhmrny fh gmhefhn j` tcfh pdatms [trafh mhmrny fh pdahm strmss mdastfkfty [trafh mhmrny fh `jrb j` gfdfhmar `uhktfjhad
1>? 1>1 1>1 1>7 1>;
>.5
Pfrtuad wjri
1>;
>.=
Bfhfbub pjtmhtfad mhmrny
1>;
>.> >.>.7 >.>.5
[tatfjhary vadum j` a em`fhftm dfhm fhtmnrad Gjuheary vadum prjgdmb Nmhmradfzatfjh
1>= 1>0 1>3
>. 0 >.3.1 >. 6
[tatfjhary vadum j` a em`fhftm arma fhtmnrad Batcmbatfkad prmdfbfhArfms [tatfjhary vadum j` a `uhktfjhad wftc `frst jremr emrfvatfvms [tatfjhary vadum j` a `uhktfjhad wftc smkjhe jremr emrfvatfvms
1>6 1>5 1>6 1>6
>.1?
[tatfjhary vadum j` a em`fhftm vjdubm fhtmnrad
135
>.11
Hjhdfhmarftfms Batmrfad hjhdfhmarfty Nmjbmtrfk hjhdfhmarfty
135 13= 135
>.17
Waydmfnc-Wftz Bmtcje
130
>.1;
@fhftm mdmbmht bmtcje Danrahnm pjdyhjbfads fh jhm efbmhsfjh Haturad kjjrefhatms Pmktjr arftcbmtfk Danrahnm pjdyhjbfads fh twj efbmhsfjhs Mdmbmht stf``hmss batrfx ahe rfnct cahe sfem vmktjr Danrahnm pjdyhjbfads fh tcrmm efbmhsfjhs
133 136 161 161 167 16; 16;
>.15
Kdjsurm
165
>.1=
[yhjpsfs j` mquatfjhs
160
KCA_UMW 3
FHEFKFAD HJUAUFJH
1;;
3.1
Gasfk em`fhftfjhs [ubbatfjh kjhvmhtfjh Irjhmkimr `u ` uhktfjh _mrbutatfjh `uhktfjh Emrfvatfvm hjtatfjh
1;; 1;; 1;5 1;5 1;=
Mquatfjhs j` mdastfkfty ahe tcfh pdatm fh fhefkfad hjtatfjh
1;=
3.5.1 3.5.7
Parfatfjhad kadkudus [tatfjhary vadum j` a `uhktfjhad wftc `frst jremr emrfvatfvms [tatfjhary vadum j` a `uhktfjhad wftc smkjhe jremr emrfvatfvms
1;0 1;0 1;>
3.=.1 3.=.7 3.=.; 3.=.5
Hjhdfhmar strafhs A pmrspmktfvm j` rm`mrmhkm `rabm Danrahnfah strafh fh twj efbmhsfjh Mudmrfah strafh fh twj efbmhsfjh Fhefkfad hjtatfjh ahe hjh-dfhmar strafh fh tcrmm efbmhsfjh
1;> 1;3 1;3 1;6 1;6
3.0
Kdjsurm
15?
3.>
[yhjpsfs j` mquatfjhs
151
>.1.1 >.1.7 >.7 >.; >.;.1 >.;.7 >.;.; >.;.5
>.3
>.11.7 >.11.1
>.1;.1 >.1;.7 >.1;.; >.1;.5 >.1;.= >.1;.0
3.1.1 3.1.7 3.1.; 3.1.5 3.7 3.5
3.=
A__MHEFS A FHEMS
G FG DFJNWA_CV
157 15;
KJHUMHU[
fx
_WM@AKM Fh 7?1?, F taunct Aevahkme taunct Aevahkme Bmkcahfks j` Batmrfads `jr Batmrfads `jr tcm `frst tfbm. Ucm gjjis avafdagdm `jr tmakcfhn tcm kjursm wmrm sfbfdar tj tcm jhms fh usm wcmh F was a nraeuatm stuemht bjrm tcah `jrty ymars anj. Ucm qumstfjh F hmmeme tj ahswmr was< ^cat arm tcm hmmes j` tcm stuemhts tjeay tcat scjude gm aeermssme gy a nraeuatm dmvmd kjursm fh bmkcahfks j` batmrfads: Ucfs gjji fs by ahswmr as mdagjratme hmxt. Ucm mardy emvmdjpbmhts j` strukturad tcmjrfms wmrm vmry fhtuftfvm. Ucmsm emvmdjpbmhts wmrm erfvmh gy a emsfrm tj nmt ah ahadytfkad sjdutfjh `jr usm fh emsfnh j` strukturad bmbgmrs. Fh tcm past `jrty ymars wm cavm puscme tcm emsfnh mhvmdjpm tj krmatm bjrm m``fkfmht strukturms. Ucfs m``fkfmhky cas gmmh jgtafhme gy bjef`yfhn jur kdassfkad strukturad tcmjrfms wftc kjbpdmxftfms sukc as ahfsjtrjpy, fhmdastfk strafhs, batmrfad hjh-cjbjnmhmfty, batmrfad hjh-dfhmarfty, nmjbmtrfk hjh-dfhmarfty, scmar fh gmabs ahe pdatms, mtk. Ucjunc nmttfhn ah ahadytfkad sjdutfjh fs stfdd a emsfragdm njad, ft fs hj djhnmr a dfbftatfjh ft was fh tcm past. Hjw wm kah `fhe hubmrfkad sjdutfjhs a`tmr fhkjrpjratfhn kjbpdmxftfms j` fhtmrmst fhtj jur tcmjry. Cjwmvmr, tcfs hmme-erfvmh fhkjrpjratfjh j` kjbpdmxftfms cas krmatme a vmry darnm ahe garjqum ihjwdmenm gasm tcat fs gmkjbfhn a kcaddmhnm tj bastmr `jr bjst stuemhts j` bmkcahfks. [yhtcmsfzfhn tcfs trmbmhejus nrjwtc j` ihjwdmenm fhtj a djnfkad `rabmwjri fs tcm prfbary jglmktfvm j` tcfs gjji . Ucm djnfk scjwh sybgjdfkaddy jh tcm kjvmr j` tcm gjji prjvfems tcm `rabmwjri j` aeefhn kjbpdmxftfms. Mxabpdms ahe pjst-tmxt prjgdmbs cfncdfnct tcm bjeudarfty j` tcm djnfk ahe embjhstratm tcm aeeftfjh j` kjbpdmxftfms tj tcm kdassfkad tcmjrfms j` axfad bmb gmrs, tjrsfjh j` kfrkudar sca`ts, ahe gmhefhn j` gmabs ahe pdatms. Kjbpdmxfty Baps j` Baps j` [mktfjhs 7.;.; ahe 7.;.; ahe =.7.6 dfsts =.7.6 dfsts kjbpdmxftfms ahe assjkfatme mxabpdms ahe prjgdmbs fh tcm kcaptmrs j` jhm efbmhsfjhad strukturad bmbgmrs ahe gmhefhn j` pdatms. ]hemrstahefhn tcm bmtcjejdjny j` fhkjrpjratfhn kjbpdmxfty fhtj tcm kdassfkad tcmjrfms jpmhs tcm ejjr `jr fhkrmefgdm hubgmr j` pjssfgfdftfms tcat arfsm `rjb kjbgfhatfjhs j` kjbpdmxftfms. Ucm jtcmr tjpfks wcfkc baim by gjji bjrm kjhtmbpjrary arm emskrfgme gmdjw fh jremr j` nrmatmst ef``mrmhkms `rjb mxfstfhn gjjis. Ucm dmarhfhn jglmktfvms arm emskrfgme at tcm gmnfhhfhn j` makc kcaptmr. Ucfs gjji cas bjrm batmrfad tcmh kah gm kjvmrme fh a tcrmm krmeft nraeuatm kjursm. Fhstruktjr‖s kcjfkm efktatms wcfkc tjpfks tj fhkduem fh tcmfr kjursm. Hmwtjhfah bmkcahfks ahe ef``mrmhtfad kadkudus cavm gmmh tcm bafh stay j` emvmdjpbmht j` strukturad tcmjrfms. [trahes j` Mudmrfah pmrspmktfvm ahe varfatfjhad kadkudus cavm adways gmmh prmsmht fh strukturad bmkcahfks fh `jrb j` mhmrny bmtcjes. Gut mhmrny bmtcjes arm a sbadd sugsmt j` varfatfjhad kadkudus. Parfatfjhad kadkudus fs a v mry mdmnaht ahe nmhmrad way j` jgtafhfhn gjuheary vadum prjgdmbs gy bfhfbfzfhn tcm pjtmhtfad mhmrny j` a strukturad bmbgmr. _jtmhtfad mhmrny mxprmssfjhs arm masy tj jgtafh mvmh wftc a`jrmbmhtfjhme kjbpdmxftfms, partfkudardy wftc darnm strafhs. Ucus, stuemhts dmarh ah adtmrhatfvm way j` jgtafhfhn gjuheary vadum prjgdmbs tcat bay gm ef``fkudt tj jgtafh wftc ef``mrmhtfad kadkudus. Ah aeeme gjhus fs tcat tcm prfhkfpdms j` varfatfjhad kadkudus arm tcm gmerjkis j` jptfbfz atfjh bmtcjes ahe apprjxfbatm bmtcjes. Apprjxfbatm bmtcjes j` Waydmfnc -Wftz ahe `fhftm mdmbmht bmtcje arm efskussme as ahjtcmr embjhstratfjh j` varfatfjhad kadkudus fh Kcaptmr >. >. Fhefkfad hjtatfjh fs a kjbpakt way j` emrfvfhn mquatfjhs ahe `jrbudas. Ft usms arm embjhstratme `jr pdatm gmhefhn, mdastfkfty, ahe varfatfjhad kadkudus. Ucm kjbgfhatfjh j` fhefkfad hjtatfjh wftc varfatfjhad kadkudus rmeukm emrfvatfjhs j` gjuheary vadum prjgdmbs tj dmss tcah a panm wcmh tcm sabm emrfvatfjh fh mxpaheme hjtatfjh gy ef``mrmhtfad kadkudus bay taim bahy panms. Fhefkfad hjtatfjh fs efskussme fh Kcaptmr 3. 3. Add Aevahkme Add Aevahkme Bmkcahfks j` Batmrfads gjjis Batmrfads gjjis cavm a kcaptmr jh gmabs jh mdastfk `juheatfjhs. Fh Kcaptmr ;, ;, fh`dumhkm `uhktfjhs arm usme tj `jrbudatm ahe sjdvm prjgdmbs j` kdassfkad gmab tcmjry, fh`fhftm gmab jh mdastfk `juheatfjhs ahe `fhftm gmabs jh mdastfk `juheatfjhs. ^cmh djaefhn fs sfbpdm tcmh ahadytfk ad sjdutfjhs arm jgtafhme. ^cmh djaefhn fs kjbpdmx tcmh hubmrfkad fhtmnratfjh fs usme `jr sjdvfhn tcm prjgdmb. Hubmrfkad fhtmnratfjh skcmbm rmqufrms usm j` sprmaescmmts. Add stuemhts arm `abfdfar wftc sprmaescmmts ahe wjude cavm hj ef``fkudty fh `jddjwfhn tcm sjdutfjh prjkmeurm mxpdafhme fh tcm tmxt. Kcaptmrs 1, 5, ahe 0 arm rmdatfvmdy traeftfjhad. Kcaptmr 1 efskussms 1 efskussms strmss ahe strafh ahe batmrfad bjemds―ft smrvms gjtc as a rmvfmw ahe tj fhtrjeukm kjhkmpts usme fh sugsmqumht kcaptmrs. Kcaptmr 5 efskussms 5 efskussms Mudmr gukidfhn tcmjry j` kjdubh gukidfhn wftc varfjus kjbpdmxftfms. Kcaptmr 0 fhtrjeukms 0 fhtrjeukms mdastfkfty ahe Afry strmss `uhktfjh ahe tcmfr usanm tj jgtafh sjdutfjhs tj bmkcahfks j` batmrfads prjgdmb. Add aevahkme bmkcahfks j` batmrfads gjjis kjhtafh djts j` mquatfjhs. Uj cmdp stuemhts, a syhjpsfs j` mquatfjhs fs prmsmhtme at tcm mhe j` makc kcaptmr gm`jrm tcm pjsttmxt prjgdmbs. Jh by wmgsftm baecuvagdm.jrn tcmrm arm smvmrad grfm` hjtmgjjis rmvfmwfhn tjpfks hmmeme fh tcfs gjji. Ujpfks sukc as statfks, bmkcahfks j` batmrfads, batrfx bmtcjes, ahe hubmrfkad bmtcjes―add ah mhnfhmmrfhn stuemht cas smmh fh ah uhemrnraeuatm kurrfkudub. Adtcjunc kjhsfstmht fh fts emsfnh ahe hjtatfjh wftc by fhtrjeuktjry ahe fhtmrbmefatm bmkcahfks j` batmrfads gjjis, tcfs gjji ejms hjt empmhe upjh ahy gjji usme fh tcm uhemrnraeuatm kurrfkudub. F wmdkjbm ahy kjbbmhts, sunnmstfjhs, kjhkmrhs, jr kjrrmktfjhs tcat wfdd cmdp bm fbprjvm tcm gjji. Vju kah rmakc bm tcrjunc by wmgsftm baecuvagdm.jrn.
AKIHJ^DMENBMHU[ _rj`mssjr Fgracfb Bfsifjndu rmvfmwme tcm mhtfrm gjji ahe baem bahy vaduagdm sunnmstfjhs `jr wcfkc F ab vmry nratm`ud. By tcahis adsj tj _rj`mssjr ^fddfab Guddmft `jr rmvfmwfhn Kcaptmr ; ahe cfs sunnmstfjhs arm adsj nratm`uddy akihjwdmenme.
1
[trmss ahe [trafh
DMAWHFHN JGLMKUFPM[ 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5;
1. Uj uhemrstahe tcm kjhkmpts j` strmss ahe strafh. 7. Uj uhemrstahe strmss ahe strafh trahs`jrbatfjhs fh tcrmm efbmhsfjhs. ;. Uj uhemrstahe tcm batmrfad bjemds rmdatfhn strmssms ahe strafhs.
A gufdefhn kjddapsms gmkausm ft was hjt strjhn mhjunc tj suppjrt tcm djaes aktfhn jh ft. Ucm dfe j` a saukm gjttdm kahhjt gm jpmhme gmkausm tcm gjhe gmtwmmh tcm gjttdm ahe tcm dfe fs tjj strjhn. A krahisca`t vfgratms ahe kausms hjfsm gmkausm ft fs hjt stf`` mhjunc. A efvmr kahhjt lubp cfnc mhjunc tj mxmkutm cmr efvms gmkausm tcm efvfhn gjare fs tjj stf``. [trmhntc ahe stf``hmss arm gjtc fbpjrtaht fh tcm emsfnh j` strukturad mdmbmhts. [trmss, a bmasurm j` tcm fhtmhsfty j` fhtmrhad `jrkms, ahe strafh, a bmasurm j` tcm fhtmhsfty j` em`jrbatfjh, arm twj `uheabmhtad varfagdms fh tcm bmkcahfks j` batmrfads tcat arm usme fh assmssfhn strmhntc ahe stf``hmss. Fh tcfs kcaptmr wm start gy grfm`dy rmvfmwfhn tcm kjhkmpts j` strmss ahe strafh at a pjfht. Ucm hjtatfjh j` ejugdm sugskrfpts jh strmssms ahe strafh fs emskrfgme. J` partfkudar fbpjrtahkm fs tcm usm j` tcm sugskrfpts fh emtmrbfhfhn tcm efrmktfjh j` a strmss kjbpjhmht jh a sur`akm. Ahadysfs j` `jrkms ahe em`jrbatfjh, cmhkm j` strmssms ahe strafhs, fs kjheuktme fh a kjjrefhatm systmb tcat fs kcjsmh tj sfbpdf`y ahadysfs. Gut tcm baxfbub strmssms ahe strafhs tcat bay kausm `afdurm fh a batmrfad bay mxfst jh pdahms ahe fh efrmktfjhs tcat arm ef``mrmht `rjb tcm kcjsmh kjjrefhatm systmb. [trmss ahe strafh trahs`jrbatfjh mquatfjhs prjvfem a bmahs j` bjvfhn `rjb jhm kjjrefhatm systmb tj ahjtcmr. Fh tcfs kcaptmr wm start wftc tcm mquatfjhs j` strmss ahe strafh trahs`jrbatfjh fh twj efbmhsfjhs tcat wmrm stuefme fh tcm fhtrjeuktjry kjursm jh tcm bmkcahfks j` batmrfads. ^m kast tcmsm `abfdfar mquatfjhs fh batrfx `jrb ahe usm tcm batrfx bmtcje tj emrfvm `abfdfar kjhkdusfjhs f h twj efbmhsfjhs. ^m tcmh nmhmradfzm tcm batrfx bmtcje apprjakc tj sjdvm prjgdmbs fh strmss ahe strafh trahs`jrbatfjh fh tcrmm efbmhsfjhs. [trmss ahe strafhs arm twj em`fhftfjhs rmdatme gy mquatfjhs jgtafhme mxpmrfbmhtaddy. Ucm mxpmrfbmhtaddy emtmrbfhme mquatfjhs empmhe upjh tcm batmrfad bjemd kjhstruktme tj apprjxfbatm tcm batmrfad gmcavfjr. Fh tcfs kcaptmr wm grfm`dy rmvfmw dfhmar ahe hjh-dfhmar batmrfad bjemds, tcm m``mkt j` tmbpmraturm, ahe `afdurm tcmjrfms. ^m wfdd usm tcmsm batmrfad bjemds fh sugsmqumht kcaptmrs tj emvmdjp tcmjrfms j` strukturad bmbgmrs.
1.1
[UWM[[ [UWM[[ JH A []W@AKM
Ucm `jrkms j` attraktfjh ahe rmpudsfjh rmpudsfjh gmtwmmh twj partfkdms partfkdms (atjbs jr bjdmkudms) fh a gjey arm assubme tj akt adjhn tcm dfhm tcat ljfhs tcm twj partfkdms. @jrkms tcat akt adjhn tcm dfhm ljfhfhn twj partfkdms arm kaddme kmhtrad `jrkms. Ucm kjhkmpt j` kmhtrad `jrkms startme wftc Hmwtjh‖s uhfvmrsad nravftatfjh daw. Kmhtrad `jrkms vary fhvmrsmdy as ah mxpjhmht j` tcm raefad efst ahkm smparatfhn tcm twj partfkdms. At atjbfk dmvmds tcm mxpjhmht fs a pjwmr j` 3 jr 1?.Ucus, mvmry partfkdm mxmrts a `jrkm jh mvmry jtcmr partfkdm, as scjwh sybgjdfkaddy jh ah fbanfhary sur`akm j` a gjey fh @fnurm 1.1a 1.1a. Ucmsm `jrkms gmtwmmh tcm partfkdms cjde tcm gjey tjnmtcmr ahe arm rm`mrrme tj as tcm fhtmrhad `jrkms. Ucm gjey kcahnms scapm wcmh wm appdy mxtmrhad `jrkms jh tcm gjey. Ucm kcahnm fh scapm fbpdfms tcat tcm efstahkm gmtwmmh tcm partfkdms bust kcahnm, wcfkc `urtcmr fbpdfms tcat tcm `jrkms gmtwmmh tcm partfkdms (fhtmrhad `jrkms) bust kcahnm. ^cmh tcm kcahnm fh tcm fhtmrhad `jrkms mxkmmes sjbm kcaraktmrfstfk batmrfad vadum, tcm gjey wfdd grmai. Ucus, tcm strmhntc j` tcm batmrfad kah gm kcaraktmrfzme gy tcm bmasur m j`kcahnm j` kcahnm fh tcm fhtmhsfty j` fhtmhsfty j` fhtmrhad `jrkms. Ucfs bmasurm j` kcahnm fh tcm fhtmhsfty j` fhtmrhad `jrkms fs wcat wm kadd strmss. ^m kah rmpdakm add tcm `jrkms tcat arm mxmrtme jh ahy sfhndm partfkdm fh @fnurm 1.1a 1.1a gy tcm rmsudtaht j` tcmsm `jrkms jh tcat partfkdm as scjwh fh @fnurm 1.1g 1.1g. Ucm banhftuem ahe efrmktfjh j` tcmsm rmsudtaht `jrkms wfdd vary wftc tcm djkatfjh j` tcm partfkdm (pjfht), as scjwh fh @fnurm 1.1g 1.1g. Fh jtcmr wjres, wcmh mxtmrhad `jrkms arm appdfme, ah fhtmrhad efstrfgutme `jrkm systmb fs nmhmratme fh tcm batmrfad. Ucm fhtmrhad efstrfgutme `jrkm jh ah fbanfhary kut sur`akm j` a gjey fs kaddme tcm strmss jh a sur`akm. @urtcmrbjrm, wm hjtm tcat `jrkm fs a vmktjr9 cmhkm tcm fhtmrhad efstrfgutme `jrkms (strmss jh a sur`akm) kah gm rmsjdvme fhtj hjrbad 1
[trmss at a pjfht 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
7
Hjrbad tj pdahm A @ G G
A
G
K M
@ K A
E
(a)
@ A
A
M
K @ M
@ E E
Uahnmht fh pdahm
(g)
@fnurm 1.1 Fhtmrhad `jrkms gmtwmmh partfkdms jh tcm twj sfems j` ah fbanfhary kut. (a) @jrkms gmtwmmh partfkdms fh a gjey scjwh jh partfkdm A. A. (g) Ucm rmsudtaht `jrkm jh makc partfkdm.
(pmrpmhefkudar tj tcm sur`akm) ahe tahnmhtfad (paraddmd tj tcm sur`akm) efstrfgutme `jrkms as scjwh fh @fnurm 1.1g 1.1g. Ucm fhtmrhad efstrfgutme `jrkm tcat fs hjrbad tj tcm sur`akm j` ah fbanfhary kut fs kaddme tcm hjrbad strmss jh a sur`akm. Ucm fhtmrhad efstrfgutme `jrkm tcat fs paraddmd tj tcm sur`akm j` ah fbanfhary kut sur`akm fs kaddme tcm scmar strmss jh tcm sur`akm. Uagdm 1.1 scjws 1.1 scjws tcm varfjus uhfts j` strmss usme fh tcfs gjji. Ft scjude gm hjtme tcat, jhm psf fs psf fs mquad tj 0.6= i_a jr apprjxfbatmdy > i_a. i_a. Adtmrhatfvmdy, Adtmrhatfvmdy, 1 i_a fs mquad tj ?.15= psf jr apprjxfbatmdy ?.1= psf. psf. Uagdm 1.1 ]hfts j` [trmss ]hfts
Emskrfptfjh
Gasfk ]hfts
psf
_juhes pmr squarm fhkc fhkc
dg/fh7
isf
Ifdjpjuhes (ifps) pmr squarm fhkc
1?; dg/fh7
_a
_askads
H/b7
i_a
Ifdjpaskads
1?; H/b7
B_a
Bmnapaskads
1?0 H/b7
N_a
Nfnapaskads
1?6 H/b7
Hjrbad strmss jh a sur`akm bay gm vfmwme as tcm fhtmrhad `jrkms tcat arm e mvmdjpme jwfhn tj tcm batmrfad rmsfstahkm tj tcm puddfhn tcm puddfhn apart jr puscfhn tjnmtcmr j` twj aeljfhfhn pdahms j` pdahms j` ah fbanfhary kut. Dfim prmssurm, hjrbad strmss fs adways pmrpmhefkudar tj tcm sur`akm j` tcm fbanfhary kut. Gut uhdfim prmssurm, wcfkc kah jhdy gm kjbprmssfvm, hjrbad strmss kah gm tmhsfdm. A hjrbad strmss tcat pudds tcm sur`akm away `rjb a gjey fs kaddme a tmhsfdm strmss. A hjrbad strmss tcat puscms tcm sur`akm fhtj a gjey fs kaddme a kjbprmssfvm strmss. Fh jtcmr wjres, tmhsfdm strmss akts fh tcm efrmktfjh j ` tcm jutware hjrbad tj tcm sur`akm, wcfdm kjbprmssfvmstrmss kjbprmssfvm strmss fs jppjsftm tj tcm efrmktfjh j` tcm jutware hjrbad tj tcm sur`akm. Hjrbad strmss fs usuaddy rmpjrtme as tmhsfdm (U) jr kjbprmssfvm (K), hjt as pjsftfvm jr hmnatfvm. Ucus, 1?? B_a (U) ahe 1? isf (K) arm tc m kjhvmhtfjhad ways j` rmpjrtfhn tmhsfdm ahe kjbprmssfvm hjrbad strmssms, rmspmk tfvmdy. [cmar strmss jh a sur`akm bay gm vfmwme as tcm fhtmrhad `jrkms tcat arm emvmdjpme jwfhn tj tcm rmsfstahkm j` tcm batmrfad tj tcm sdfefhn tcm sdfefhn j` twj aeljfhfhn pdahms adjhn pdahms adjhn tcm fbanfhary kut. Dfim `rfktfjh, scmar strmssms akt tahnmhtfaddy tj tcm pdahm fh tcm efrmktfjh jppjsftm tcm fbpmhefhn bjtfjh j` tcm sur`akm. Gut uhdfim `rfktfjh, scmar strmss fs hjt rmdatme tj tcm hjrbad strmss.
1.7
[UWM[[ [UWM[[ AU A _JFHU
Ucm grmaifhn j` a strukturm starts at a pjfht wcmrm tcm fhtmrhad `jrkm fhtmhsfty (f.m., strmss) mxkmmes sjbm batmrfad kcaraktmrfstfk vadum. Ucfs fbpdfms tcat wm hmme tj rm`fhm jur em`fhftfjh j` —strmss jh a sur`akm‒ tj —strmss at a pjfht.‒ Gut ah fh`fhftm hubgmr j` pdahms (sur`akms) kah pass tcrjunc a pjfht. ^cfkc fbanfhary sur`akm e j wm scrfhi tj zmrj: ^cmh wm scrfhi tcm sur`akm arma tj zmrj, wcfkc fhtmrhad `jrkm kjbpjhmht scjude wm usm: Gjtc ef``fkudtfms kah gm aeermssme gy assfnhfhn efrmktfjhs tj tcm jrfm htatfjh j` tcm fbanfhary sur`akm ahe tj tcm efrmktfjh j` tcm fhtmrhad `jrkm jh tcfs sur`akm ahe tcmh karryfhn tcm emskrfptfjh j` tcm efrmktfjhs as tcm sugskrfpts j` tcm strmss kjbpjhmhts lust as wm karrfme x karrfme x,, y, y, ahe z ahe z as as sugskrfpts tj emskrfgm tcm kjbpjhmhts j` a vmktjr. @fnurm 1.7 scjws 1.7 scjws a gjey kut gy ah fbanfhary pdahm tcat cas ah jutware hjrbad fh tcm f efrmktfjh. Jh tcfs sur`akm wm cavm a ef``mrmhtfad arma Af jh wcfkc a rmsudtaht `jrkm1 akts. cm kjbpjhmht j` tcm `jrkm fh tcm l tcm l ef efr mktfjh mktfjh fs @ l. A kjbpjhmht 1. F` a rmsudtaht rmsudtaht bjbmht bjbmht fs fhkdueme, fhkdueme, tcmh tcmh tcm strmss strmss fs rm`mrrme rm`mrrme tj as kjupdm strmss. Kjupdm strmss fs fbpjrtaht f` strmss ahadysfs fs kjheuktme at sukc a sbadd skadm tcat tcm bjbmht trahsbftt me gy tcm gjhes gmtwmmh bjdmkudms bust gm fhkjrpjratme. [mm @uhn Y160=R `jr aeeftfjhad aeeftfjhad emtafds.
[trmss at a pjfht 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
Jutware hjrbad
;
f
@ l Fhtmrhad `jrkm
Af
@fnurm 1.7 [trmss at a pjfht.
j` avmranm strmss fs ( @ l / Af). F` wm scrfhi Af tj zmrj, wm nmt tcm em`fhftfjh j` a strmss kjbpjhmht at a pjfht as scjwh gy Mquatfjh (1.1). (1.1).
fl Efrmktf Efrmktfjh jh j` jutwa jutware re hjrba hjrbadd tj tcm fbanfha fbanfhary ry kut sur`ak sur`akmm
2
@ l -------- Af ? A f dfb
(1.1)
Efrmk Efrmktf tfjh jh j` tcm tcm fhtmrha fhtmrhadd `jrkm `jrkm kjbpjhm kjbpjhmht ht
Hjw wcmh wm djji at a strmss kjbpjhmht, tcm `frst sugskrfpt tmdds us tcm jrfmhtatfjh j` tcm fbanfhary sur`akm ahe tcm smkjhe sugskrfpt tmdds us tcm efrmktfjh j` tcm fhtmrhad `jrkm. Fh tcrmm efbmhsfjhs, makc sugskrfpt f ahe l ahe l kah kah rm`mr tj tcm x tcm x,, y, jr z efrmktfjh. efrmktfjh. Fh jtcmr wjres, tcmrm arm hfhm pjssfgdm y, jr z kjbgfhatfjhs j` tcm twj sugskrfpts, as scjwh fh tcm strmss batrfx fh @fnurm 1.;. 1.;. Ucm efanjhad mdmbmhts fh tcm strmss batrfx arm hjrbad strmssms, ahe add j``-efanjhad mdmbmhts fh tcm strmss batrfx rmprmsmht scmar strmssms.
xx xy xz yx yy yz zx zy zz @fnurm 1.; [trmss batrfx fh tcrmm efbmhsfjhs.
Hjtm tcat tj spmkf`y tcm strmss at a pjfht, wm bust cavm a banhftuem a he twj efrmktfjhs.Uagdm efrmktfjhs. Uagdm 1.7 scjws 1.7 scjws tcm hubgmr j` kjbpjhmhts hmmeme tj spmkf`y a skadar, a vmktjr, ahe a strmss. Hjw `jrkm, bjbmht, vmdjkfty, ahe akkmdmratfjh arm add ef``mrmht quahtftfms, gut add arm kaddme vmktjrs. Fh a sfbfdar bahhmr, strmss gmdjhns tj a katmnjry kaddme tmhsjrs ([mm tmhsjrs ([mm [yhnm ahe [kcfde Y16>3R) jr bjrm spmkf`fkaddy, strmss spmkf`fkaddy, strmss fs a smkjhe-jremr tmhsjr , wcmrm —smkjhe jremr‒ rm`mrs tj tcm mxpjhmht fh tcm dast rjw. Fh tcfs tmrbfhjdjny, a vmktjr fs a tmhsjr j` jremr jhm, ahe a skadar fs a tmhsjr j` jremr zmrj. Uagdm 1.7 Kjbparfsjh j` Hubgmr j` Kjbpjhmhts Efbmhsfjhs
Quahtfty
[kadar Pmktjr
Jhm
Uwj ?
[trmss
Ucrmm
1 1
?
1 7
1 ;?
1 11
7 71
; ;1
7
7
6 ;7
1 1
5 7
Fh tcm fhtrjeuktjry kjursm jh tcm bmkcahfks j` batmrfads, ft was mxpdafhme tcat scmar strmss fs sybbmtrfk, as scjwh gy Mquatfjh (1.7). (1.7).
xy yx
yz zy
zx xz
(1.7)
Ucm sybbmtry j` scmar strmss nfvmh gy Mquatfjhs (1.7) fbpdfms tcat strmss tcat strmss at a pjfht cas hfhm strmss kjbpjhmhts, gut jhdy sfx arm fhempmhemht . [fbfdardy, fh twj efbmhsfjhs strmss at a pjfht cas `jur hjhzmrj kjbpjhmhts fh nmhmrad, gut jhdy tcrmm arm fhempmhemht.
1.7.1
[fnh kjhvmhtfjh `jr strmss ^m wfdd kjhsfemr Af tj gm pjsftfvm f` tcm jutware hjrbad tj tcm sur`akm fs fh tcm pjsftfvm f efrmktfjh. efrmktfjh. F` tcm jutware hjrbad fs fh tcm hmnatfvm f efrmktfjh, tcmh Af wfdd gm kjhsfemrme tj gm hmnatfvm
^ftc tcfs kjhvmhtfjh fh bfhe, wm kah fbbmefatmdy emeukm tcm sfnh `jr strmss. A strmss kjbpjhmht kah gm pjsf tfvm fh twj ways. Gjtc tcm hubmratjr ahe emhjbfhatjr arm pjsftfvm fh Mquatfjh (1.1), (1.1), jr gjtc tcm hubmratjr ahe emhjbfhatjr arm hmnatfvm fh Mquatfjh (1.1). (1.1). A strmss kjbpjhmht fs pjsftfvm f` tcm hubmratjr ahe emhjbfhatjr cavm tcm sabm sfnh fh Mquatfjh (1.1). (1.1).
[trmss at a pjfht 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
1.7.7
5
[trmss mdmbmhts
Ucm efskussfjh fh tcm prmkmefhn smktfjh scjws tcat strmss at a pjfht fs ah agstrakt quahtfty, ahe emvmdjpfhn ah fhtuftfvm `mmd `jr ft fs ef``fkudt. ef``fkudt. [trmss jh a sur`akm, cjwmvmr, cjwmvmr, fs masfmr tj vfsuadfzm as a efstrfgutme `jrkm jh a sur`akm. Astrmss mdmbmht fs ah fbanfhary jglmkt tcat cmdps us vfsuadfzm strmss at a pjfht gy addjwfhn us tj kjhstrukt sur`akms tcat cavm jutware hjrbad fh tcm efrmktfjhs j` tcm kjjrefhatms. Ft scadd gm smmh fh tcm efskussfjh tcat `jddjws tcat a strmss mdmbmht fs a paraddmdmpfpme fh Kartmsfah kjjrefhatms, a `ranbmht j` a kydfhemr fh kydfherfkad kjjrefhatms, ahe a `ranbmht j` a spcmrm fh spcmrfkad kjjrefhatms. ^m start jur efskussfjh fh tcm hmxt smktfjh wftc tcm kjhstruktfjh j` a strmss mdmbmht fh Kartmsfah kjjrefhatms. ^m kah usm a sfbfdar prjkmss tj eraw strmss mdmbmhts fh kydfherfkad ahe spcmrfkad kjjrefhatm systmbs.
1.7.;
Kjhstruktfjh j` a strmss mdmbmht
Kjhsfemr tcm pjfht at wcfkc wm waht tj emskrfgm strmss. Arjuhe tcfs pjfht, fbanfhm ah jglmkt tcat cas sfems wcjsm jutware hjrbad arm fh efrmktfjhs j` tcm kjjrefhatms. A paraddmdmpfpme cas sfx sur`akms wftc jutware hjrbad tcat arm mftcmr fh tcm pjsftfvm jr hmnatfvm kjjrefhatm efrmktfjh. Ucus, wm cavm akkjuhtme `jr tcm `frst sugskrfpt fh jur strmss em`fhftfjh. ^m ihjw tcat `jrkm fs fh tcm pjsftfvm jr hmnatfvm efrmktfjh j` tcm smkjhe sugskrfpt. ^m usm jur sfnh kjhvmhtfjh tj scjw tcm strmss fh tcm efrmktfjh j` tcm `jrkm jh makc j` tcm sfx sur`akms. Uj embjhstratm tcm kjhstruktfjh lust emskrfgme, wm wfdd assubm tcat add hfhm strmss kjbpjhmhts scjwh fh tcm strmss batrfx fh @fnurm 1.; arm 1.; arm pjsftfvm. Dmt us kjhsfemr tcm `frst rjw fh tcm strmss batrfx fh @fnurm 1.;. 1.;. Ucm `frst sugskrfpt nfvms us tcm efrmktfjh j` tcm jutware hjrbad, wcfkc fs tcm x tcm x efrmktfjh. efrmktfjh. [ur`akms A [ur`akms A ahe ahe G G fh fh @fnurm 1.5 cavm 1.5 cavm tcm jutware hjrbad fh tcm x tcm x efrmktfjh, efrmktfjh, ahe ft fs jh tcmsm sur`akms tcat tcm strmss kjbpjhmht j` tcm `frst rjw wfdd gm scjwh. Ucm jutware hjrbad jh sur`akm A sur`akm A fs fs fh tcm pjsftfvm x pjsftfvm x efrmktfjh efrmktfjh Yemhjbfhatjr fs pjsftfvm fh Mquatfjh (1.1)R. (1.1)R. @jr tcm strmss kjb pjhmht tj gm pjsftfvm jh sur`akm A sur`akm A,, tcm `jrkm bust gm fh tcm pjsftfvm efrmktfjh Yhubmratjr bust gm pjsftfvmR, as scjwh fh@fnurm fh@fnurm 1.5. 1.5. Ucm jutware hjrbad jh sur`akm G sur`akm G fs fs fh tcm hmnatfvm x hmnatfvm x efrmktfjh efrmktfjh Yemhjbfhatjr fs hmnatfvm fh Mquatfjh (1.1)R. (1.1)R. @jr tcm strmss kjbpjhmht tj gm pjsftfvm jh sur`akm G sur`akm G,, tcm `jrkm bust gm fh tcm hmnatfvm efrmktfjh Yhubmratjr bust gm hmnatfvmR, as scjwh fh @fnurm 1.5. 1.5. Dmt us hjw kjhsfemr tcm smkjhe rjw fh tcm strmss batrfx fh @fnurm 1.;. 1.;. @rjb tcm `frst sugskrfpt, wm ihjw tcat tcm hjrbad tj tcm sur`akm fs fh tcm y tcm y efrmktfjh. efrmktfjh. [ur`akm K cas cas ah jutware hjrbad fh tcm pjsftfvm y pjsftfvm y efrmktfjh9 efrmktfjh9 tcmrm`jrm add `jrkms jh sur`akm K arm fh tcm pjsftfvm efrmktfjh j` tcm smkjhe sugskrfpt, as scjwh fh @fnurm 1.5. 1.5. [ur`akm E [ur`akm E cas cas ah jutware hjrbad fh tcm hmnatfvm y hmnatfvm y efrmktfjh. Ucmrm`jrm, add `jrkms jh sur`akm E sur`akm E arm arm fh tcm hmnatfvm efrmktfjh j` tcm smkjhe sugskrfpt, as scjwh fh @fnurm 1.5. 1.5. y yy K
yx zz
yz zx M
xy A xx
xz
xx G
zy
zy
xy
@
xz
zx
x
yz
zz yx
E yy
@fnurm 1.5 strmss mdmbmht scjwfhn add pjsftfvm strmss kjbpjhmhts.
Ucm kjbpjhmhts j` tcm tcfre rjw fh tcm strmss batrfx arm scjwh jh sur`akms M sur`akms M ahe @ ahe @ fh fh @fnurm 1.5, 1.5, fh akkjreahkm wftc tcm `jrmnjfhn djnfk. Ucm pjsftfvm hjrbad strmss kjbpjhmhts (m.n., xx) arm puddfhn tcm paraddmdmpfpme fh jppjsftm efrmktfjhs9 tcat fs, tcm paraddmdmpfpme fs fh tmhsfjh jwfhn tj a pj sftfvm hjrbad strmss kjbpjhmht. As bmhtfjhme mardfmr, hjrbad strmssms arm rmpj rtme as tmhsfjh jr kjbprmssfjh, hjt as pjsftfvm jr hmnatfvm. @fnurm 1.5 scjws 1.5 scjws tcat tcm sybbmtrfk pafr j` scmar strmss kjbpjhmhts mftcmr pjfhts tjware ah menm jr away `rjb ft―tcfs jgsmrvatfjh kah gm usme fh erawfhn scmar strmssms jh tcm tcrmm sur`akms j` tcm strmss mdmbmht jhkm tcm scmar strmss jh jhm j` fts sur`akms cas gmmh erawh.
[trmss trahs`jrbatfjh fh t wj efbmhsfjhs 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
1.7.5
=
_dahm strmss
_dahm strmss fs jhm j` tcm twj typms j` twj-efbmhsfjhad sfbpdf`fkatfjh fh tcm bmkcahfks j` batmrfads. Fh [mktfjh 1.0.1, 1.0.1, wm wfdd stuey pdahm strafh, tcm jtcmr typm j` twj-efbmhsfjhad sfbpdf`fkatfjh, ahe fh [mktfjh 1.3.;, 1.3.;, wm wfdd smm tcm ef``mrmhkm gmtwmmh tcm twj typms j` sfbpdf`fkatfjh. _dahm strmss fbpdfms add strmssms jh a pdahm arm zmrj. ^m kcjjsm z kcjjsm z tj tj gm tcm kjjrefhatm ahe smt add strmssms wftc sugskrfpt z tj tj gm zmrj. @fnurm 1.=a 1.=a scjws tcm strmss batrfx fh pdahm strmss, @fnurm 1.=g 1.=g tcm assjkfatme strmss mdmbmht `jr tcm pjfht, ahe @fnurm 1.=k 1.=k tcm sfbpdf`fme twj-efbmhsfjhad rmprmsmhtatfjh j` tcm strmss mdmbmht as vfmwme `rjb tcm z tcm z axfs. axfs. ^m hjtm tcat tcm pdahm wftc jutware hjrbad fh tcm z tcm z efrmktfjh efrmktfjh fs strmss `rmm. [trmss-`rmm sur`akms [trmss-`rmm sur`akms arm adsj kaddme `rmm sur`akms. y yy yy yx yx xy ey xx xy ? xy xx xx yx yy ? ey x xx xx ? ? ? ex xy ez xy yx J yx ex z yy yy @fnurm 1.= strmss mdmbmht fh pdahm strmss.
1.;
[UWM[[ [UWM[[ UWAH[@JWBAUFJH FH U ^J EFBMH[FJH[
Ucm strmss trahs`jrbatfjh mquatfjhs `rjb tcm fhtrjeuktjry kjursm jh tcm bmkcahfks j` batmrfads arm grfm`dy rmvfmwme fh tcfs smktfjh ahe kast fh a batrfx `jrb. Ucfs nfvms ahjtcmr pmrspmktfvm jh `abfdfar kjhkmpts. Fh [mktfjh 1.5, 1.5, tcfs hmw pmrspmktfvm wfdd gm mxtmheme tj strmss trahs`jrbatfjh fh tcrmm efbmhsfjhs. y t
F h k d f h m e _ d a h m
m h a d _ d a k f t r m P
y
h Jutware hjrbad tj t fhkdfhme pdahm
h
x
Cjrfzjhtad _dahm
x
z z (a)
(g)
@fnurm 1.0 Djkad ahe ndjgad kjjrefhatm systmbs.
Ucm rmdatfjhscfp j` strmssms at a at a pjfht fh fh ef``mrmht kjjrefhatm systmbs, tcat fs, trahs`jrbatfjh j` strmss kjbpjhmhts wftc kjjrefhatm systmbs, fs kaddme strmss trahs`jrbatfjh. [trmss trahs`jrbatfjh kah adsj gm vfmwme as rmdatfhn strmssms jh ef``mrmht pdahms tcat pass tcrjunc a pjfht . Ucm hjrbad j` tcm pdahm em`fhm tcm axfs j` a kjjrefhatm systmb tj wcfkc wm arm trahs`jrbfhn tcm strmss kjbpjhmhts. Ucm `fxme rm`mrmhkm kjjrefhatm systmb fh wcfkc tcm mhtfrm prjgdmb fs emskrfgme fs kaddme tcm ndjgad kjjrefhatm systmb. A kjjrefhatm systmb tcat kah gm `fxme at ahy pjfht jh tcm gjey ahe cas ah jrfmhtatfjh tcat fs em`fhme wftc rmspmkt tj tcm ndjgad kjjrefhatm systmb fs kaddme tcm djkad kjjrefhatm systmb. Fh tcfs gjji tcm ndjgad kjjrefhatm systmb wfdd bjst j`tmh gm tcm x tcm x-- y y- z z Kartmsfah Kartmsfah kjjrefhatm systmb. Wmdatfhn fhtmrhad `jrkms ahe bjbmhts tj mxtmrhad `jrkms ahe bjbmhts fs usuaddy ejhm fh a ndjgad kjjrefhatm systmb. Ucm fhtmrhad quahtftfms arm tcmh usme tj jgtafh strmssms fh tcm ndjgad kjjrefhatm systmb. ^m assubm tcat tcm pjfht fs fh pdahm strmss, ahe wm ihjw tcm strmssms fh tcm x tcm x-- y y- z z kjjrefhatm kjjrefhatm systmb, tcat fs, jh tcm cjrfzjhtad ahe vmrtfkad pdahms scjwh fh @fnurm 1.0a 1.0a. ^m smmi tj `fhe strmssms jh fhkdfhme pdahms, wcfkc kah gm jgtafhme gy rjtatfjh agjut tcm z tcm z axfs. axfs. Adtmrhatfvmdy, wm smmi tj `fhe strmssms fh tcm h-t - z z kjjrefhatm kjjrefhatm systmb, wcmrm h fs tcm hjrbad efrmktfjh tj a pdahm tcrjunc tcat pjfht as scjwh fh @fnurm 1.0a 1.0a ahe g, ahe t fs fs tcm tahnmht efrmktfjh tj tcm pdahm sukc tcat h-t - z z fs fs a rfnct-caheme kjjrefhatm systmb as scjwh fh @fnurm 1.0g 1.0g.
[trmss trahs`jrbatfjh fh t wj efbmhsfjhs 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
0
Fh tcm fhtrjeuktjry kjursm jh tcm bmkcahfks j` batmrfads, tcm strmssms fh tcm h-t - z z kjjrefhatm kjjrefhatm systmb wmrm rmdatme tj strmssms fh tcm Kartmsfah kjjrefhatm systmb gy bmahs j` tcm wmenm bmtcje, wcfkc fs grfm`dy mdagjratme cmrm. @fnurm 1.>a 1.>a scjws tcm strmssms aktfhn jh tcm vmrtfkad, cjrfzjhtad, ahe fhkdfhme pdahms tcat kjbpjsm tcm strmss wmenm. Ucm strmssms arm kjhvmrtme tj `jrkms gy budtfpdyfhn gy tcm armas j` makc pdahm jh wcfkc tcm strmssms akt, as scjwh jh tcm `jrkm wmenm fh @fnurm 1.>g 1.>g. Gy mqufdfgrfub j` `jrkms fh tcm h ahe t efrmktfjhs, efrmktfjhs, wm jgtafh
hh xx kjs 7 yy sfh 7 7 xy sfh
kjs
ht ‛ xx kjs sfh yy sfh
kjs
xy kjs 7 ‛ sfh 7
(1.7a)
ez hh
ht
ht et ez )
hh et ez )
xy xx
xxet kjs kjs ez )
et t
h
xy (et kjs kjs ez )
yx (et sfh sfh ez )
yx yy (et sfh sfh ez ) (g)
yy (a) ( g) @jrkm wmenm. @fnurm 1.> (a) [trmss wmenm. (g
Gy sugstftutfhn 6?j `jr fh Mquatfjh (1.7a), (1.7a), wm jgtafh tcm hjrbad strmss fh tcm t efrmktfjh efrmktfjh as
tt xx sfh 7 yy kjs 7 ‛ 7 xy kjs sfh
(1.7g)
Jh aeefhn Mquatfjh (1.7a) ahe (1.7a) ahe (1.7g), (1.7g), wm jgtafh
hh tt xx yy
(1.;)
Mquatfjh (1.;) fbpdfms (1.;) fbpdfms tcat tcm sub j` hjrbad strmssms fh ah jrtcjnjhad systmb fs fhempmhemht j` kjjrefhatm trahs`jrbatfjh.
1.;.1
Batrfx bmtcje fh twj efbmhsfjhs
@fnurm 1.0k 1.0k scjws tcm hjrbad (h (h) ahe tahnmht (t (t ) efrmktfjhs j` a pdahm. Ucm efrmktfjh kjsfhms j` tcm uhft vmktjrs fh tcm h ahe t efrmktfjhs kah gm wrfttmh as fh Mquatfjh (1.5a). (1.5a). h x Hjtfhn tcat
kjs
h y
sfh
t x
t y
h x
kjs
t y
sfh
(1.5a)
6? + , wm jgtafh Mquatfjh (1.5g). (1.5g).
‛ h y
t x
(1.5g)
Uj emvmdjp strmss trahs`jrbatfjh mquatfjhs fh batrfx `jrb, wm fhtrjeukm tcm batrfx hjtatfjh nfvmh fh Mquatfjh (1.5k). (1.5k).
h x h h y
t x t t y
xx xy yx yy
(1.5k)
Ucm sybbmtry j` scmar strmssms fbpdfms tcat tcm strmss batrfx fs sybbmtrfk. Fh batrfx hjtatfjh, tcfs sybbmtry fs mxprmssme as
U wcmrm
U
(1.=)
fbpdfms tcm trahspjsm j` tcm batrfx. Fh batrfx hjtatfjh , Mquatfjhs (1.7a) ahe (1.7a) ahe (1.7g) kah (1.7g) kah gm wrfttmh as
hh h U h
ht t U h
tt t U t
(1.0)
U
wcmrm fbpdfms tcm trahspjsm j` tcm kjdubh. Mquatfjh (1.0) scjw (1.0) scjw tcat wm kah jgtafh a strmss kjbpjhmht gy prm- ahe pjstbudtfpdyfhn tcm strmss batrfx gy tcm uhft vmktjrs fh tcm efrmktfjh j` tcm sugskrfpts. sugskrfpts. Gmkausm tcm scmar strmss strmss fs sybbmtrfk, wm kjude cavm prmbudtfpdfme gy tcm uhft hjrbad ahe pjstbudtfpdfme gy tcm uhft tahnmht. Fh jtcmr wjres, tcm jremr j` budtfpdfkatfjh fs fbbatmrfad fh tcm kadkudatfjh j` scmar strmss. [trmss jh a sur`akm fs kaddme traktfjh jr a strmss vmktjr. Hjtm tcat strmss at a pjfht fs a smkjhe-jremr tmhsjr. ^cmh wm spmkf`y a sur`akm, tcm jrfmhtatfjh j` tcm sur`akm fs em`fhme9 cmhkm tcm strmss jh tcm sur`akm hmmes jhdy jhm efrmktfjh tj spmkf`y, wcfkc fbpdfms tcat strmss jh a sur`akm fs a vmktjr quahtfty. Batcmbatfkaddy tcm strmss vmktjr {[ {[ } fs em`fhme fh Mquatfjh (1.>). (1.>).
[ h
(1.>)
Mquatfjh (1.>) fh (1.>) fh mxpaheme hjtatfjh kah gm wrfttmh as [ x
xx h x + xy h y
[ y
yx h x + yy h y
(1.3)
[trmss trahs`jrbatfjh fh t wj efbmhsfjhs 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
>
[fhkm efrmktfjh kjsfhms ej hjt cavm uhfts, tcm strmss vmktjr cas tcm uhfts j` strmss. Ucus, wm hjw ihjw tcrmm ef``mrmht quahtftfms tcat cavm uhfts j` `jrkm pmr uhft arma< • _rmssurm, _rmssurm, wcfkc fs a skadar skadar quahtfty. quahtfty. • Uraktfjh, Uraktfjh, wcfkc wcfkc fs a vmktjr quahtfty quahtfty.. • [trmss, [trmss, wcfkc fs a smkjhe-jremr smkjhe-jremr tmhsjr tmhsjr.. Uj gmttmr apprmkfatm tcm kjhkmpt j` traktfjh, wm jhkm bjrm kjhsfemr tcm `jrkm wmenm scjwh fh @fnurm 1.>g 1.>g. ^m rmprmsmht tcm fhkdfhm arma et ez 2 2 eA ahe eA ahe usm tcm efrmktfjh kjsfhm hjtatfjh j` Mquatfjh (1.5a) tj (1.5a) tj scjw tcm twj-efbmhsfjhad rmprmsmhtatfjh j` tcm `jrkm wmenm, as fh @fnurm 1.3a 1.3a. Ucm `jrkms scjwh jh tcm fhkdfhme pdahm kjude gm rmsjdvme fhtj `jrkms fh tcm x tcm x ahe ahe y kjjref y kjjrefhatms, as scjwh fh @fnurm 1.3g 1.3g. Gy mqufdfgrfub j` `jrkms fh @fnurm 1.3g 1.3g, wm jgtafh Mquatfjh (1.3). (1.3). Adtmrhatfvmdy, tcm strmss vmktjr {[ {[ } kjude gm wrfttmh as
[ hh h + ht t
(1.6)
Mquatfjhs (1.>) ahe (1.>) ahe (1.6) arm (1.6) arm rmprmsmhtatfjhs j` tcm vmktjr {[ {[ } fh twj ef``mrmht kjjrefhatm systmbs, as scjwh fh @fnurm 1.3k 1.3k. [ eA) y (eA)
y
ht eA) eA)
hh{h} hh eA eA))
xx h x eA) eA)
[ eA) x (eA)
eA) xx h x eA)
{[ }
[ y
ht {t } [ x eA) xy (h x eA)
xy (h x eA) x eA) yx (h y eA)
eA) yy (h y eA)
eA) yx (h y eA) eA) yy (h y eA)
(a)
(g)
(k)
@fnurm 1.3 [tatfkaddy mqufvadmht `jrkm wmenms.
1.;.7
_rfhkfpad strmssms
Wmkadd tcm `jddjwfhn em`fhftfjhs `rjb tcm fhtrjeuktjry kjursm jh tcm bmkcahfks j` batmrfads. _dahms jh wcfkc tcm scmar strmssms arm zmrj arm kaddme tcm prfhkfpad pdahms. Ucm hjrbad efrmktfjh tj tcm prfhkfpad pdahms fs rm`mrrme tj as tcm prfhkfpad efrmktfjh, jr tcm prfhkfpad axfs. Ucm ahndms tcm prfhkfpad axfs baims wftc tcm ndjgad kjjrefhatm systmb arm kaddme prfhkfpad ahndms. Hjrbad strmss jh a prfhkfpad pdahm fs kaddme prfhkfpad strmss. Ucm nrmatmst prfhkfpad strmss fs kaddme prfhkfpad strmss 1. Fh pdahm strmss, tcm pdahm wftc jutware hjrbad fh tcm z tcm z efrmktfjh efrmktfjh cas zmrj scmar strmss. Ucmrm`jrm gy em`fhftfjh, tcm z tcm z efrmktfjh fs a prfhkfpad efrmktfjh. [fhkm tcmrm arm tcrmm prfhkfpad pdahms, tcmrm arm tcrmm prfhkfpad strmssms at a pjfht. Ucm tcrmm prfhkfpad strmssms arm dagmdme 1 7 ahe ;. Dmt tcm uhft hjrbad vmktjr j` a prfhkfpad pdahm gm nfvmh gy { p} p} ahe tcm kjrrmspjhefhn prfhkfpad strmss gy p. Gy em`fhftfjh, tcm pdahm wftc hjrbad { p} p} wfdd cavm zmrj scmar strmss ahe tcm hjrbad strmss wfdd tcm prfhkfpad strmss ahe Mquatfjhs (1.>) ahe (1.>) ahe (1.6) kah (1.6) kah gm wrfttmh as
[ p p p
(1.1?)
Mquatfjh (1.1?) fh (1.1?) fh mxpaheme hjtatfjh kah gm wrfttmh as `jddjws<
[
xx xy p x ? p x p ? p p y yx yy p y
(1.11a)
Adtmrhatfvmdy, Mquatfjh (1.11a) kah (1.11a) kah gm wrfttmh as Mquatfjh (1.11g). (1.11g).
xx ‛ p xy p x yx yy ‛ p p y
?
(1.11g)
Mquatfjh (1.11g) rmprmsmhts (1.11g) rmprmsmhts twj mquatfjhs fh tcm twj uhihjwhs p uhihjwhs p x ahe p ahe p y. @jr a hjhtrfvfad (hjhzmrj (hjhzmrj)) sjdutfjh tj mxfst, tcm emtmrbfhaht j` tcm batrfx bust gm zmrj. Ucfs fs a kdassfk statmbmht j` ah mfnmhvadum prjgdmb. Uj scjw tcat tcm mfnmhvadums j` tcm strmss batrfx arm tcm prfhkfpad strmssms, wm smt tcm emtmrbfhaht j` tcm batrfx fh Mquatfjh (1.11g) tj (1.11g) tj zmrj ahe jgtafh 7 p7 ‛ p xx + yy + xx yy ‛ xy
?
(1.17)
Mquatfjh (1.17) fs (1.17) fs kaddme tcm kcaraktmrfstfk mquatfjh fh ah mfnmhvadum prjgdmb. Ucm agjvm mquatfjh fs a quaeratfk wcjsm rjjts arm
1,7
2
7 xx + yy xx + yy 7 ‛ 5 xx yy ‛ xy 7
(1.1;a)
[trmss trahs`jrbatfjh fh tcrmm efbmhsfjhs 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
3
Ucm tmrbs fh Mquatfjh (1.1;a) kah (1.1;a) kah gm rmarrahnme tj jgtafh
1,7
2
xx + yy xx ‛ yy 7 7 -------------------- -------------------- 7 7 + xy
(1.1;g)
Mquatfjh (1.1;g) nfvms (1.1;g) nfvms tcm `jrbudas `jr prfhkfpad strmssms wm saw fh tcm fhtrjeuktjry kjursm jh tcm bmkcahfks j` batmrfads. ^m kjhkduem, tcm mfnmhvadums j` tcm strmss batrfx arm tcm prfhkfpad strmssms ahe tcm mfnmhvmktjrs j` tcm strmss batrfx arm tcm prfhkfpad efrmktfjhs. Uj emtmrbfhm tcm mfnmhvmktjrs, wm kah usm mftcmr j` tcm twj mquatfjhs rmprmsmhtme fh (1.11g), (1.11g), adjhn wftc tcm `akt tcat tcm sub j` tcm squarm j` efrmktfjh kjsfhms fs 1.
1.5
[UWM[[ [UWM[[ UWAH[@JWBAUFJH FH UCWMM EFBMH[FJH[
Ucm kjhkmpts ahe `jrbudas emvmdjpme fh twj efbmhsfjhs kah gm mxtmheme tj tcrmm efbmhsfjhs gy hjtfhn tcat tcm batrfx rmdatfjhscfps emskrfgme fh [mktfjh 1.; ej hjt empm he upjh tcm sfzm j` tcm batrfx. @fnurm 1.6(a) 1.6(a) scjws tcm efrmktfjh kjsfhms j` t cm uhft hjrbad fh tcrmm efbmhsfjhs. y
y
A A x
hh{h}
h
h x 2 kjs
y
A z
h y 2 kjs
x
z
x
h z 2 kjs
A y
x y z
ht {t } {[ } x
z
z
(g)
a ( g) Mqufdfgratfhn scmar strmss. @fnurm 1.6 (a) Efrmktfjh kjsfhms j` a uhft hjrbad. (g
^m em`fhm tcm `jddjwfhn quahtftfms fh batrfx hjtatfjh
h x h 2 h y h z
[ x [ 2 [ y [ z
2
xx xy xz yx yy yz zx zy zz
(1.15)
Ucm hjrbad strmss kah gm `juhe `rjb Mquatfjh (1.0). (1.0). Fh twj efbmhsfjhs, tcm hjrbad ahe tahnmht e frmktfjhs wmrm rmdatme. Fh tcrmm efbmhsfjhs, cjwmvmr, tcmrm arm ah fh`fhftm hubgmr j` tahnmhts fh tcm fhkdfhme p dahm scjwh fh@fnurm fh@fnurm 1.6a 1.6a. Ucm kadkudatfjh j` scmar strmss fh tcrmm efbmhsfjhs mhtafds prjgdmbs j` twj typms. 1. Ucm tahnmht efrmktfjh fs ihjwh. Ucm scmar strmss fh tcat efrmktfjh bay gm `juhe `rjbMquatfjh `rjbMquatfjh (1.0). (1.0). [cmar strmss `juhe `rjb Mquatfjh (1.0) rmprmsmhts (1.0) rmprmsmhts tcm kjbpjhmht j` tcm strmss vmktjr fh tcm nfvmh tahnmht efrmktfjh9 tcfs bay hjt gm tcm scmar strmss hmkmssary `jr mqufdfgrfub j` tcm wmenm scjwh fh @fnurm 1.6a 1.6a. 7. Ucm scmar strmss tcat fs hmkmssary `jr mqufdfgrfub bay gm `juhe `rjb Mquatfjh (1.6), (1.6), as scjwh fh @fnurm 1.6g 1.6g. ^m kah `fhe tcm strmss vmktjr {[ {[ } `rjb Mquatfjh (1.>) ahe (1.>) ahe hh `rjb Mquatfjh (1.0). (1.0). @rjb Mquatfjh (1.6) wm jgtafh ht t 2 [ ‛ hh h . Ucm banhftuem ahe efrmktfjh j` tcm scmar strmss kah gm emtmrbfhme `rjb tcm vmktjr ht {t }. }.
1.5.1
_rfhkfpad strmssms
Ucm mqufvadmht `jrb j` Mquatfjh (1.11a) ahe (1.11a) ahe (1.11g) fh (1.11g) fh tcrmm efbmhsfjhs kah gm wrfttmh as `jddjws<
xx xy xz p x yx yy yz p y zx zy zz p z
2
p
?
?
p
?
?
p x ? p y p p z
xx ‛ p yx zx
?
jr
xy yy ‛ p zy
p x p y zz ‛ p p z xz yz
2 ?
(1.1=)
Ucm batrfx Mquatfjh (1.1=) rmprmsmhts (1.1=) rmprmsmhts tcrmm mquatfjhs fh tcm tcrmm uhihjwh efrmktfjh kjsfhms. @jr a hjhtrfvfad sjdutfjh, tcm emtmrbfhaht j` tcm batrfx bust gm zmrj. Jhkm bjrm wm jgsmrvm tcat tcm mfnmhvadums j` tcm strmss batrfx arm tcm prfhkfpad strmssms, ahe tcm mfnmhvmktjrs arm tcm prfhkfpad efrmktfjhs. Uj emtmrbfhm tcm mfnmhvmktjrs kjrrmspjhefhn tj ah mfnmhvadum, wm kah usm ahy twj j` tcm tcrmm mquatfjhs fh Mquatfjh (1.1=), (1.1=), adjhn wftc tcm ihjwdmenm tcat tcm squarm j` tcm efrmktfjh kjsfhms fs 1, as scjwh fh Mquatfjh (1.10). (1.10). 7
7
7
p x + p y + p z 2 1
(1.10)
[trmss trahs`jrbatfjh fh tcrmm efbmhsfjhs 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
1.5.7
6
_rfhkfpad strmss kjhvmhtfjh
_rfhkfpad strmssms bay gm jremrme jr uhjremrme. Nmhmraddy spmaifhn, tcm `jddjwfhn kjhvmhtfjh fs jgsmrvme `jr jremrfhn j` prfhkf pad strmssms. Fh tcrmm efbmhsfjhs tcm prfhkfpad strmssms arm rmpjrtme su kc tcat 1 7 ; . Fh twj efbmhsfjhs tcm prfhkfpad strmssms arm rmpjrtme sukc tcat 1 7 , gut ; fs hjt njvmrhme gy ahy jremr. Ucfs fs gmkausm ; fs mquad tj zz , wcfkc empmhes upjh wcmtcmr tcm statm j` strmss fs pdahm strmss jr pdahm strafh. Gy kjhvmh tfjh, kadkudatme prfhkfpad strmssms arm rmpjrtme fh jremrme `jrb. Ucm prfhkfpad ahndms arm kadkudatme `rjb tcm efrmktfjh kjsfhms j` tcm prfhkfpad efrmktfjhs. Gy kjhvmhtfjh, tcm ahndms rmpjrtme arm gmtwmmh zmrj ahe 13?¿, tcat fs, ? x y z 13?
1.5.;
(1.1>)
Kcaraktmrfstfk mquatfjh ahe strmss fhvarfahts
^m smt tcm emtmrbfhaht j` tcm batrfx fh Mquatfjh (1.1=) tj (1.1=) tj zmrj tj jgtafh tcm kcaraktmrfstfk mquatfjh gmdjw.
p; F 1 p7 + F 7 p F ; F 1 2
xx + yy + zz
F 7
xx xy yx yy
+
yy yz zy zz
+
2 ?
xx xz zx zz
(1.13)
F ;
xx xy yx yy zx zy
xz yz zz
(1.16)
wcmrm, | | fs tcm emtmrbfhaht j` tcm quahtfty mhkdjsme fh tcm vmrtfkads. Ucm tcrmm rjjts7 j` tcm kcaraktmrfstfk mquatfjh yfmde tcm prfhkfpad strmssms, as jgsmrvme fh [mktfjh 1.;.7. 1.;.7. _rfhkfpad strmssms at a pjfht arm uhfqum ahe uhfqum ahe ej hjt empmhe upjh tcm kjjrefhatm systmb fh wcfkc tcm strmss batrfx was spmkf`fme. F` rjjts j` tcm kcaraktmrfstfk mquatfjh arm fhempmhemht j` tcm kjjrefhatm systmb tcmh tcm kjm``fkfmhts fh tcm kcaraktmrfstfk mquatfjh scjude gm fhempmhemht j` tcm kjjrefhatm systmb. Cmhkm, tcm kjm``fkfmhts F kjm``fkfmhts F 1, F 7, ahe F ahe F ; arm kaddme strmss fhvarfahts gmkausm tcm vadums j` tcmsm quahtftfms arm fhempmhemht j` kjjrefhatm trahs`jrbatfjh. Hjtfkm tcat tcm sfnh fh tcm tmrbs kjhtafhfhn jee strmss fhvarfahts fs hmnatfvm. • Ucm `frst `frst strmss strmss fhvarfaht fhvarfaht F F 1 fs tcm sub j` efanjhad tmrbs j` tcm strmss batrfx. • Ucm smkjhe smkjhe strmss fhvarfaht fhvarfaht F F 7 fs tcm sub j` bfhjr emtmrbfhahts j` tcm strmss batrfx. • Ucm tcfre tcfre strmss strmss fhvarfaht fhvarfaht F F ; fs tcm emtmrbfhaht j` tcm mhtfrm strmss batrfx. [fhkm strmss fhvarfahts arm fhempmhemht j` tcm kjjrefhatm systmb, tcm vadum j` tcms m quahtftfms wfdd wfdd gm tcm sabm sabm fh a prfhkfpad kjjrefhatm kjjrefhatm systmb. Ucm strmss batrfx batrfx fh a prfhkfpad kjjrefhatm systmb kah gm wrfttmh as
1
?
?
?
7
?
?
?
;
(1.7?)
Ucm strmss fhvarfahts kah gm wrfttmh fh tmrbs j` prfhkfpad strmss as F 1 2
1 + 7 + ;
F 7 2
1
?
?
7
+
7
?
?
;
+
1
?
?
;
1 2
1 7 + 7 ; + ; 1
F ; 2 ? ?
?
?
7
?
?
;
2
1 7 ;
(1.71)
Ucm strmss fhvarfahts, partfkudardy tcm `frst ahe tcm tcfre, prjvfem a qufki ahe masy kcmki jh tcm dagjrfjusdy kadkudatme prfhkfpad strmssms.
1.5.5
Baxfbub scmar strmss
Fh tcm fhtrjeuktjry kjursm jh tcm bmkcahfks j` batmrfads, wm saw tcat tcm baxfbub scmar strmss mxfsts jh a pdahm tcat fs 5= tj tcm prfhkfpad pdahms. Ucm banhftuem j` tcm baxfbub scmar strmss fs nfvmh gy
bax
1 ‛ 7 7 ‛ ; ; ‛ 1 2 bax --------------------------------- 7 ----------------7 7
(1.77)
wcmrm tcm tmrb jh tcm rfnct-cahe sfem fs tcm banhftuem j` baxfbub ef``mrmhkm gmtwmmh twj prfhkfpad strmssms. @fnurms 1.1? tcrjunc 1.17 scjw 1.17 scjw tcm varfjus pdahms jh wcfkc tcm baxfbub scmar strmss wfdd mxfst, empmhefhn upjh tcm rmdatfvm vadums j` tcm prfhkfpad strmssms.
7.
Ucm rjjts rjjts j` j` mquat mquatfjh fjh x x; F 1 x7 F 7 x F ; ? arm x arm x1 7 A kjs A kjs F 1/;, ahe x ahe x7,; ‛7 A kjs A kjs ( 0?¿) F 1/;, wcmrm A 2 ;
kjs ; 2 7 F 1 ; ‛ F 1 ; F 7 + F ; 7 A
;
.
F 1 ;
7
‛ F 7 ; ahe
[trmss trahs`jrbatfjh fh tcrmm efbmhsfjhs 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
1?
p7
p
7
7 ‛ ; 2 ‛ ;7 2 ----------------7
7; p1
p1
p;
p;
@fnurm 1.1? _dahms j` baxfbub scmar strmss tcat arm 5= tj prfhkfpad pdahms 7 ahe ;.
p7
p7
p1
p;
p1
;1
; ‛ 1 2 ‛ 1; 2 ----------------7
17
1 ‛ 7 2 ‛ 71 2 ----------------7
p;
@fnurm 1.11 _dahms j` baxfbub scmar strmss tcat arm 5= tj prfhkfpad pdahms 1 ahe ;.
p7
p7
p1
p1
p;
p;
@fnurm 1.17 _dahms j` baxfbub scmar strmss tcat arm 5= tj prfhkfpad pdahms 1 ahe 7.
1.5.5
Jktacmerad strmssms
A pdahm tcat baims mquad ahndms wftc tcm prfhkfpad pdahms fs kaddme ah jktacmerad pdahm. Ucm strmssms jh tcm jktacmerad pdahm arm kaddme jktacmerad strmssms. Fh [mktfjh 1.1? wm 1.1? wm wfdd usm tcm jktacmerad strmssms tj `jrbudatm a `afdurm tcmjry. 7
1
; @fnurm 1.1; Ucm mfnct jktacmerad pdahms.
@fnurm 1.1; scjws 1.1; scjws mfnct (jktad) pdahms tcat baim mquad ahndms wftc tcm prfhkfpad pdahms. ^m usm h1, h7, ahe h; tj rmprmsmht tcm efrmktfjh kjsfhms j` tcm jutware hjrbad j` a pdahm bmasurme `rjb tcm prfhkfpad axfs. Ucm hjrbad strmss ahe scmar strmss jh a pdahm wftc efrmktfjh kjsfhms h1, h7, ahe h; kah gm scjwh tj gm nfvmh gy Mquatfjh (1.7;). (1.7;).
hh
2
1 h 17 + 7 h 77 + ; h ;7
ht
2
7 17 h 17 + 77 h 77 + ;7 h;7 ‛ hh
(1.7;)
Ucjunc tcm sfnh j` tcm efrmktfjh kjsfhms kcahnms wftc makc j` tcm mfnct jktacmerad pdahms, tcmfr banhftuem fs tcm sabm `jr add mfnct pdahms, tcat fs, h 1 2 h7 2 h ; 2 1 ; . [ugstftutfhn tcm efrmktfjh kjsfhms fhtj Mquatfjh (1.7;) wm (1.7;) wm jgtafh tcm jktacmerad hjrbad strmss as
jk t
2
1 + 7 + ; ;
(1.75a)
Avmranm strafhs 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
Mquatfjh (1.75a) scjws
jk t
11
2 F 1 ; , wcmrm F wcmrm F 1 fs tcm `frst strmss fhvarfaht fhvarfaht.. A bjrm fhtmrmstfhn fhtmrprmtatfjh j` jktacm-
erad hjrbad strmss fs jgtafhme gy kjhsfemrfhn a pjfht fh `dufe. @jr a pjfht fh `dufe, tcm scmar strmssms arm zmrj ahe add hjrbad strmssms strmss ms (prfhk fpad strmssms) arm kjbprmssfvm ahe mquad tj tcm prmssurm at tcat pjfht, tcat fs, 1 2
jk t
cyerjstatfk prmssurm. @rjb Mquatfjh (1.75a) wm jgtafh
7
2
;
2 ‛ p , wcmrm p fs p fs tcm
2 ‛ p 9 fh jtcmr wjres, tcm jktacmerad hjrbad strmss fs sfbfdar tj tcm
cyerjstatfk prmssurm. [ugstftutfhn tcm efrmktfjh kjsfhms `jr tcm jktacmerad pdahms fh Mquatfjh (1.7;) wm (1.7;) wm jgtafh tcm banhftuem j` scmar strmss, wcfkc kah gm sfbpdf`fme as scjwh fh Mquatfjh (1.75g). (1.75g).
jk t
17 + 77 + ;7 1 + 7 + ; 7
------------------------------------ ‛ ------------------------------------; 6
jk t 1.5.=
1 -;
1 -;
7 17 + 7 77 + 7 ;7 ‛ 7 1 7 + 7 ; + ; 1
jr
1 ‛ 7 7 + 7 ‛ ; 7 + ; ‛ 1 7
(1.75g)
Emvfatjrfk strmss
A emvfatjrfk strmss batrfx fs a strmss batrfx `rjb wcfkc tcm m``mkt j` cyerjstatfk statm j` strmss ( F F 1;) fs rmbjvme. Ft pdays ah fbpjrtaht rjdm fh tcmjry j` pdastfkfty ([mm Bmhemdsjh Y1603R). Ucm emvfatjrfk strmss batrfx kah gm wrfttmh fh tcm Kartmsfah kjjrefhatms jr fh prfhkfpad kjjrefhatms as scjwh fh @fnurm 1.15. 1.15.
E
2
xx ‛ F 1 ; xy xz yx yy ‛ F 1 ; yz zx zy zz ‛ F 1 ;
E
jr
2
1 ‛ F 1 ;
?
?
?
7 ‛ F 1 ;
?
?
?
; ‛ F 1 ;
(g)
(a) ( a) Kartmsfah kjjrefhatms. (g ( g) _rfhkfpad kjjrefhatms. @fnurm 1.15 [trmss emvfatjrfk batrfx. fh (a
^m kah `fhe tcm fhvarfahts j` emvfatjrfk strmss batrfx fh tcm sabm bahhmr as wm `juhe `jr tcm jrefhary strmss batrfx. F` wm ejh‖t cavm tcm strmss batrfx fh prfhkfpad kjjrefha tms, tcmh ft fs adnmgrafkaddy dmss tmefjus tj `f he tcm emvfatjrfk strmss fhvarfahts efrmktdy `jr tcm batrfx fh Kartmsfah (jr kydfherfkad / spcmrfkad) kjjrefhatms. Cjwmvmr, `jr emrfvatfjh purpjsms ft fs masfmr tj wjri wftc tcm batrfx fh prfhkfpad kjjrefhatms. Ft kah gm scjwh (smm prjgdmbs 1.1? ahe 1.1? ahe 1.11) 1.11) tcat tcm tcrmm strmss emvfatjrfk strmss fhvarfahts arm as scjwh gmdjw. L 1 2 ?
1 7 L 7 2 ‛ -- Y 1 ‛ 7 + +
0
1 L ; 2 ------ 7 1 ‛ 7 ‛ ; 7 7 ‛ ; ‛ 1 7 ; ‛ 1 ‛ 7 (1.7=) 7>
7 ‛ ; 7 + ; ‛ 1 7
[ugstftutfhn Mquatfjh (1.75g) fhtj (1.75g) fhtj Mquatfjh (1.7=) wm (1.7=) wm jgtafh 7
L 7 2 ‛ ; 7 jk t
(1.70)
MSAB_DM 1.1 [cjw tcat tcm prfhkfpad efrmktfjhs arm jrtcjnjhad f` tcm prfhkfpad strmssms arm hjt mquad.
MSAB_DM 1.7 Emtmrbfhm tcm smkjhe ahe tcfre emvfatjrfk strmss fhvarfahts `jr tcm strmss batrfx nfvmh gmdjw. ;? ?
?
7?
;? ‛ 1? B_ a
7? ‛ 1?
? AH[.
1.=
L 7 2 ‛ 3??B_a
7
L ; 2 ‛ >???B_a
;
APMWANM [UWAFH[
[trafh fs a bmasurm j` tcm fhtmhsfty j` kcahnm fh tcm scapm j` a gjey. A kcahnm fh scapm kah gm emskrfgme gy efspdakmbmhts j` pjfhts jh tcm gjey ahe fs tcus a prjgdmb fh nmjbmtry (ifhmbat fks) as empfktme fh @fnurm 1.1=. 1.1=. Ucm tjtad bjvmbmht j` j` a pjfht wftc rmspmkt tj a `fxme rm`mrmhkm kjjrefhatms fs kaddme efspdakmbmht. Ucm rmdatfvm bjvmbmht j` j` a pjfht wftc rmspmkt tj ahjtcmr pjfht jh tcm gjey fs kad dme em`jrbatfjh. Ucus efspdakmbmht j` a pjfht rmprmsmhts kcahnms eum tj em`jrbatfjh pdus rfnfe gjey bjtfjh. Danrahnfah strafh fs Fh emskrfgfhn kcahnms, wm bust spmkf`y tcm rm`mrmhkm vadum `rjb wcfkc wm arm bmasurfhn kcahnm. Danrahnfah kjbputme `rjb em`jrbatfjh gy usfhn tcm jrfnfhad uhem`jrbme nmjbmtry as nmjbmtry as tcm rm`mrmhkm nmjbmtry. Mudmrfah strafh fs kjbputme `rjb em`jrbatfjh gy usfhn tcm `fhad tcm `fhad em`jrbme nmjbmtry nmjbmtry as tcm rm`mrmhkm n mjbmtry. Ucm Danrahnfah emskrfptfjh fs usuaddy usme fh
[trafh at a pjfht 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
17
Ifhmbatfks Efspdakmbmhts
[trafhs
@fnurm 1.1= [trafhs ahe efspdakmbmhts.
sjdfe bmkcahfks. Ucm Mudmrfah emskrfptfjh fs usuaddy usme fh `dufe bmkcahfks. ^cmh a batmrfad uhemrnjms vmry darnm em`jrbatfjhs, sukc as fh a sj`t ruggmr jr fh prjlmktfdm pmhmtratfjh j` bmtads, mftcmr emskrfptfjh bay gm usme, empmhefhn upjh tcm ahadysfs. Hjh-dfhmar strafh em`fhftfjhs arm efskussme fh Kcaptmr 3. 3. ^m wfdd usm Danrahnfah strafh uhdmss ft fs statme jtcmrwfsm. Wmkadd tcat tcmrm arm twj ifhes j` strafh―hjrbad strafhs ahe scmar strafhs. Hjrbad strafhs arm bmasurms j` kcahnm j` dmhntc ahe scmar strafhs arm bmasurms j` kcahnm j` ahndms. Hjrbad strafh fs usuaddy emsfnhatme gy tcm Nrmmi dmttmr mpsfdjh () Ucm kcahnm j` dmhntc, adsj kaddme em`jrbatfjh, fs emsfnhatme gy tcm djwmrkasm Nrmmi dmttmr emdta (). Ucm avmranm hjrbad Danrahnfah strafh j` a dfhm fs tcm ratfj j` em`jrbatfjh j` tcm dfhm tj tcm jrfnfhad dmhntc D dmhntc D? as `jddjws<
2
D ?
(1.7>)
Mdjhnatfjhs ( ?) rmsudt fh pjsftfvm fh pjsftfvm hjrbad strafhs. Kjhtraktfjhs ( ?) rmsudt fh hmnatfvm hjrbad strafhs. Fh sbadd-strafh Fh sbadd-strafh apprjxfbatfjhs apprjxfbatfjhs ( ?.?1), tcm em`jrbatfjh kah gm apprjxfbatme gy tcm em`jrbatfjh kjbpjhmht kjbpjhmht fh tcm efrmktfjh j` tcm uhem`jrbme dfhm mdmbmht. @fnurm 1.10 scjws 1.10 scjws a gar A_ gar A_ tcat tcat em`jrbs tj tcm pjsftfjh A_ pjsftfjh A_ 1. @jr sbadd-strafh kadkuda-
tfjhs, tcm em`jrbme dmhntc j` A_ j` A_ 1 fs apprjxfbatme gy dmhntc A_ dmhntc A_ 7, wcfkc fs jgtafhme gy erawfhn a pmrpmhefkudar `rjb _ `rjb _ 1 tj tcm jrfnfhad efrmktfjh A_ efrmktfjh A_ . Ahjtcmr pmrspmktfvm jh sbadd-strafh apprjxfbatfjh fs tcat tcm em`jrbatfjh fs tcm kjbpjhmht j` em`jrbatfjh vmktjr __ __ 1 fh tcm jrfnfhad efrmktfjh. Ucm sbadd-strafh apprjxfbatfjh rmsudts fh dfhmar ahadysfs ahe nrmatdy sfbpdf`fms kadkudatfjhs. _ 1
h t f j f s p j _ e m b j r m ` E j h f f t ft j s j p h a d f h n n f J r
_ 7
A
@fnurm 1.10 Apprjxfbatfjh j` a sbadd hjrbad strafh.
[cmar strafh fs usuaddy emsfnhatme gy tcm Nrmmi dmttmr nabba (). Ucm avmranm Danrahnfah scmar strafh fs em`fhme as a kcahnm j` ahndm `rjb tcm rfnct ahndm as
2 -- ‛ 7
(1.73)
wcmrm, tcm Nrmmi dmttmr adpca () fs tcm `fhad ahndm bmasurme fh raefahs (rae), ahe tcm Nrmmi dmttmr pf () ;.151=6 rae. Emkrmasms rae. Emkrmasms fh tcm ahndm ( rmsudt fh pjsftfvm scmar strafh. Fhkrmasms strafh. Fhkrmasms fh fh tcm ahndm ( rmsudt fh hmnatfvm scmar strafh. @jr sbadd scmar strafh ( 4 ?.?1), tcm `jddjwfhn apprjxfbatfjh j` tcm trfnjhjbmtrfk `uhktfjhs bay gm usme. tah
1.=.1
sfh
kjs 1
]hfts j` avmranm strafhs
Mquatfjh (1.7>) scjws (1.7>) scjws tcat hjrbad strafh fs efbmhsfjhdmss, cmhkm scjude cavm hj uhfts. Cjwmvmr, tj ef``mrmhtfatm avmranm strafh ahe strafh at a pjfht (efskussme fh [mktfjh 1.0), 1.0), uhfts j` dmhntc pmr uhft dmhntc arm usme `jr avmranm hjrbad strafhs. Ucus, avmranm hjrbad strafhs arm rmpjrtme wftc tcm uhfts j` fhkc/fhkc (jr kb/kb, b/b, mtk.). Avmranm scmar strafhs arm rmpjrtme fh raefahs. Fh rmpjrtfhn mxpmrfbmhtad rmsudts ahe `jr emskrfgfhn vmry darnm em`jrbatfjhs, strafhs arm nfvmh gy bmahs j` a pmrkmhtanm kcahnm, tcat fs, tcm rfnct-cahe sfem j` Mquatfjh (1.7>) ahe (1.7>) ahe (1.73) fs (1.73) fs budtfpdfme gy 1?? gm`jrm tcm rmsudts arm rmpjrtme. Ucus, a hjrbad strafh j` ?.=% fs mquad tj a strafh j` ?.??=. Ucm Nrmmi dmttmr bu () fs j`tmh usme fh rmpjrtfhn sbadd strafhs gmkausm ft stahes `jr bfkrj ( bfkrj ( 2 1?0). Ucus, a strafh j` 1??? fhfh rmprmsmhts a hjrbad strafh j` ?.??1 fhkc pmr fhkc.
1.0
[UWAFH AU A _JFHU
Dmt u, v, ahe w gm efspdakmbmhts fh tcm x tcm x,, y, y, ahe z ahe z efrmktfjhs, efrmktfjhs, rmspmktfvmdy. Ucm mhnfhmmrfhn strafhs at a pjfht kah gm vfsuadfzme gy kjhsfemrfhn tcm em`jrbatfjh j` a paraddmdmpfpme paraddmdmpfpme j` fh`fhftmsfbad efbmhsfjhs x, x, y, y, ahe z , scjwh fh @fnurm 1.1>. 1.1>.
[trafh at a pjfht 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
y
1;
y
u v y
y w
u
x
x
z
z
-- ‛ 7 xy
z
(a) y
x
(g) y
-- ‛ 7 zx
w
z
v
x
y
-- ‛ 7 yz
v z
x
z
x
w
x
u z
(k)
(e )
(a) hjrbad strafhs, (g (g) scmar strafh xy, (k) scmar strafh yz , ahe (e) scmar strafh zx. @fnurm 1.1> [trafh kjbpjhmhts< (a
As tcm efbmhsfjhs j` tcm paraddmdmpfp me tmhe tj zmrj, wm nmt tcm batcmbatfkad em`fhftfjh j` mhnfhmmrfhn strafh at a pjfht. Gmkausm tcm dfbftfhn jpmratfjh fs fh a nfvmh efrmktfjh, wm jgtafh partfad emrfvatfvms, hjt tcm jrefhary emrfvatfvms. Ucm ejugdm sugskrfpt fh tcm kasm j` scmar strafh fhefkatms tcm kcahnm j` ahndm gmtwmmh twj kjjrefhatm dfhms em`fhme gy tcm sugskrfpts. ^m adsj rmkjre ejugdm sugskrfpts `jr hjrbad strafh `jr saim j` kjhsfstmhky, as wmdd as `jr jur sugsmqumht batrfx em`fhftfjh j` strafh. Ucm batcmbatfkad em`fhftfjhs j` mhnfhmmrfhn strafhs at a pjfht arm as nfvmh fh Mquatfjh (1.76a) tcrjunc (1.76a) tcrjunc (1.76e). (1.76e).
xx
u- ----- x ? x
2 dfb
2
u x xy
yy 2
yx
-----v-- y ? y
zz
u- + ----- v- ----- x
2
u + v y x
(1.76g)
v- + ------ w- ----- y ? z y
2
v + w z y
(1.76k)
w- + -----u-- ------ x ? x z
2
w + u x z
(1.76e)
2 dfb
2
x ? y y ?
yz zy
dfb
z ?
zx
2
xz
w w 2 dfb -------- 2 z ? z z
v y
2 dfb
2 dfb
z ?
(1.76a)
Mquatfjh (1.76g) tcrjunc (1.76g) tcrjunc (1.76e) scjw (1.76e) scjw tcat scmar strafh fs sybbmtrfk ahe tcat tcm jremr j` sugskrfpts fs fbbatmrfad. Ucm sybbmtry j` scmar strafh baims fhtuftfvm smhsm. Ucm kcahnm j` ahndm gmtwmmh tcm x tcm x ahe ahe y y efrmktfjhs efrmktfjhs fs jgvfjusdy tcm sabm as gmtwmmh tcm y tcm y ahe ahe x x efrmktfjhs. efrmktfjhs. Hjtm tcat tcm sfnh j` scmar strafh empmhes jhdy upjh wcmtcmr tcm ahndm fhkrmasms jr emkrmasms, hjt upjh tcm jremr fh wcfkc wm kjhsfemr x kjhsfemr x ahe ahe y y.. @fnurm 1.13 scjws 1.13 scjws tcm mhnfhmmrfhn strafh kjbpjhmhts as ah mhnfhmmrfhn strafh batrfx. Ucm batrfx fs sybbmtrfk gmkausm j` tcm sybbmtry j` scmar strafh. Ucus, fh tcrmm-efbmhsfjhad prjgdmbs strafh cas hfhm kjbpjhmhts, gut jhdy sfx j` tcm hfhm kjbpjhmhts arm fhempmhemht gmkausm j` sybbmtry j` scmar strafh.
xx xy xz yx yy yz zx zy zz @fnurm 1.13 Mhnfhmmrfhn strafh batrfx.
^m smm tcat dfim strmss, mhnfhmmrfhn strafh cas twj sugskrfpts, wcfkc wjude smmb tj sunnmst tcat mhnfhmmrfhn strafh fs adsj a smkjhe-jremr tmhsjr dfim strmss. Cj wmvmr uhdfim strmss, mhnfhmmrfhn strafh ejms hjt satfs` y kmrtafh kjjrefhatm trahs`jrbatfjh daws, wcfkc wm wfdd stuey fh [mktfjh 1.>. 1.>. Cmhkm ft fs hjt a smkjhe-jremr tmhsjr gut fs rmdatme tj ft, as scjwh fh Mquatfjh (1.;?). (1.;?). tmhs tmhsjr jr hjrb hjrbad ad stra strafh fh 2 mhnf mhnfhm hmmr mrfhn fhn hjr hjrba badd straf strafh h
tmhs tmhsjr jr scm scmar ar stra strafh fh 2 mhnf mhnfhmm hmmrf rfhn hn scm scmar ar str straf afh/7 h/7
(1.;?)
Fh [mktfjh 1.>, 1.>, wm scadd smm tcat tcm `aktjr j` 1 7 tcat kcahnms tcm mhnfhmmrfhn scmar strafh tj tmhsjr scmar strafh pmrbfts tcm mxtmhsfjh j` strmss trahs`jrbatfjh rmsudts tj strafh trahs`jrbatfjh.
[trafh at a pjfht 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
1.0.1
15
_dahm strafh
_dahm strafh fs tcm jtcmr typm j` twj-efbmhsfjhad prjgdmbs fh tcm stuey j` tcm bmkcahfks j` batmrfads. Ucm ef``mrmhkms gmtwmmh pdahm strmss ahe pdahm strafh arm efskussme fh[mktfjh fh [mktfjh 1.3.;. 1.3.;. Gy —twj-efbmhsfjhad,‒ wm fbpdy tcat jhm j` tcm kjjrefhatms ejms hjt pday a rjdm fh tcm emskrfptfjh j` tcm prjgdmb, tcus wm smt add strafhs wftc sugskrfpt z z tj tj gm zmrj as scjwh fh tcm s trafh batrfx fh @fnurm 1.16. 1.16. Adtcjunc `jur kjbpjhmhts j` strafh arm hmmeme fh pdahm strafh, gmkausm j` tcm sybbmtry j` scmar strafh, jhdy tcrmm j` tcm `jur kjbpjhmhts arm fhempmhemht.
xx xy yx yy ?
? ?
?
?
@fnurm 1.16 _dahm strafh batrfx.
Ucm assubptfjh j` pdahm strafh fs j`tmh baem fh ahadyzfhn vmry tcfki gjefms sukc as pjfhts arjuhe tuhhmds jr bfhm sca`ts, jr a pjfht fh tcm bfeedm j` a tcfki kydfhemr sukc as a sugbarfhm cudd. Fh tcfki gjefms, ft fs arnume tcat tj em`jrb, a pjfht bust pusc a djt j` batmrfad fh tcm tcfkihmss efrmktfjh9 cmhkm tcm strafhs kah gm mxpmktme tj gm sbadd fh tcm tcfkihmss efrmktfjh. Ucus, tcm strafh fh tcm tcfkihmss efrmktfjh fs hjt mxaktdy zmrj gut fs sbadd mhjunc tj hmndmkt. _dahm strafh fs a batcmbatfkad apprjxfbatfjh baem tj sfbpdf`y tcm ahadysfs.
1.0.7
_jdar kjjrefhatms
AGKE fs tcm ef``mrmhtfad mdmbmht fh pjdar kjjrefhatm fh uhem`jrbme statm scjwh @fnurm 1.7?. 1.7?. Ucm ef``mrmhtfad mdmbmht AGKE mdmbmht AGKE em`jrbs tj A tj A1 G1K 1 E1 eum tj raefad efspdakmbmht ur fh @fnurm 1.7?a 1.7?a. _jfht A _jfht A bjvms bjvms tj pjfht A pjfht A1 eum tj tcm raefad efspdakmbmht ur. _jfht G _jfht G bjvm bjvm tj pjfhts G pjfhts G1 gy ah fhkrmasme abjuht as ur fhkrmasms fh tcm r efrmktfjh. efrmktfjh. Bjvmbmht j` pjfht E pjfht E tj tj E E1 fs kausme gy tcm raefad efspdakmbmht ur pdus tcm fhkrmasm j` ur fh tcm efrmktfjh wcfkc adsj kcahnms tcm ahndm gy gmtwmmh tcm raefad ahe tahnmhtfad efrmktfjh. Ucfs kcahnm j` ahndm wfdd kjhtrfgutm tjwares tcm scmar strafh. Ucm raefad hjrbad strafh fs nfvmh gy
rr
u r + u r r r ‛ u r A 1 G 1 ‛ AG 2 ------------------------2 dfb --------------------------------------------------- 2 r r ? AG
rr
2
u r jr r
ur r
(1.;1a)
K 1 y K
ur u + r
K 1
y
K
v v +
v r r
v +
E1
G1
G1
E1
E r A1 r A
u
r
r
G
r + ur
ur r u + r r x
(a)
E r A1
r
A7 G
r
v
r + r --------------r
A v
x (g)
( a) lust raefad efspdakmbmht ur. (g (g) lust tahnmhtfad efspdakmbmht v . @fnurm 1.7? [trafhs fh pjdar kjjrefhatms `rjb (a
Ucm ef``mrmhtfad mdmbmht AGKE mdmbmht AGKE em`jrbs tj A tj A1 G1K 1 E1 eum tj tahnmhtfad efspdakmbmht v fh @fnurm 1.7?g 1.7?g. _jfhts A _jfhts A ahe ahe E E bjvm tj pjfhts A pjfhts A1 ahe E ahe E1 eum tj tcm tahnmhtfad efspdakmbmht v. Bjvmbmht j` pjfht G pjfht G tj tj G G1 fs eum tj tcm fhkrmasm fh v fh tcm efrmktfjh. Ucm kcahnm j` ahndm gmtwmmh tcm raefad ahe tahnmhtfad efrmktfjh rmqufrms us tj sugtrakt tcm efstahkm A efstahkm A7 G wcfkc G wcfkc fs eum tj fhkrmasm j` raefus `rjb r tj tj r + r . Ucm efstahkm A efstahkm A7 G kah gm `juhe usfhn sfbfdar trfahndm ahe fs as scjwh. Ucm dmhntc AE dmhntc AE kcahnms kcahnms eum tj gjtc efspdakmbmhts as smmh fh gjtc `fnurms, cmhkm tcm tahnmhtfad hjrbad strafh fs
A 1 E 1 ‛ AE r + ur ‛ r v + v ‛ v 2 --------------------------2 dfb ----------------------------------------+ ------------------------------------------------------ jr ? AE r r
u 1 v 2 ----r + -r r
(1.;1g)
[trafh trahs`jrbatfjh 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
1=
^m kah `fhe makc j ` tcm ahndm as `jddjws tah 2 dfb
?
u r + u r ‛ u r --------------------------------------------------- r + ur
dfb
?
u r 1 ur ------------------------------ -r r
v + v r r ‛ v r + r r tah 2 dfb ------------------------------------------------------------------------------- 2 r r ?
v v ‛ ---- r r
Ucm scmar strafh fs tcm kcahnm j` ahndm EAG ahndm EAG,, cmhkm fs tcm sub j` tcm twj ahndms, tcat fs, r 2
r 1.>
+.
1 u r v v 2 -+ ‛ ----r r r
(1.;1k)
[UWAFH UWAH[@JWBAUFJH
[trafh trahs`jrbatfjh fs rmdatfhn strafhs fh twj kjjrefhatm systmbs. ^m jhkm bjrm start wftc tcm twj-efbmhsfjhad prjgdmb j` pdahm strafh. Jgtafh mquatfjhs tcat wmrm jgtafhme fh tcm fhtrjeuktjry kjursm jh tcm bmkcahfks j` batmrfads ahe scjw cjw tcm strmss trahs`jrbatfjh mquatfjhs kah gm usme `jr strafh trahs`jrbatfjh. @fnurm 1.71 scjws 1.71 scjws twj kjjrefhatm systmbs. Ucm h, t , z kjjrefhatm kjjrefhatm systmb fs jgtafhme `rjb x `rjb x,, y, y, z kjjrefhatm kjjrefhatm systmb gy rjtatfhn kjuhtmrkdjkiwfsm agjut tcm z tcm z axfs axfs gy ah ahndm j` . Ucm kjjrefhatms kah gm rmdatme as scjwh gmdjw. x 2 h kjs ‛ t sfh
y 2 h sfh + t kjs
(1.;7a)
Ucm efspdakmbmhts fh Kartmsfah kjjrefhatms (u (u, v) kah gm rmdatme tj efspdakmbmhts (u (uh, vt ) as scjwh gmdjw. u h 2 u kjs + v sfh
v t 2 ‛ u sfh + v kjs
y
t
(1.;7g)
h
x
z @fnurm 1.71 Kjjrefhatm trahs`jrbatfjh.
]sfhn kcafh rudm wm kah wrftm tcm hjrbad strafh fh h efrmktfjh as
uh u h x u h y (1.;7k) 2 + x h y h h kjs ahe y h 2 sfh . [ugstftutfhn tcmsm rmsudts ahe Mquatfjh (1.;7g) wm (1.;7g) wm jgtafh hh
@rjb Mquatfjh (1.;7a) wm cavm x h 2 tcm `jddjwfhn.
hh hh
2
2
2
u kjs + v sfh kjs + u kjs + v sfh sfh jr x y
u kjs 7 + v sfh kjs + u kjs sfh + v sfh 7 x x y y hh
2
xx kjs 7 + yy sfh 7 + xy sfh kjs
jr (1.;;a)
Fh a sfbfdar bahhmr wm jgtafh tcm `jddjwfhn mquatfjhs
tt ht
2
2 ‛ 7 xx sfh
xx sfh 7 + yy kjs 7 xy sfh
kjs
+ 7 yy sfh
kjs
(1.;;g)
+ xy kjs 7 ‛ sfh 7
(1.;;k)
kjs
Ucm agjvm mquatfjhs arm sfbfdar tj tcm strmss trahs`jrbatfjh rmdatfjhs YMquatfjhs YMquatfjhs (1.7a) tcrjunc (1.7a) tcrjunc (1.7g)R (1.7g)R wftc tcm ef``mrmhkm tcat tcm kjm``fkfmht j` scmar strmss tmrb fs twfkm tcm kjm``fkfmht j` tcm scmar strafh tmrb. Ucfs ef``mrmhkm mxfsts gmkausm wm ar m usfhn mhnfhmmrfhn strafh fhstmae j` tmhsjr strafh. F` wm kjhvmrt tcm mhnfhmmrfhn strafhs tj tmhsjr strafhs, gy efvfefhn add mhnfhmmrfhn scmar strafhs gy 7, tcmh tcm rmsudts `jr tcm batrfx bmtcje fh strmss trahs`jrbatfjh kah gm aeaptme `jr strafh trahs`jrbatfjh. @fnurm 1.77 scjws 1.77 scjws tcm tmhsjr strafh batrfx tcat fs jgtafhme `rjb tcm mhnfhmmrfhn strafhs.
2
xx yx 2 yx 7 zx 2 zx 7
xy 2 xy 7 yy zy 2 zy 7
@fnurm 1.77 Umhsjr strafh batrfx jgtafhme `rjb ah mhnfhmmrfhn strafh batrfx.
xz 2 xz 7 yz 2 yz 7 zz
Batmrfad bjemd 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
10
Ucm twj pmrpmhefkudar efrmktfjhs h ahe t fh fh @fnurm 1.71 kah 1.71 kah gm rmprmsmhtme gy uhft vmktjrs {h {h} ahe {t {t }. }. Ahadjnjus tj Mquatfjh (1.0) wm (1.0) wm jgtafh tcm strafh trahs`jrbatfjh mquatfjhs `jr tcm hjrbad ahe scmar strafh fh tcm h-t efrmktfjhs efrmktfjhs as scjwh gmdjw.
hh
2
h U h
ht
2
t U h
tt
2
t U t
(1.;5)
Ucm scmar strafh ht fh Mquatfjh (1.;5) fs (1.;5) fs tmhsjr scmar strafh ahe bust gm budtfpdfme gy 7 tj jgtafh tcm mhnfhmmrfhn scmar strafh ht . Ucm strafh vmktjr fh tcm efrmktfjh j` h fs ahadjnjus tj tcm strmss vmktjr ahe kah gm em`fhme as scjwh fh Mquatfjh (1.;=). (1.;=).
M 1.>.1
h
2
(1.;=)
_rfhkfpad strafhs
^m rmkadd tcm `jddjwfhn em`fhftfjhs `rjb tcm fhtrjeuktjry kjursm jh tcm bmkcahfks j` batmrfad. Ucm prfhkfpad kjjrefhatm efrmktfjhs `jr strafh arm tcm axms fh wcfkc tcm scmar tcm scmar strafh fs zmrj. zmrj. Hjrbad strafhs fh tcm prfhkfpad efrmktfjhs arm kaddme prfhkfpad Ucm nrmatmst prfhkfpad prfhkfpad strafh fs kaddme prfhkfpad strafh 1. Ucm ahndms tcm prfhkfpad axfs baims wftc tcm ndjgad kjjrefhatm strafhs. Ucm nrmatmst systmb arm kaddme prfhkfpad kaddme prfhkfpad ahndms. ahndms. Fh pdahm strafh, tcm scmar strafhs wftc sugskrfpts z sugskrfpts z arm arm zmrj, wcfkc fbpdfms tcat jhm prfhkfpad efrmktfjh dfms adjhn tcm z tcm z axfs. Fh pdahm strafh prjgdmbs wm wfdd dagmd tcm z tcm z efrmktfjh efrmktfjh as prfhkfpad efrmktfjh ;. Ucmrm arm tcrmm prfhkfpad strafhs at a pjfht, wcfkc arm dagmdme as 1 7 ahe ;. Ucm prfhkfpad strafhs p arm tcm mfnmhvadums j` tcm tmhsjr strafh batrfx, ahe tcm prfhkfpad efrmktfjhs { p} p} arm tcm mfnmhvmktjrs j` tcm strafh batrfx, wcfkc arm `juhe fh mxaktdy tcm sabm bahhmr as tcm prfhkfpad strmssms ahe tcm assjkfatme prfhkfpad efrmktfjhs. Ucm kcaraktmrfstfk mquatfjh ahadjnjus tj Mquatfjh (1.13) fs (1.13) fs
p; ‛ F 1 p7 + F 7 p ‛ F ;
2 ?
(1.;0)
ahe fts rjjts j` tcm kcaraktmrfstfk yfmde tcm prfhkfpad strafhs. Ucm kjm``fkfmhts F 1, F 7, ahe F ahe F ; arm kaddme tcm strafh fhvarfahts gmkausm tcmfr vadum fs fhempmhemht j` kjjrefhatm trahs`jrbatfjh. Hjtfkm tcat tcm sfnh fh tmrbs kjhtafhfhn tcm je e strmss fhvarfahts fs hmnatfvm. Ucm strafh fhvarfahts F fhvarfahts F 1, F 7, ahe F ahe F ;, ahadjnjus tj strmss fhvarfahts fh Mquatfjh (1.16) arm (1.16) arm F 1 2
xx + yy + zz
F 7 2
xx xy yx yy
+
yy yz zy zz
+
xx xz zx zz
F ; 2
xx xy xz yx yy yz zx zy zz
(1.;>)
Ucm strafh fhvarfahts fh tmrbs j` prfhkfpad strafhs arm F 1 2
1 + 7 + ;
F 7 2
1
?
?
7
+
7
?
?
;
+
1
?
?
;
2
1 7 + 7 ; + ; 1
F ; 2
1
?
?
?
7
?
?
?
;
2
17 ;
(1.;3)
Ucm strafh fhvarfahts, partfkudardy tcm `frst ahe tcm tcfre, prjvfem a qufki ahe masy kcmki jh tcm dagjrfjusdy kadkudatme prfhkfpad strafhs. Ucm baxfbub mhnfhmmrfhn scmar strafh fs rmdatme tj tcm ef``mrmhkm fh prfhkfpad strafhs ahe kah gm mxprmssme as
bax
1 ‛ 7 7 ‛ ; ; ‛ 1 --------- 2 bax ----------------------------- 7 --------------7 7 7
1.3
(1.;6)
BAUMWFAD BJEMD
Fh bmkcahfks j` batmrfads, tcm mxpmrfbmhtaddy emtmrbfhme batmrfad rmdatfjhscfp gmtwmmh strmssms ahe strafhs fs fhkjrpjratme fhtj tcm djnfkad `rabmwjri j` bmkcahfks tj prjeukm `jrbudas `jr tcm ahadysfs ahe emsfnh j` strukturad bmbgmrs. bmbgmrs.@fnurm @fnurm 1.7; sybgjdf1.7; sybgjdfkaddy scjws tcm rmdatfjhscfp gmtwmmh strmss ahe strafh. F` a batmrfad bjemd ejms hjt `ft tcm mxpmrfbmhtad eata wmdd, tcmrm wfdd gm a cfnc emnrmm j` mrrjr fh tcm tcmjrmtfkad prmefktfjhs. Vmt a batmrfad bjemd tcat `fts tcm mxpmrfbmhtad e ata vmry akkuratmdy bay gm sj kjbpdmx tcat hj ahadytfkad bjemd (tcmjry) kah gm gufdt. Ucus, tcm kcjfkm j` batmrfad bjemd fs efktatme gjtc gy tcm mxpmrfbmhtad eata ahe gy tcm akkuraky hmmes j` tcm ahadysfs. Batmrfad [trafhs
[trmssms Bjemd
@fnurm 1.7; Batmrfad bjemd.
1.3.1
Dfhmar batmrfad bjemd
Ucm sfbpdmst batmrfad bjemd fs a dfhmar rmdatfjhscfp gmtwmmh strmssms ahe strafhs. ^ftc hj aeeftfjhad assubptfjhs, tcm dfhmar rmdatfjhscfp j` tcm sfx fhempmhemht strafh kjbpjhmhts tj sfx fhempmhemht strmss kjbpjhmhts kah gm wrfttmh as scjwh fh Mquatfjh (1.5?). (1.5?). Ucm batrfx K fl fh Mquatfjh (1.5?) fs (1.5?) fs kaddme tcm kjbpdfahkm batrfx. Mquatfjh (1.5?) fbpdfms (1.5?) fbpdfms tcat wm hmme ;0 batmrfad kjhstahts tj emskrfgm tcm bjst nmhmrad dfhmar rmdatfjhscfp gmtwmmh strmssms ahe strafhs. Cjwmvmr, ft kah gm scjwh tcat tcm batrfx
Batmrfad bjemd 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
1>
`jrbme gy tcm kjhstahts K fl fs a sybbmtrfk batrfx9 tcat fs, K fl 2 K ahe l kah kah gm ahy hubgmr `rjb 1 tj 0. Ucfs sybbmtry lf, wcmrm f ahe l kah gm prjvme gy usfhn tcm rmqufrmbmht tcat tcm strafh mhmrny―mhmrny stjrme fh a batmrfad jwfhn tj em`jrbatfjh, bust ad ways gm pjsftfvm. Ucm sybbmtry rmqufrmbmht rmeukms tcm hubgmr j` fhempmhemht kjhstaht s tj 71 `jr tcm bjst nmhmrad dfhmar rmdatfjhscfp gmtwmmh strmss ahe strafh.
xx yy zz yz zx xy
K 11 xx
K 17 yy K 1; zz K 15 yz K 1= zx K 10 xy K 71 xx + K 77 yy + K 7; zz + K 75 yz + K 7= zx + K 70 xy K ;1 xx + K ;7 yy + K ;; zz + K ;5 yz + K ;= zx + K ;0 xy K 51 xx + K 57 yy + K 5; zz + K 55 yz + K 5= zx + K 50 xy K =1 xx + K =7 yy + K =; zz + K =5 yz + K == zx + K =0 xy K 01 xx + K 07 yy + K 0; zz + K 05 yz + K 0= zx + K 00 xy
xx yy zz yz zx xy
jr
K 11 K 17 K 1; K 15 K 1= K 10 K 17 K 77 K 7; K 75 K 7= K 70 K 1; K 7; K ;; K ;5 K ;= K ;0 K 51 K 57 K 5; K 55 K 5= K 50 K =1 K =7 K =; K =5 K == K =0 K 01 K 07 K 0; K 05 K 0= K 00
xx yy zz yz zx xy
(1.5?)
Mquatfjh (1.5?) prmsuppjsms (1.5?) prmsuppjsms tcat tcm rmdatfjh gmtwmmh tcm strmss ahe tcm strafh fh tcm x tcm x efrmktfjh efrmktfjh fs ef``mrmht `rjb tcm rmdatfjh j` strmss ahe strafh fh tcm y tcm y jr jr z z efrmktfjhs. efrmktfjhs. Adtmrhatfvmdy statme, Mquatfjh (1.5?) fbpdfms (1.5?) fbpdfms tcat f` wm appdy a `jrkm (strmss) fh tcm x tcm x efrmktfjh efrmktfjh ahe jgsmrvm tcm em`jrbatfjh (strafh), tcfs em`jrbatfjh wfdd gm ef``mrmht `rjb tcm em`jrbatfjh tcat wjude gm prjeukme f` wm appdfme tcm sabm `jrkm fh tcm y tcm y efrmktfjh. efrmktfjh. Ucfs pcmhjbmhjh fs hjt jgsmrvagdm gy tcm haime mym `jr bjst bmtads, gut `jr bmtads at tcm krystad-sfzm dmvmd, tcm hubgmr j` kjhstahts hmmeme tj emskrfgm tcm strmss‛strafh rmdatfjhscfp ejms empmhe upjh tcm krystad strukturm. Ucus, arm wm kjheuktfhn tcm ahadysfs at tcm mym dmvmd jr at tcm krystad-sfzm dmvmd: @jr ahadysfs at tcm mym dmvmd, wm avmranm tcm fbpakt j` tcm krystad strukturm, tj jgtafh tcm sfbpdmst batmrfad emskrfptfjh. Ah fsjtrjpfk batmrfad cas strmss‛strafh rmdatfjhscfps tcat arm fhempmhemht j` tcm jrfmhtatfjh j` tcm kjjrefhatm systmb at a pjfht. Ah ahfsjtrjpfk batmrfad fs a batmrfad tcat fs hjt fsjtrjpfk. Ucm bjst nmhmrad ahfsjtrjpfk batmrfad rmqufrms 71 fhempmhemht batmrfad kjhstahts tj kjhstahts tj emskrfgm fts dfhmar strmss‛strafh rmdatfjhscfps. Ah fsjtrjpfk g fsjtrjpfk gjey jey rmqufrms jhdy twj fhempmhemht batmrfad kjhstahts tj emskrfgm fts dfhmar strmss‛strafh rmdatfjhscfps. [jbm j` tcm `aktjrs tcat fh`dumhkm wcmtcmr wm trmat a batmrfad as fsjtrjpfk jr ahfsjtrjpfk arm tcm emnrmm j` ef``mrmhkm fh batmrfad prjpmrtfms wftc jrfmhtatfjh, tcm skadm at wcfkc tcm ahadysfs fs gmfhn kjheuktme, ahe tcm ifhe j` fh`jrbatfjh tcat fs emsfrme `rjb tcm ahadysfs. Cjbjnmhmfty fs ahjtcmr apprjxf batfjh tcat fs j`tmh usme tj emskrfgm a batmrfad gmcavfjr. A batmrfad fs safe safe tj gm cjbjnmhmjus f` tcm batmrfad prjpmrtfms arm tcm sabm at add pjfhts jh tcm gjey. Adtmrhatfvmdy, f` tcm batmrfad kjhstahts K fl arm `uhktfjhs j` tcm kjjrefhatms x kjjrefhatms x,, y, y, jr z jr z , tcm batmrfad fs kaddme hjhcjbjnmhmjus. Bjst batmrfads at tcm atjbfk dmvmd, krystaddfhm dmvmd, jr nrafh-sfzm dmvmd arm hjhcjbjnmhmjus. Ucm trmatbmht j` a batmrfad as cjbjnmhmjus jr hjhcjbjnmhmjus empmhes jhkm bjrm jh tcm typm j` fh`jrbatfjh tcat fs rmqufrme `rjb tcm ahadysfs. Cjbjnmhfzatfjh j` batmrfad prjpmrtfms fs tcm prjkmss j` avmranfhn ef``mrmht batmrfad prjpmrtfms gy ah jvmradd batmrfad prjpmrty. Ahy gjey kah gm trmatme as a cjbjnmhmjus gjey f` tcm skadm at wcfkc ahadysfs fs kjheuktme fs baem su``fkfmhtdy darnm. Fsjtrjpfk‛cjbjnmhm Fsjtrjpfk‛cjbjnmhmjus, jus, ahfsjtrjpfk‛cjbjnm ahfsjtrjpfk‛cjbjnmhmjus, hmjus, fsjtrjpfk‛hjhcjbjnm fsjtrjpfk‛hjhcjbjnmhmjus, hmjus, ahe ahfsjtrjpfk‛hjhcjbjnmhmjus arm add pjssfgdm emskrfptfjhs j` batmrfad gmcavfjr. Ucmrm arm ;1 typms j` krystad. Krystaddfhm gjefms kah gm nrjupme fhtj kdassms `jr tcm purpjsm j` em`fhfhn tcm fhempmhemht batmrfad kjhstahts hmmeme fh tcm dfhmar strmss‛strafh rmdatfjhscfp. Fh gmtwmmh tcm fsjtrjpfk batmrfad ahe tcm bjst nmhmrad ahfsjtrj pfk batmrfad, tcmrm arm batmrfad nrjups j` smvmrad typms, sjbm j` wcfkc arm efskussme gr fm`dy gmdjw. 1. Bjhjkdfhfk batmrfads< Ucm xy Ucm xy pdahm pdahm fs tcm pdahm j` sybbmtry fh bjhjkdfhfk batmrfads. Ucfs fbpdfms t cat tcm strmss‛strafh rmdatfjhscfp fs tcm sabm fh tcm pjsftfvm z pjsftfvm z ahe hmnatfvm z efrmktfjhs. A bjhjkdfhfk batmrfad rmqufrms 1; fhempmhemht batmrfad kjhstahts fh tcrmm efbmhsfjhs. efbmhsfjhs. 7. Jrtcjtrjpfk batmrfads< Jrtcjtrjpfk batmrfads cavm twj axfs j` sybbmtry. Ucat fs, f` wm rjtatm gy 6?j agjut tcm x x jr tcm y y axfs, wm wfdd jgtafh tcm sabm strmss‛strafh rmdatfjh. Ah jrtcjtrjpfk batmrfad rmqufrms hfhm fhempmhemht kjhstahts fh tcrmm efbmhsfjhs ahe `jur fhempmhemht kjhstahts fh twj efbmhsfjhs. efbmhsfjhs. ;. Urahsvmrsmdy fsjtrjpfk batmrfads< Batmrfads arm fsjtrjpfk fh a pdahm. Ucat fs, rjtatfjh gy ah argftrary ahndm agjut tcm z axfs ejms hjt kcahnm tcm strmss‛strafh rmdatfjhs, ahe tcm batmrfad fs fsjtrjpfk fh tcm x tcm x-- y y pdahm. Urahsvmrsmdy fsjtrjpfk batmrfad rmqufrms `fvm rmqufrms `fvm fhempmhemht batmrfad kjhstahts fh kjhstahts fh tcrmm efbmhsfjhs.
1.3.7
Fsjtrjpfk batmrfad
Ah fsjtrjpfk batmrfad rmqufrms jhdy twj fhempmhemht batmrfad kjhstahts. kjhstahts. Ucmrm arm bahy kjhstahts usme tj emskrfgm fsjtrjpfk batmrfad, gut jhdy twj arm fhempmhemht. Wjtatfjh agjut tcm x tcm x,, y, y, jr z jr z axfs axfs gy ahy argftrary ahndm rmsudts fh tcm sabm strmss‛strafh rmdatfjhscfp. Fh batrfx hjtatfjh, tcm strmss‛strafh rmdatfjhscfp `jr fsjtrjpfk batmrfad fs nfvmh gy Mquatfjh (1.51). (1.51).
xx yy zz yz zx xy
K 11
K 17
K 17
K 17
K 11
K 17
K 17
K 17
K 11
?
?
?
?
?
?
?
?
?
xx yy ? ? ? ? ? ? zz 7 K 11 ‛ K 17 ? ? yz 7 K 11 ‛ K 17 ? ? zx 7 K 11 ‛ K 17 ? ? xy ?
?
?
(1.51)
Batmrfad bjemd 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
13
Mquatfjh (1.57) adjhn (1.57) adjhn wftc Mquatfjh (1.5;), (1.5;), arm kaddme tcm Nmhmradfzme Cjjim‖s daw tcat was stuefme fh tcm fhtrjeuktjry kjursm j` bmkcahfks j` batmrfads. Mquatfjh (1.51) kah (1.51) kah gm wrfttmh usfhn tcrmm kjhstahts< tcm bjeudus j` mdastfkfty M mdastfkfty M , tcm _jfssjh ratfj , ahe tcm scmar bjeudus j` mdastfkfty N as scjwh fh Mquatfjh (1.57). (1.57).
xx xx ‛ yy + zz M xy xy N
yy yy ‛ zz + xx M yz yz N
Kjbparfhn Mquatfjh (1.57) wftc (1.57) wftc Mquatfjh (1.51), (1.51), wm jgtafh< K 11 1 M M
zz zz ‛ xx + yy M zx zx N K 17 ‛ M M
(1.57)
7(K 7(K 11 ‛ K 17) 1N
[ugstftutfhn K 11 ahe K 17 fh tcm dast rmdatfjhscfp, wm jgtafh N
M -------------------7 1 +
(1.5;)
Ah adtmrhatfvm `jrb `jr Mquatfjh (1.57) tcat (1.57) tcat bay gm masfmr tj rmbmbgmr fs tcm batrfx `jrb nfvmh gmdjw.
xx 1 1 --- ‛ yy M ‛ zz
xx 1 ‛ yy ‛ 1 zz ‛ ‛
(1.55)
Ucm Nmhmradfzme Cjjim‖s daw `jr fsjtrjpfk batmrfad fs fs vadfe `jr ahy ahy jrtcjnjhad kjjrefhatm systmb at a pjfht, fhkduefhn a pjfht fh a hjhcjbjnmhmjus batmrfad. Ucus, Ucus, wm kjude wrftm wrftm mqufvadmht mqufvadmht `jrbs `jr `jrMquatfjh Mquatfjh (1.57) `jr (1.57) `jr kydfherfkad (pjdar) kjjrefhatms (r , , z ) ahe `jr spcmrfkad kjjrefhatms (r (r , , ). Ucm prfhkfpad kjjrefhatm systmbs `jr strmssms ahe strafhs arm adsj jrtcjnjhad. @jr jhdy fsjtrjpfk batmrfads, batmrfads, tcm prfhkfpad efrmktfjhs `jr strmssms ahe strafhs arm femhtfkad9 cm hkm wm kah wrftm tcm nmhmradfzme Cjjim‖s daw fh prfhkfpad kjjrefhatm `jrb as scjwh gmdjw.
1 1 ‛ 7 + ; M
7 7 ‛ ; + 1 M
; ; ‛ 1 + 7 M
(1.5=)
_rfhkfpad strmssms ahe prfhkfpad strafhs arm ef``mrmht `jr jrtcjtrjpfk ahe jtcmr ahfsjtrjpfk batmrfads ahe sfbfdar mquatfjhs kahhjt gm wrfttmh `jr tcmsm batmrfads. Ucm hjrbad strmssms kah gm sjdvme fh tmrbs j` hjrbad strafhs gy usfhn tcm Mquatfjh (1.57) tj (1.57) tj jgtafh. M 1 ‛ xx + yy + zz M xx + 1 ‛ yy + zz M xx + yy + 1 ‛ zz (1.50) xx ---------------------------------------------------------------- yy ---------------------------------------------------------------- zz ---------------------------------------------------------------- 1 ‛ 7 1 + 1 ‛ 7 1 + 1 ‛ 7 1 +
1.3.;
_dahm strmss ahe pdahm strafh
F` wm taim tcm em`fhftfjhs j` pdahm strmss ahe pdahm strafh ahe appdy tcmb tj Mquatfjh (1.57), (1.57), wm wfdd jgtafh tcm rfnct-bjst batrfkms fh @fnurm 1.75. 1.75. Ucm ef``mrmhkms gmtwmmh tcm pafr j` twj-efbmhsfjhad femadfzatfjhs j` batmrfad gmcavfjr arm fh tcm zmrj ahe hjhzmrj vadums j` tcm hjrbad strafh ahe hjrbad strmss fh tcm z tcm z efrmktfjh. efrmktfjh. Fh pdahm strmss, zz 2 ? , wcfkc fbpdfms tcat tcm hjrbad strafh fh tcm z tcm z efrmktfjh efrmktfjh fs zz 2 ‛ xx + yy M . Fh pdahm strafh, zz 2 ? , wcfkc fbpdfms tcat tcm hjrbad strmss fh tcm z tcm z efrmk efrmktfjh fs zz 2 xx + yy .
_dahm strmss
_dahm strafh
xx xy yx yy
? ?
?
?
?
xx yx
xy yy
?
?
?
?
?
Nmhmradfzme Cjjim‖s daw
xx xy yx yy ?
Nmhmradfzme Cjjim‖s daw
?
xx xy yx yy ?
?
? ?
zz 2 ‛ --- xx + yy M
? ?
zz 2 xx + yy
@fnurm 1.75 [trmss ahe strafh batrfkms fh pdahm strmss ahe pdahm strafh.
Uj gmttmr apprmkfatm tcm ef``mrmhkm gmtwmmh pdahm strmss ahe pdahm strafh, kjhsfemr tcm twj pdatms scjwh fh@fnurm fh @fnurm 1.7=, 1.7=, jh wcfkc jhdy kjbprmssfvm hjrbad strmssms fh tcm x tcm x ahe ahe y y efrmktfjh efrmktfjh arm appdfme. Ucm tjp ahe gjttjb sur`akms jh tcm pdatm fh @fnurm 1.7=a 1.7=a arm `rmm sur`akms (pdahm strmss)9 gut gmkausm tcm pdatm fs `rmm tj mxpahe, tcm em`jrbatfjh (strafh) fh tcm tcm z z efrmktfjh efrmktfjh fs hjt zmrj. Ucm pdatm fh @fnurm 1.7=g 1.7=g fs kjhstrafhme `rjb mxpahefhn fh tcm z tcm z efrmktfjh efrmktfjh gy tcm rfnfe pdatms. As tcm batmrfad puscms jh tcm pdatm, a rmaktfjh `jrkm emvmdjps, ahe tcfs rmaktfjh `jrkm rmsudts fh a hjhzmrj vadum j` hjr bad strmss fh tcm z efrmktfjh. efrmktfjh. Ucjunc tcm mxabpdm fh @fnurm 1.7= cmdps 1.7= cmdps mxpdaf h tcm ef``mrmhkm, ft scjude gm mbpcasfzme tcat pdahm pdahm strmss jr pdahm strafh strafh j`tmh fs ah apprjxfbatfjh apprjxfbatfjh baem baem tj sfbpdf`y ahadysfs. ahadysfs. Ucm apprjxfbatfjh j` pdahm str mss fs j`tmh baem fh tcfh pdatms ahe scmdds, sukc as tcm sifh j` ah afrkra`t jr tcm wadds j` prmssurm vmssmds. Ucm apprjxfbatfjh j` pdahm strafh fs j`tmh baem fh tcfki pdatms ahe scmdds, sukc as sugbarfhm cudds jr pjfhts gmhmatc tcm martc‖s sur`akm. A ef``mrmhkm j` ah jremr j` banhftuem jr bjrm gmtwmmh tcm tcfkihmss ahe jtcmr efbmhsfjhs fs nmhmraddy usme tj kdassf`y a gjey as —tcfh —tcfh.‒ .‒
Batmrfad bjemd 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
@rmm [ur`akm
Em`jrbatfjh
zz 2 ?
zz ?
Wfnfe [ur`akm @rmm [ur`akm
zz ? Wfnfe [ur`akm
zz 2 ?
zz 2 ?
zz ?
Em`jrbatfjh
zz 2 ?
Wmaktfjh @jrkm
16
Wmaktfjh @jrkm
zz ?
( g) _dahm strafh. @fnurm 1.7= (a) _dahm strmss. (g
Ft scjude gm rmkjnhfzme tcat fh pdahm strafh ahe pdahm strmss tcmrm arm jhdy tcrmm fhempmhemht quahtftfms―mvmh tcjunc tcm hubgmr j` hjhzmrj quahtftfms fs bjrm tcah tcrmm. @jr mxabpdm, f` wm ihjw xx , yy, ahe xy, wm kah kadkudatm xx , yy, xy, zz , ahe zz `jr pdahm strmss ahe pdahm strafh. [fbfdardy, f` wm ihjw xx , yy, ahe xy, wm kah kadkudatm xx , yy, xy, zz , ahe zz `jr pdahm strmss ahe pdahm strafh. Ucus, fh gjtc pdahm strmss ahe pdahm strafh, tcm hubgmr j` fhempmhemht strmss jr strafh kjbpjhmhts fs tcrmm, adtcjunc tcm hubgmr j` hjhzmrj kjbpjhmhts fs nrmatmr tcah tcrmm.
MSAB_DM 1.; Ucm mhnfhmmrfhn strafhs at a pjfht jh a gufdefhn `rabm wmrm emtmrbfhme as xx 2 0?? yy 2 5?? zz 2 7?? xy 2 3?? yz 2 5?? zx 2 0?? Ucm bjeudus j` mdastfkfty j` tcm `rabm batmrfad fs 7?? N_a ahe _jfssjh‖s ratfj fs 1/;. Emtmrbfhm tcm prfhkfpad strmssms, baxfbub scmar strmss ahe prfhkfpad strmss efrmktfj h 1. AH[. AH[.
1.3.5
1
2 ;;6.5 B_a(U) 9 7 2 16=.? B_a(U) 9
x 2 57.3 ¿ 99 y 2 =0 ¿ 9 z 2 0>.5 ¿
jr
;
2 13=.0 B_a(U) 9
ba x
2 >0.3= B_a
x 2 1;>.7 ¿ 9 y 2 y 2 175 ¿ 9 z 2 117.0 ¿
Hjhdfhmar batmrfad bjemds
Wuggmr, pdastfks, ahe buskdms ahe jtcmr jrnahfk tfssums mxcfgft hjhdfhmarfty fh tcm strmss‛s trafh rmdatfjhscfp, mvmh at sbadd strafhs. Bmtads adsj tcm mxcfgft hjhdfhmarfty a`tmr yfmde strmss. Fh tcfs smktfjh, wm kjhsfemr varfjus hjhdfhmar batmrfad bjemds< tcat fs, varfjus `jrbs j` mquatfjhs tcat arm usme `jr rmprmsmhtfhn tcm strmss‛strafh hjhdfhmar rmdatfjhscfp. Ucm kjhstahts fh tcm mquatfjhs rmdatfhn strmss ahe strafh arm batmrfad kjhstahts. ^m scadd kjhsfemr tcrmm batmrfad bjemds< 1. Mdastfk‛pmr`mktdy pdastfk, fh wcfkc tcm hjhdfhmarfty fs apprjxfbatme gy a kjhstaht. 7. Dfhmar strafh caremhfhn, fh wcfkc tcm hjhdfhmarfty fs apprjxfbatme gy a dfhmar `uhktfjh. ;. _jwmr daw, fh wcfkc tcm hjhdfhmarfty fs apprjxfbatme gy jhm tmrb, a hjhdfhmar `uhktfjh. Ucm kcjfkm j` batmrfad bjemd tj usm empmhes hjt jhdy upjh tcm batmrfad strmss‛strafh kurvm, gut adsj upjh tcm emnrmm j` akkuraky hmmeme ahe tcm rmsudtfhn kjbpdmxfty j` tcm ahadysfs.
1.3.=
Mdastfk‛pmr`mktdy pdastfk batmrfad bjemd
@fnurm 1.70 scjws 1.70 scjws tcm strmss‛strafh kurvms emskrfgfhn ah mdastfk‛pmr`mktdy pdastfk gmcavfjr j` a batmrfad. Ft fs assubme tcat tcm batmrfad cas tcm sabm gmcavfjr fh tmhsfjh ahe fh kjbprmssfjh. [fbfdardy `jr scmar strmss‛strafh, tcm batmrfad gmcavfjr fs sabm `jr pjsftfvm ahe hmnatfvm strmssms ahe strafhs. 7
yf md e
yf md e
2 ‛ yf md e
2
2 M
yf md e
yf md e
yf md e
2
yf md e
2 N
yf md e
yf md e
yf md e
yf md e
2 ‛ yf md e (g)
(a)
Mdastfk‛pmr`mktdy pdastfk batmrfad gmcavfjr< ( a) hjrbad ahe (g (g) scmar. @fnurm 1.70 Mdastfk‛pmr`mktdy
Gm`jrm yfmde strmss, tcm strmss‛strafh rmdatfjhscfp fs nfvmh gy Cjjim‖s daw, a`tmr yfmde strmss, tcm strmss fs a kjhstaht. Mdastfk‛pmr`mktdy pdastfk batmrfad gmcavfjr fs a sfbpdf`yfhn apprjxfbatfjh usme tj kjheukt ah mdastfk‛pdastfk ahadysfs. Ucm apprjxfbatfjh fs a kjhsmrvatfvm apprjxfbatfjh, sfhkm ft fnhjrms tcm kapakfty j` tcm batmrfad tj karry strmssms cfncmr tcah tcm yfmde strmss.
Batmrfad bjemd 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
7?
Ucm mquatfjhs emskrfgfhn tcm strmss‛strafh kurvm arm as `jddjws<
yfmde ‛ yfmde yfmde ‛ yfmde
yfmde M ‛ yfmde
ahe
2
yfmde N ‛ yfmde
yfmde ‛ yfmde yfmde ‛ yfmde
(1.5>)
Ucm smt j` pjfhts `jrbfhn tcm gjuheary gmtwmmh tcm mdastfk ahe pdastfk rmnfjhs jh a gjey fs kaddme tcm mdastfk‛pdastfk gjuheary. Emtmrbfhfhn tcm djkatfjh j` tcm mdastfk‛pdastfk gjuheary fs jhm j` tcm krftfkad fssums fh mdastfk‛pdastfk ahadysfs. Ucm djkatfjh j` tcm mdastfk‛pdastfk gjuheary fs emtmrbfhme usfhn tcm `jddjwfhn jgsmrvatfjhs< 1. Jh tcm mdastfk‛pdastfk gjuheary, strafh bust gm mquad tj yfmde strafh, ahe strmss mquad tj yfmde strmss. 7. Em`jrbatfjhs ahe strafhs arm kjhtfhujus at add pjfhts, fhkduefhn pjfhts at tcm mdastfk‛ mdastfk‛ pdastfk pdastf k gjuheary. gjuh eary.
1.3.0
Dfhmar strafh-caremhfhn batmrfad bjemd
@fnurm 1.7>, 1.7>, scjws strmss‛strafh kurvms `jr tcm dfhmar strafh-caremhfhn bjemd, adsj rm`mrrme tj as tcm gfdfhmar batmrfad bjemd. Ft fs assubme tcat tcm batmrfad cas tcm sabm gmcavfjr fh tmhsfjh ahe fh kjbprmssfjh. [fbfdardy `jr scmar strmss‛strafh, tcm batmrfad gmcavfjr fs sabm `jr pjsftfvm ahe hmnatfvm strmssms ahe strafhs.
yf md e
yf md e ‛
yf mde md e
yf md e
yfmde
+N
7
2 N 1
yf md e
yf mde md e
2 ‛
yfmde + N7 ‛ yfmde
2 M 1
+ M + yfmde 7 yfmde
yf mde md e
yfmde + M 7 ‛ yfmde
2
+ yfmde
yf md e (g)
( a) hjrbad ahe (g (g) scmar. @fnurm 1.7> Dfhmar strafh caremhfhn bjemd< (a
Ucfs fs ahjtcmr kjhsmrvatfvm, sfbpdf`yfhn apprjxfbatfjh j` batmrfad gmcavfjr fh wcfkc wm jhkm bjrm fnhjrm tcm agfdfty j` tcm batmrfad tj karry strmssms cfncmr tcah tcjsm scjwh gy tcm strafnct dfhms. Emtmrbfhfhn tcm djkatfjh j` tcm mdastfk‛pdastfk gjuheary fs jhkm bjrm a krftfkad fssum fh tcm ahadysfs ahe fs emtm rbfhme as emskrfgme fh[mktfjh fh [mktfjh 1.3.=. 1.3.=. Ucm mquatfjhs emskrfgfhn tcm strmss‛strafh kurvm arm as `jddjws<
2
1.3.>
yfmde + M 7 ‛ yfmde M 1 ‛ yfmde + M 7 + yfmde
yfmde ‛ yfmde yfmde yfmde
ahe
2
yfmde + N7 ‛ yfmde N1 ‛ yfmde + N 7 + yfmde
yfmde ‛ yfmde yfmde yfmde
(1.53)
_jwmr daw batmrfad bjemd
@fnurm 1.73 scjws 1.73 scjws a pjwmr daw rmprmsmhtatfjh j` tcm hjhdfhmar strmss‛strafh kurvm. Ft fs assubme tcat tcm batmrfad cas tcm sabm gmcavfjr fh tmhsfjh ahe fh kjbprmssfjh. [fbfdardy, [fbfdardy, `jr scmar strmss‛strafh, tcm batmrfad gmcavfjr fs sabm `jr pjsftfvm ahe hmnatfvm strmssms ahe strafhs. Ucm mquatfjhs emskrfgfhn tcm strmss‛strafh rmdatfjhscfp arm as `jddjws<
M h 2 ‛ M ‛ h
? ?
ahe
N h 2 ‛ N ‛ h
? ?
?4h41
2 M
h
2 ‛ M ‛
h81
h
?4h41
strmss‛strafh kurvms< (a (a) hjrbad ahe (g (g) scmar. @fnurm 1.73 Hjhdfhmar strmss‛strafh
h
2 ‛ N ‛
h
?4h41 (a)
2 N
h81
?4h41
h81
h81
(1.56)
(g)
M``mkts j` tmbpmraturm 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
71
Ucm kjhstahts M kjhstahts M ahe ahe h, kaddme tcm strmhntc kjm``fkfmht ahe tcm strafh-caremhfhn kjm``fkfmht. Bjst bmtads fh tcm pdastfk rmnfjh ahe bjst pdastfks arm rmprmsmhtme gy tcm sjdfe kurvm, wftc a strafh-caremhfhn kjm``fkfmht dmss tcah 1. [j`t ruggmr, as wmdd as buskdms ahe jtcmr jrnahfk batmrfads, rmprmsmhtme gy tcm eascme dfhm fh @fnurm 1.73, 1.73, cavm a strafh-caremhfhn kjm``fkfmht nrmatmr tcah 1. @rjb Mquatfjh (1.56) wm (1.56) wm hjtm tcat wcmh strafh fs hmnatfvm, tcm tmrb jh tcm rfnct gmkjbms pjsftfvm, pmrbfttfhn mvaduatfjh j` tcm hubgmr tj `raktfjhad pjwmrs. @urtcmrbjrm, wftc hmnatfvm strafh wm jgtafh hmnatfvm strmss, as wm scjude. ^m kjude kjbgfhm a dfhmar mquatfjh `jr tcm dfhmar part ahe hjhdfhmar mquatfjh `jr tcm hjhdfhmar part as wm efe fh [mktfjh 1.3.09 1.3.09 jr wm kjude kjbgfhm twj hjhdfhmar mquatfjhs, tcus krmatfhn aeeftfjhad batmrfad bjemds. ^cfkc batmrfad bjemd tj usm: Ucm ahswmr empmhes upjh tcm batmrfad strmss‛strafh kurvm, tcm emnrmm j` akkuraky hmmeme, ahe tcm rmsudtfhn kjbpdmxfty j` tcm ahadysfs.
MSAB_DM 1.5 Adubfhub cas a yfmde strmss j` yfmde 5? isf fh tmhsfjh, a yfmde strafh j` yfmde ?.??5, ah udtfbatm strmss j` udt 5= isf, ahe tcm kjrrmspjhefhn udtfbatm strafh j` udt ?.?1>. Emtmrbfhm tcm batmrfad kjhstahts `jr (a) tcm mdastfk‛pmr`mktdy pdastfk bjemd, (g) tcm dfhmar strafh-caremhfhn bjemd, ahe (k) tcm hjhdfhmar pjwmr daw bjemd.
1?,??? isf 5? isf
AH[.
?.??5 ?.??5
9
??? isf 1? ??? 5? ;35.0 ‛ ?.??5 isf AH[.
1.6
07.> ?.?315 isf ‛ 07.> ‛ ?.?315
?.??5 ?.??5
isf
9
? ?
M@@MKU[ J@ UMB_MWAU]WM
A batmrfad mxpahes wftc ah fhkrmasm fh tmbpmraturm ahe kjhtrakts wftc a emkrmasm fh tmbpmraturm. F` tcm kcahnm fh tmbpmraturm fs uhf`jrb, uhf`jrb, ahe f` tcm batmrfad fs fsjtrjpfk ahe fsjtrjpfk ahe cjbjnmhmjus, cjbjnmhmjus, tcmh add dfhms jh tcm batmrfad wfdd kcahnm efbmhsfjhs gy mquad abjuhts. Ucm mxpahsfjh wfdd rmsudt fh a hjrbad strafh, gut hj scmar strafh wfdd gm prjeukme gmkausm tcmrm wfdd gm hj kcahnm fh tcm ahndms gmtwmmh ahy twj dfhms. Mxpmrfbmhtad jgsmrvatfjhs kjh`frb tcfs emeuktfjh. Mxpmrfbmhts adsj scjw tcat tcm kcahnm fh tmb pmraturm (U ) U ) fs rmdatme tj tcm tcmrbad hjrbad strafh (U ) as
U U
(1.=?)
wcmrm tcm Nrmmi dmttmr adpca ( fs kaddme tcm dfhmar kjm``fkfmht j` tcmrbad mxpahsfjh. Ucm dfhmar rmdatfjhscfp nfvmh gy Mquatfjh (1.=?) fs (1.=?) fs vadfe `jr bmtads fh tcm rmnfjh `ar `rjb tcm bmdtfhn pjfht. Fh tcfs dfhmar rmnfjh, tcm strafhs `jr bjst bmtads arm sbadd ahe tcm usuad uhfts `jr arm @ jr K. Ucrjuncjut tcm efskussfjh fh tcfs smktfjh wm scadd assubm tcat wm arm fh tcm dfhmar rmnfjh. Umst pmr`jrbme at U j
s s m r t [ d a b r j H
Umst pmr`jrbme at U j+EU
J1
J
U
-- M
Hjrbad [trafh
@fnurm 1.76 M``mkt j` tmbpmraturm jh strmss‛strafh kurvm.
Ucm tmhsfjh tmst tj emtmrbfhm batmrfad kjhstahts fs kjheuktme at ah abgfmht tmbpmraturm. ^m mxpmkt tcm strmss‛strafh kurvm tj cavm tcm sabm kcaraktmr at twj ef``mrmht abgfmht tmbpmraturms. F` wm rafsm tcm tmbpmraturm gm`jrm wm appdy tcm `jrkm _ `jrkm _ jh tcm spmkfbmh, tcmh tcm spmkfbmh wfdd mxpahe ahe rmsudt fh a tcmrbad strafh as nfvmh gyMquatfjh gy Mquatfjh (1.=?). (1.=?). Gut gmkausm tcmrm fs hj mxtmrhad `jrkm, tcmrm wfdd gm hj rmsudtfhn fhtmrhad `jrkms, cmhkm hj strmssms. Ucus, tcm fhkrmasm fh tmbpmraturm gm`jrm tcm appdfkatfjh j` tcm `jrkm kausms tcm startfhn pjfht j` tcm strmss‛strafh kurvm tj bjvm `rjb J tj J1 as scjwh fh @fnurm 1.76. 1.76. Ucm tjtad strafh at ahy pjfht fs tcm sub j` bmkcahfkad strafh ahe tcmrbad strafhs, wcfkc kah gm wrfttmh as
--- U
(1.=1) M F` tcmrm arm hj fhtmrhad `jrkms nmhmratme fh a gjey jwfhn tj kcahnms fh tmbpmraturm, tcmh hj strmssms wfdd gm prjeukme. Batmrfad hjh cjbjnmhmfty, batmrfad ahfsjtrjpy, hjhuhf`jrb tmbpmraturm efstrfgutfjh, jr rmaktfjh `jrkms `rjb gjey kjhstrafhts arm tcm rmasjhs `jr tcm nmhmratfjh j` strmssms `rjb tmbpmraturm kcahnms. ^m rmkjre tcm `jddjwfhn jgsmrvatfjh.
• Hj tcmrbad strmssms arm prjeukme fh a cjbjnmhmjus, fsjtrjpfk, uhkjhstrafhme gjey as a rmsudt j` uhf`jrb uhf`jrb tmb pmraturm kcahnms.
@afdurm tcmjrfms 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
77
Ucm nmhmradfzme vmrsfjh j` Cjjim‖s daw rmdatms bmkcahfkad strafhs tj strmssms. Ucm tjtad hjrbad strafh, as smmh `rjb Mquatfjh (1.=1), (1.=1), fs tcm sub j` bmkcahfkad ahe tcmrbad strafhs. @jr fsjtrjpfk batmrfads wftc tmbpmraturm kcahnms, tcm nmhmradfzme daw kah gm wrfttmh as scjwh gmdjw. ^m ej hjt cavm tj kjbbmht jh tcm cjbjnmhmfty j` tcm batmrfad jr tcm uhf`jrbfty j` tmbpmraturm kcahnm gmkausm Cjjim‖s daw fs wrfttmh `jr a pjfht, hjt tcm wcjdm gjey.
xx
xx ‛ yy + zz M + U
(1.=7a)
yy yy ‛ zz + xx M + U
(1.=7g)
zz zz ‛ xx + yy M + U
(1.=7k)
xy xy N
(1.=7e)
yz yz N
(1.=7m)
zx zx N
(1.=7`)
2
Bmkcahfkad strafh
Ucmrbad strafh
MSAB_DM 1.= Ucm strafhs at a pjfht jh adubfhub ( M M >? N_a, N 73 N_a, ?.7=, ahe 7; /K) wmrm `juhe tj gm
xx 0=?
yy ;??
xy >=?
F` tcm tmbpmraturm fhkrmasms gy 7?K ahe tcm pjfht fs fh pdahm strmss, emtmrbfhm tcm strmssms xx yy, ahe xy ahe tcm strafh zz . AH[. xy 71B_a 9 xx 11.7 B_a U yy 3.5 B_a K 9 zz 5=?
1.1?
@AFD]WM UCMJWFM[
Ucm baxfbub strmhntc j` a batmrfad fs fts atjbfk strmhntc. Fh gudi batmrfads, cjwmvmr, tcm efstrfgutfjh j` `daws (fbpurftfms, bfkrj cjdms, bfkrjkrakis, mtk.) krmatms darnm strmss nraefmhts. Ucmsm darnm strmss nraefmhts kausm tcm gudi strmhntc j` tcm batmrfad tj gm jremrs j` banhftuem djwmr tcah tcm atjbfk strmhntc j` tcm batmrfad. @afdurm tcmjrfms assubm a cjbjnmhmjus batmrfad fh wcfkc tcm m``mkts j` `daws cavm gmmh avmranme fh sjbm bahhmr. Ucfs assubptfjh j` cjbjnmhmfty rmsudts fh avmranm batmrfad strmhntc vadums tcat fs aemquatm `jr bjst prjgdmbs fh mhnfhmmrfhn emsfnh ahe ahadysfs. @jr cjbjnmhmjus‛fsjtrjpfk-batmrfads, tcm kcaraktmrfstfk `afdurm strmss fs mftcmr tcm yfmde strmss jr tcm udtfbatm strmss, usuaddy jgtafhme `rjb a uhfaxfad tmhsfdm tmst, a tmst fh wcfkc tcmr m fs jhdy jhm hjhzmrj strmss kjbpj hmht. Cjw ej wm rmdatm tcm jhm strmss kjbpjhmht j` a uhfaxfad strmss statm tj tcm strmss kjbpjhmhts fh twj- ahe tcrmm-efbmhsfjhad statms j` strmss: A ` afdurm tcmjry fs a statmbmht agjut tcm rmdatfjhscfp j` strmss kjbpjhmhts tj batmrfad `afdurm kcaraktmrfstfks vadums. Hj jhm tcmjry fs appdfka gdm tj add batmrfads. ^m scadd kjhsfemr `jur tcmjrfms (smm Uagdm 1.;). 1.;). Baxfbub scmar strmss tcmjry ahe tcm baxfbub jktacmerad scmar strmss tcmjry arm nmhmraddy usme `jr euktfdm batmrfads. Fh euktfdm batmrfads, `afdurm fs kcaraktmrfzme gy yfmde strmss. Ucm baxfbub hjrbad strmss tcmjry ahe bjef`fme Bjcr‖s tcmjry arm nmhmraddy usme `jr grfttdm batmrfads. Fh grfttdm batmrfads, `afdurm fs kcaraktmrfzme gy udtfbatm strmss. Uagdm 1.; [yhjpsfs j` @afdurm Ucmjrfms
Euktfdm Batmrfad
1.1?.1
Grfttdm Batmrfad
Kcaraktmrfstfk `afdurm strmss
Vfmde strmss
]dtfbatm strmss
Ucmjrfms
1. Baxfbub scmar strmss 7. Baxfbub jktacmerad scmar strmss
1. Baxfbub hjrbad strmss 7. Bjef`fme Bjcr
Baxfbub scmar strmss tcmjry
@jr euktfdm batmrfads, tcm tcmjry statms< A batmrfad wfdd `afd wcmh tcm baxfbub scmar strmss mxkmmes tcm scmar strmss at tcm yfmde jgtafhme `rjb a uhfaxfad tmhsfdm tmst. Ucm `afdurm krftmrfjh emskrfgme fs wrfttmh as
ba x yfmde
(1.=;)
wcmrm yfmde fs tcm baxfbub scmar strmss at tcm yfmde pjfht `juhe fh a uhfaxfad tmhsfjh tmst. Ucm baxfbub scmar strmss fs nfvmh gy tcm Mquatfjh (1.1=). (1.1=). [ugstftutfhn tcm strmssms at tcm yfmde pjfht fh a uhfaxfad tmhsfjh tmst as 1 yfmde , 7 ? ahe ; ? fh Mquatfjh (1.1=) ahe (1.1=) ahe mquatfhn tcm rmsudt tj yfmde , wm jgtafh tcm baxfbub scmar strmss at yfmde pjfht as `jddjws<
yfmde 2 yfmde 7
(1.=5)
@afdurm tcmjrfms 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
7;
Urmska‖s yfmde krftmrfjh nfvmh gmdjw. [ugstftutfhn Mquatfjhs (1.=;) ahe (1.=;) ahe (1.1=) fhtj (1.1=) fhtj Mquatfjh (1.=5), (1.=5), wm jgtafh tcm Urmska‖s
bax
1 ‛ 7 7 ‛ ; ; ‛ 1 yf md e yfmde
(1.==)
Baxfbub jktacmerad scmar strm
yfmde
1
yfmde
Baxfbub scmar strmss yfmde @fnurm 1.;? @afdurm mhvmdjpms `jr euktfdm batmrfads fh pdahm strmss.
1.1?.7
Baxfbub jktacmerad scmar strmss tcmjry
Adsj kaddme vjh-Bfsms tcmjry. Mxpmrfbmhts cavm scjwh tcat euktfdm batmrfads kah wftcstahe vmry darnm cyerjstatfk prmssurms ahe yfmdefhn fs njvmrhme gy jkt rmakcfhn a krftfkad vadum. Ucm baxfbub jktacmerad scmar strmss tcmjry `jr euktfdm batmrfads fs< A batmrfad wfdd `afd wcmh tcm baxfbub jktacmerad scmar strmss mxkmmes tcm jktacmerad scmar strmss strmss at tcm yfmde jgtafhme `rjb a uhfaxfad tmhsfdm tmst . Ucm `afdurm krftmrfjh fs
jk t yfmde
(1.=0)
wcmrm yfmde fs tcm jktacmerad scmar strmss at yfmde pjfht fh a uhfaxfad tmhsfdm tmst. [ugstftutfhn 1 yfmde , 7 ? ahe ; ?, tcm strmssms at yfmde pjfht fh a uhfaxfad tmhsfjh tmst, fhtj tcm mxprmssfjh j` jktacmerad scmar strmss fh Mquatfjh (1.75e), (1.75e), wm jgtafh (1.75e) fhtj Mquatfjh (1.=0), (1.=0), wm jgtafh yfmde 2 7 yfmde ; . [ugstftutfhn tcfs ahe Mquatfjh (1.75e) fhtj 1 7 7 7 ------- 1 ‛ 7 + 7 ‛ ; + ; ‛ 1 yfmde (1.=>) 7 Ucm dm`t-cahe sfem j` Mquatfjh (1.=>) fs (1.=>) fs rm`mrrme tj as vjh-Bfsms strmss v jh . Gmkausm vjh Bfsms strmss vjh fs mxtmhsfvmdy usme fh tcm emsfnh j` strukturms ahe bakcfhms, wm `jrbaddy em`fhm ft as `jddjws<
vjh
1 2 ------7
1 ‛ 7 7 + 7 ‛ ; 7 + ; ‛ 1 7
Fh uhfaxfad tmhsfjh tmst, at yfmde pjfht wm cavm 1 2
jk t 2
7 ; yfmde , wcfdm
v jh
2
yfmde , ahe 7
2
;
(1.=3)
2 ? . Ucus, at yfmde pjfht
yfmde , wcfkc fs sfbpdmr ahe usme fh `afdurm krftmrfjh `jr euktfdm batmrfad ahe rm`mrrme tj as
tcm vjh Bfsms yfmde krftmrfjh.
v jh yfmde
(1.=3a)
Mquatfjhs (1.==) ahe (1.==) ahe (1.=>) arm (1.=>) arm `afdurm mhvmdjpms fh a spakm fh wcfkc tcm axms arm prfhkfpad strmssms. @afdurm mhvmdjpms smparatm tcm emsfnh spakm `rjb `afdurm spakm. Fh erawfhn `afdurm mhvmdjpms, tcm kjhvmhtfjh 1 7 fs fnhjrme. F` tcm kjhvmhtfjh wmrm mh`jrkme, tcmrm wjude gm hj mhvmdjpm fh tcm smkjhe quaeraht ahe jhdy ah mhvmdjpm gmdjw tcm 5=j dfhm wjude gm aebfssfgdm fh tcm tcfre quaeraht, rmsudtfhn fh vmry strahnm-djjifhn mhvmdjpms. @jr a pdahm strmss (; 2 ?) prjgdmb, wm kah scjw tcmsm `afdurm mhvmdjpms as fh @fnurm 1.;?. 1.;?. Hjtfkm tcat tcm baxfbub jktacmerad scmar strmss mhvmdjpm mhkjbpassms tcm baxfbub scmar strmss mhvmdjpm. Mxpmrfbmhts scjw tcat `jr bjst euktfdm batmrfads, tcm baxfbub jktacmerad scmar strmss tcmjry nfvms gmttmr rmsudts tcah tcm baxfbub scmar strmss tcmjry, gut baxfbub scmar strmss tcmjry fs sfbpdmr tj usm.
1.1?.;
Baxfbub hjrbad strmss tcmjry
Ucm baxfbub hjrbad strmss tcmjry `jr grfttdm batmrfads fs< A batmrfad wfdd `afd wcmh tcm baxfbub hjrbad strmss at a pjfht mxkmmes tcm udtfbatm hjrbad strmss (udt ) jgtafhme `rjb a uhfaxfad tmhsfjh tmst . Ucm tcmjry nfvms njje rmsudts `jr grfttdm batmrfads prjvfeme tcm prfhkfpad strmss jhm fs tmhsfdm. Ucm `afdurm krftmrfjh fs bax 1 7 ;
udt
(1.=6)
@jr kjhkrmtm ahe bahy jtcmr grfttdm batmrfads, tcm udtfbatm strmss fh tmhsfjh fs `ar dmss tcah tcm udtfbatm strmss fh kjb prmssfjh gmkausm bfkrjkrakis tmhe tj nrjw fh tmhsfjh ahe tj kdjsm fh kjbprmssfjh . Gut tcm sfbpdfkfty j` tcm `afdurm krftmrfjh baims tcm tcmjry attraktfvm, ahe ft kah gm usme f` prfhkfpad strmss 1 fs tmhsfdm ahe tmhsfdm ahe fs tcm ejbfhaht prfhkfpad ejbfhaht prfhkfpad strmss.
@afdurm tcmjrfms 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
1.1?.5
75
Bjef`fme Bjcr‖s tcmjry
Ucm Bjcr‖s tcmjry `jr grfttdm batmrfads fs< A batmrfad wfdd `afd f` a strmss statm fs jh tcm mhvmdjpm tcat fs tahnmht tj tcm tcrmm tcrmm Bjcr‖s Bjcr‖s kfrkdms kjrrmkjrrm spjhefhn tj uhfaxfad udtfbatm strmss fh tmhsfjh, uhfaxfad udtfbatm strmss fh kjbprmssfjh, ahe purm scmar. scmar. K^ @afdurm mhvmdjpm U
K
Uahnmht pjfhts
[
K (K)
U
[
[
(U)
Uahnmht pjfhts
KK^ (g)
(a) ( g) Bjcr‖s `afdurm mhvmdjpm. @fnurm 1.;1 (a) strmss mdmbmhts. (g
^m kah kjheukt tcrmm mxpmrfbmhts ahe emtmrbfhm tcm udtfbatm strmss fh tmhsfjh U, tcm udtfbatm strmss fh kjbprmssfjh 1.;1a. A K, ahe tcm udtfbatm scmar strmss fh purm scmar [. Ucm tcrmm strmss statms arm scjwh jh tcm strmss mdmbmhts fh @fnurm 1.;1a Bjcr‖s kfrkdm `jr makc j` tcm tcrmm strmss statms fs tcmh erawh. @fhaddy ah mhvmdjpm tcat fs tahnmht tj tcm tcrmm kfrkdms fs erawh, wcfkc rmprmsmhts tcm `afdurm mhvmdjpm (@fnurm (@fnurm 1.;1g 1.;1g). F` a Bjcr‖s kfrkdm kjrrmspjhefhn tj a strmss statm lust tjukcms tcm mhvmdjpm at ahy pjfht, tcm batmrfad fs at fhkfpfmht `afdurm. F` ahy part j` tcm Bjcr‖s kfrkdm `jr a strmss statm fs jutsfem tcm mhvmdjpm, tcm batmrfad cas `afdme at tcat pjfht. ^m kah adsj pdjt tcm `afdurm mhvmdjpm j` @fnurm 1.;1 gy 1.;1 gy usfhn tcm prfhkfpad strmssms as tcm kjjrefhatm axfs. Fh pdahm strmss tcfs mhvmdjpm fs rmprmsmhtme gy tcm sjdfe dfhm fh @fnurm 1.;7. 1.;7. @jr bjst grfttdm batmrfads, tcm purm scmar tmst fs j`tmh fnhjrme. Fh sukc a kasm tcm dfhm tahnmht tj tcm kfrkdms j` uhfaxfad kjbprmssfjh ahe tmhsfjh wjude gm a strafnct dfhm fh @fnurm 1.;1. 1.;1. Ucm rmsudtfhn sfbpdf`fkatfjh `jr pdahm strmss fs scjwh as a eascme dfhm fh @fnurm 1.;7 ahe 1.;7 ahe emskrfgms wcat fs kaddme Bjef`fme kaddme Bjef`fme Bjcr‖s tcmjry. tcmjry. @fnurm 1.;7 mbpcasfzms 1.;7 mbpcasfzms tcm `jddjwfhn pjfhts. 1. F` gjtc prfhkfpad strmssms arm tmhsfdm, tcmh tcm baxfbub hjrbad strmss bust gm dmss tcah tcm udtfbatm tmhsfdm strmhntc. 7. F` gjtc prfhkfpad strmssms arm hmnatfvm, tcmh tcm baxfbub hjrbad strmss bust gm dmss tcah tcm udtfbatm kjbkjb prmssfvm strmhntc. ;. F` tcm prfhkfpad strmssms arm j` ef``mrmht sfnhs, tcmh `jr tcm bjef`fme Bjcr‖s tcmjry tcm `afdurm fs njvmrhme gy Mquatfjh (1.0?), (1.0?), fh wcfkc 1 fs tmhsfdm ahe 7 fs kjbprmssfvm.
1 K U
------7 ‛ -----1-
(1.0?)
7 (-[,[)
U U
K
1
Bjcr‖s Ucmjry ([,-[) Bjef`fme Bjcr‖s Ucmjry
K @fnurm 1.;7 @afdurm mhvmdjpm `jr pdahm strmss akkjrefhn tj Bjcr‖s tcmjry. Fh erawfhn `afdurm mhvmdjpms, tcm kjhvmhtfjh tcat 1 7 fs fnhjrme.
Kdjsurm 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
7=
MSAB_DM 1.0 At a krftfkad pjfht jh a bakcfhm part baem j` stmmd, tcm strmss kjbpjhmhts arm xx 1?? B_a (U), yy =? B_a (K), ahe xy ;? B_a. Assubfhn tcat tcm pjfht fs fh pdahm strmss ahe tcm yfmde strmss fh tmhsfjh fs yfmde 2 77? B_a, emtmrbfhm tcm `aktjr j` sa`mty gy usfhn (a) tcm baxfbub scmar strmss tcmjry ahe (g) tcm baxfbub jktacmerad scmar strmss tcmjry. AH[.
i 2 1.;0 9 i 2 1.==
MSAB_DM 1.> Nfvmh a djae _, tcm strmssms at a pjfht jh a `rmm sur`akm wmrm `juhe tj gm xx ; _ isf isf (K), yy = _ isf isf (U), ahe xy ‛7 _ isf, isf, wcmrm _ wcmrm _ fs fs bmasurme fh ifps. Ucm grfttdm batmrfad cas a tmhsfdm strmhntc j` 13 isf ahe a kjbprmssfvm strmhntc j` ;0 isf. Emtmrbfhm tcm baxfbub vadum j` tcm djae _ djae _ tcat tcat kah gm appdfme jh tcm strukturm usfhn tcm bjef`fme Bjcr‖s tcmjry. AH[. _ bax 2 7.53 ifps
1.11
KDJ[]WM
Fh tcfs kcaptmr wm saw tcat strmss jh a sur`akm fs ah fhtmrhad efstrfgutme `jrkm systmb tcat fs a vmktjr quahtfty, wcfdm strmss at a pjfht fs a smkjhe-jremr tmhsjr wcjsm sfnh fs emtmrbfhme `rjb tcm efrmktfjh j` tcm jutware hjrbad j` ah fbanfhary sur`akm tcrjunc tcm pjfht ahe tcm efrmktfjh j` tcm fhtmrhad `jrkm. ^m adsj saw tcat mhnfhmmrfhn strafh, wcfkc fs hjt a smkjhe-jremr tmhsjr, bay gm trahs`jrbme fhtj a smkjhe-jremr tmhsjr gy efvfefhn add scmar strafhs gy a `aktjr j` 7. _rfhkfpad strmssms ahe strafhs arm tcm mfnmhvadums, ahe prfhkfpad efrmktfjhs arm tcm mfnmhvmktjrs, j` tcm strmss ahe tmhsjr strafh batrfkms. ^m saw tcat tcm prfhkfpad strmssms ahe strafhs at a pjfht arm uhfqum, wcfkc fbpdfms tcat tcm kjm``fkfmhts kjm``fkfmhts j` tcm kcaraktmrfstfk mquatfjh mquatfjh arm uhfqum. Ucm kjm``fkfmhts kjm``fkfmhts j` tcm kcaraktmrfstfk mquatfjhs arm tcm strmss ahe strafh fhvarfahts tcat ej hjt empmhe upjh tcm kjjrefhatm systmb―ah jgsmrvatfjh wm usme tj kcmki jur kadkudatfjhs j` prfhkfpad strmssms ahe strafhs. ^m saw tcat sbadd-strafh apprjxfbatfjh fs a dfhmar rmdatfjhscfp gmtwmmh efspdakmbmhts ahe strafhs tcat fs vadfe `jr strafhs dmss tcah 1%. [badd-strafh apprjxfbatfjhs addjw sfnhf`fkaht sfbpdf`fkatfjh, fhkduefhn kadkudatfjh j` hjrbad strafh, fh wcfkc wm apprjxfbatm tcm aktuad em`jrbatfjh gy tcm kjbpjhmht j` tcm em`jrbatfjh fh tcm jrfnfhad efrmktfjh j` tcm dfhm mdmbmht, ahe scmar strafh, fh wcfkc tcm sfhm ahe tahnmht `uhktfjhs j` scmar strafh arm apprjxfbatme gy tcmfr arnubmhts. ^m saw tcat nmhmradfzme Cjjim‖s daw `jr fsjtrjpfk batmrfad kah gm wrfttmh `jr ahy jrtcjnjha d systmbs as tcm strmss strafh rmdatfjhscfp fs sabm fh add efrmktfjh. _rfhkfpad efrmktfjhs, wcfkc arm jrtcjnjhad, `jr strmss ahe strafhs arm sabm jhdy `jr fsjtrjpfk batmrfads. ^m saw tmbpmraturm kcahnms nmhmratm jhdy tcmrbad hjrbad strafhs. F` a strukturad bmbgmr fs uhkjhstrafht (statfkaddy emtmrbfhatm) tcmh hj tcmrbad strmssms wfdd gm nmhmratme. ^m saw tcat `afdurm tcmjrfms arm statmbmhts rmdatfhn tcm sfx fhempmhemht strmss kjbpjhmhts tj yfmde strmss ahe udtfbatm strmss vadums jgtafhme `rjb tmhsfjh tmsts. ^m wfdd usm tcm kjhkmpts ahe fh`jrbatfjh fhtrjeukme fh tcfs kcaptmr fh sugsmqumht kcaptmrs `jr sjdvfhn prjgdmbs ahe gufdefhn tcmjrfms j` strukturad bmbgmrs.
[yhjpsfs j` mquatfjhs 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
1.17
70
[VHJ_[F[ J@ MQ]AUFJH[
[trmss at a pjfht
[trmss fl 2
dfb
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2
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7
hh xx kjs yy sfh 7 xy sfh Urahs`jrbatfjh fh 7-E
1,7
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1 + 7 + ;
bax
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F 7 2
1
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tt t U t 9 [ h xx xy yz xx xz + F ; yx yy zz zx zz zx zy
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2
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Emvfatjrfk strmss fhvarfahts
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1 7 ;
1 ‛ 7 7 7 ‛ ; 7 ; ‛ 1 7
kjs ; 2 7 F 1 ; ‛ F 1 ; F 7 + F ; 7 A
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+ F 1 ; 9
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F 1 ; 7 ‛ F 7 ; 9
1 ‛ 7 7 + 7 ‛ ; 7 + ; ‛ 1 7 ;
1 + 7 + ; ; 9 jk t
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7
xz
1 7 + 7 ; + ; 1
7
jk t
7
xy kjs ‛ sfh
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x ‛ F 1 x + F 7 x ‛ F ; 2 ? 9 x 1 2 7 A kjs + F 1 ; 9 x 7 ; 2 ‛ 7 A kjs 0?
Jktacmerad strmssms
kjs
7 xy 9 tah 7 p 2 -------------------------- xx ‛ yy
ht t U h
xx + yy + zz
Urahs`jrbatfjh fh 7-E F 1 2 ahe ;-E
2
xx + yy xx ‛ yy 7 7 --------------------- --------------------- 7 + xy 7
2
F 1 2
yz
7
7
L 7 2 ‛ Y 1 ‛ 7 + + 7 ‛ ; + ; ‛ 1
7
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[trafh
u x u r r
2
_jdar Kjjrefhatms
rr
2
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hh
2
xx kjs + yy sfh + xy sfh kjs
hh
2
h U h
2
zz
w z
xx
F 1 2
yy
v y
Kartmsfah Kjjrefhatms Kjjrefhatms
u 1 v 2 ----r + -r r
7
ht
xx + yy + zz
2
Mdastfk-pmr`mktdy Mdastfk-pmr`mktdy pdastfk
Dfhmar strafh caremhfhn
_jwmr daw Umbpmraturm m``mkt
r
xx xy
F 7 2
yx yy
1 + 7 + ;
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1
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7
+
+
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2
u + v y x
yz zy v + w z y
zx
2
xz
2
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Batmrfad Bjemds xx xx ‛ yy + zz M Fsjtrjpfk Batmrfad
xy
7
Urahs`jrbatfjh fh 7-E ahe ;-E F 1 2
2
2 ‛ 7 xx sfh kjs + 7 yy sfh kjs + xy kjs
tt
yy yz zy zz
7
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+
+
7
7
‛ sfh
t U t 9
2
xx xz zx zz
1
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;
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xx xy xz F ; 2
yx yy yz zx zy zz
1 7 + 7 ; + ; 1
yy yy ‛ zz + xx M
F ; 2
1
?
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7
?
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;
2
zz zz ‛ xx + yy M
1 7 ;
M
N -------------------7 1 + xy xy N yz yz N zx zx N M 1 ‛ xx + yy + zz M xx + 1 ‛ yy + zz M xx + yy + 1 ‛ zz ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- xx ----------------------------- yy ---------------------------- zz ----------------------------- 1 ‛ 7 1 + 1 ‛ 7 1 + 1 ‛ 7 1 + yfmde yfmde yfmde yfmde ‛ yfmde yfmde ‛ yfmde yfmde 9 2 N M ‛ ‛ ‛ yfmde ‛ yfmde yfmde yfmde
2
yfmde + M 7 ‛ yfmde M 1 ‛ yfmde + M 7 + yfmde
yfmde ‛ yfmde yfmde 9 yfmde
h h ? M N 92 2 ? ‛ M ‛ h ‛ N ‛ h xx 2 xx ‛ yy + zz M + U
2
yy yy ‛ zz + xx M + U
1 ‛ 7 7 ‛ ; ; ‛ 1 yf md e
Pjh-bfssms tcmjry.
vjh 2 1 ‛ 7
Baxfbub hjrbad strmss tcmjry Bjef`fme Bjcr‖s tcmjry.
bax 1 7 ;
7
7
+ 7 ‛ ; + ; ‛ 1
udt
7 K ‛ 1 U 1
yfmde ‛ yfmde yfmde yfmde
? ?
@afdurm Ucmjrfms Baxfbub scmar strmss bax tcmjry.
yfmde + N 7 ‛ yfmde N1 ‛ yfmde + N7 + yfmde
7
7 yfmde
zz zz ‛ xx + yy M + U
7
Jhm-Efbmhsfjhad [trukturad Bmbgmrs
DMAWHFHN JGLMKUFPM[ 1 1. Uj uhemrstahe uhemrstahe cjw kjbpdmxftfms kjbpdmxftfms arm aeeme tj tcm kdassfkad kdassfkad tcmjrfms tcmjrfms j` axfad bmbgmrs, bmbgmrs, tjrsfjh tjrsfjh j` kfrkudar kfrkudar sca`ts, ahe sybsybbmtrfk gmhefhn j` gmabs. 7 ; 7. Uj uhemrstahe uhemrstahe tcm kjhkmpt ahe usm j` efskjhtfhufty efskjhtfhufty `uhktfjhs `uhktfjhs fh tcm ahadysfs ahadysfs j` jhm-efbmhsfjhad jhm-efbmhsfjhad strukturad strukturadbmbgmrs bmbgmrs sug5 lmktme tj efskjhtfhujus djaes. = 0 Kcafr dmns, utfdf ty pjdms, erfdds, gjjiscmdvms, efvfhn gjares, `rabm j` a gufdefhn jr a kar, simdmtjh j` a cubah gmfhn, arm abjhn tcm > kjuhtdmss appdfkatfjhs j` jhm-efbmhsfjhad strukturad bmbgmrs. Ucm tcmjrfms ahe `jrbudas `jr jhm-efbmhsfjhad strukturad bmb3 gmrs vary fh kjbpdmxfty empmhefhn upjh tcm djaefhn, batmrfad prjpmrtfms, ahe krjss-smktfjhad nmjbmtrfms, bahy j` wcfkc wm wfdd 6 kjhsfemr fh tcfs kcaptmr. 1? Fh Kcaptmr 1, wm efskussme tcm rmdatfjhscfps j` efspdakmbmhts tj strafhs, ahe strafhs tj strmssms. Fh [mktfjh 7.1, 7.1, wm wfdd 11 smm tcm twj stmp prjkmss gy wcfkc strmssms arm rmdatme tj mxtmrhad `jrkms ahe bjbmhts. Fh [mktfjh 7.7, 7.7, wm syhtcmsfs add tcmsm rmda17 tfjhscfps fhtj a djnfk gy wcfkc wm rmdatm efspdakmbmhts tj mxtmrhad `jrkms ahe bjbmhts. Fh [mktfjh 7.;, 7.;, wm usm tcm djnfk tj 1; 15 emvmdjp tcm kdassfkad tcmjrfms j` axfad bmbgmrs, tjrsfjh j` kfrkudar sca`ts, ahe sybbmtrfk gmhefhn j` gmabs wcfkc wm saw fh tcm 1= fhtrjeuktjry kjursm j` bmkcahfks j` batmrfads. A syhjpsfs j` add tcrmm tcmjrfms fs prmsmhtme fh a tagudar `jrb tj mbpcasfzm tcm 10 kjbbjhadftfms ahe ef``mrmhkms fh tcm emrfvatfjh j` strmss ahe em`jrbatfjh `jrbudas. Ucm dfbftatfjhs ahe assubptfjhs tcat arm 1> baem tj nmt tcm sfbpdmst pjssfgdm `jrbudas arm femhtf`fme wftc tcm fhtmht j` erjppfhn tcmb tj fhkduem kjbpdmxftfms fh nmjbmtry, 13 djaefhn, ahe batmrfads. 16 Efskjhtfhujus djaefhn jh a strukturad bmbgmr fs `afrdy kjbbjh. Fh tcm fhtrjeuktjry kjursm tcmsm efskjhtfhuftfms fh 7? djaefhn wmrm akkjuhtme `jr gy smnbmhtfhn tcm strukturad bmbgmr sukc tcat fh makc smnbmht tcm djaefhn was kjhtfhujus. Wmdatfvm 71 em`jrbatfjhs j` smnbmht mhes wmrm `juhe, ahe tcm jvmradd em`jrbatfjh was jgtafhme gy usfhn tcm jgsmrvatfjh tcat em`jrbatfjh 77 was kjhtfhujus at tcm pjfht j` djaefhn efskjhtfhufty. Ucfs apprjakc fs adnmgrafkaddy tmefjus, partfkudardy `jr gmab-gmhefhn prjg7; dmbs ahe statfkaddy fhemtmrbfhatm strukturad bmbgmrs. Fh tcfs kcaptmr tcm kjhkmpt j` efskjhtfhufty `uhktfjhs fs fhtrjeukme ahe 75 m``mktfvmdy usme `jr ahadyzfhn tcm em`jrbatfjh j` statfkaddy emtmrbfhatm ahe fhemtmrbfhatm strukturad bmbgmrs tcat arm suglmktme 7= tj efskjhtfhujus djaes. 70 7> 7.1 FHUMWHAD @JWKM[ AHE BJBMHU[ 73 76 [trmss rmsudtaht fs ahjtcmr habm `jr fhtmrhad `jrkms ahe bjbmhts. [trmssms gmfhn ah fhtmrhad efstrfgutme `jrkm systmb kah adways ;? gm rmpdakme gy mqufvadmht fhtmrhad `jrkms ahe bjbmhts `jr purpjsm j` mqufdfgr fub jh a `rmm gjey efanrab. Ucm twj stmp prjkmss 7.1. ;1 j` rmdatfhn strmssms tj mxtmrhad `jrkms fs mbpcasfzme fh @fnurm 7.1. ;7 ;; Mqufdfgrfub Mxtmrhad [tatfk Fhtmrhad ;5 @jrkms ahe [trmssms @jrkms ahe ;= Bjbmhts Mqufvadmhky Bjbmhts ;0 ;> ;3 @fnurm 7.1 Uwj stmp prjkmss j` rmdatfhn strmssms tj mxtmrhad `jrkms ahe bjbmhts ;6 @fnurm 7.7a 7.7a scjws tcm strmssms jh a ef``mrmhtfad mdmbmht wftc tcm jutware hjrbad fh tcm x tcm x efrmktfjh. efrmktfjh. @fnurm 7.7g 7.7g scjws 5? 51 tcm statfkaddy mqufvadmht fhtmrhad `jrkms ahe bjbmhts. Ucm statfk mqufvadmhky j` hjrbad strmss xx tj tcm fhtmrhad `jrkms ahe 7.7 rmsudts fh tcm Mquatfjh (7.1).Ucm (7.1).Ucm statfk mqufvadmhky j` scmar strmssms tj tcm fhtmrhad `jrkms ahe 57 bjbmhts scjwh fh @fnurm 7.7 rmsudts bjbmhts scjwh fh @fnurm 7.7 rmsudts 7.7 rmsudts fh Mquatfjh (7.7). (7.7) . 5; 7>
73
7
Jhm-Efbmhsfjhad [trukturad Bmbgmrs
H
e A
xx
B z
‛ y xx
A
P y
e A
‛ z xx
B y
A
xy
e A
P z
(7.1)
e A
(7.7)
A
xz e A
A
e A
A
U 2
y
xz ‛ z xy
A
eP y 2 xy eA y
y B y
z y
eP z 2 xz eA
P y
J e H 2 2 xx eA
z
P z
B z
H
z
x
J
U
(a)
x
(g )
@fnurm 7.7 [tatfk mqufvadmhky.
@jr tjrsfjh j` kfrkudar krjss smktfjhs, tcm jhdy hjh-zmrj strmss fs scmar strmss x . @fnurm 7.; scjws 7.; scjws tcat tcm scmar `jrkms kah gm wrfttmh as P z 2 e P
x eA kjs
P y 2 ‛ x eA sfh e P
y 2
kjs
z 2 sfh
(7.;)
[ugstftutfhn Mquatfjh (7.;) fhtj (7.;) fhtj tjrqum mxprmssfjh fh Mquatfjh (7.7), (7.7), wm jgtafh tcm fhtmrhad tjrqum jh kfrkudar sca`ts as< U 2
kjs kjs ‛ sfh ‛ sfh e A 2 kjs + sfh e A jr 7
x
x
A
7
x
A
U 2
e A
(7.5)
x
A
y
z x eA
y
z x
U
x
@fnurm 7.; [tatfk mqufvadmhky `jr tjrsfjh j` kfrkudar krjss smktfjhs.
[trmssms arm pjsftfvm jr hmnatfvm akkjrefhn tj sfnh kjhvmhtfjh efskussme fh [mktfjh 1.7.1, 1.7.1, tcus tcm fhtmrhad `jrkms ahe bjbmhts wfdd gm pjsftfvm jr hmnatfvm akkjrefhn tj sfnh kjhvmhtfjh efskussme fh [mktfjh 7.;.7. 7.;.7.
7.1.1
Emkjupdfhn axfad, gmhefhn, ahe tjrsfjh ahadysfs
Fh statfks wm saw tcat ahy efstrfgutme `jrkm kah gm rmpdakme gy a `jrkm ahe a bjbmht at ahy pjfht, jr gy a sfhndm `jrkm (ahe hj bjbmht) at a spmkf`fk pjfht. Ucmsm spmkf`fk pjfhts cmdp emkjupdm axfad `rjb gmhefhn, ahe gmhefhn `rjb tjrsfjh `jr purpjsms j` sfbpdmr ahadysfs. Fh tcm fhtrjeuktjry bmkcahfks j` batmrfads kjursm wm saw tcat `jr dfhmar-mdastfk-fsjtrjpfk-cjb jnmhjus batmrfad, tcm kmhtrjfe j` j` tcm krjss smktfjh emkjupdms tcm gmhefhn `rjb axfad prjgdmbs. Fh tcm sabm bahhmr, scmar bahhmr, scmar kmhtmr emkjupdms emkjupdms tcm scmar `jrkms (cmhkm scmar strmssms) eum tj gmhefhn `rjb tcjsm eum tj tjrsfjh. @jr bjrm kjbpdmx batmrfad bjemds, tcm emtmrbfhatfjh j` tcm spmkf`fk pjfhts tcat emkjupdms axfad, gmhefhn, ahe tjrsfjh ahadysfs rmqufrms bjrm m``jrt. Ucm scmar kmhtmr fs a pjfht fh spakm at wcfkc tcm scmar strmss eum tj gmhefhn kah gm rmpdakme gy statfkaddy mqufvadmht fhtmrhad scmar `jrkms ahe hj fhtmrhad tjrqum. @jr dfhmar prjgdmbs ahe `jr hjh-dfhmar batmrfads wm wfdd gm agdm tj emkjupdm axfad, gmhefhn, ahe tjrsfjh ahadysfs. Cjwmvmr, `jr nmjbmtrfk hjh-dfhmarfty, tcat fs, `jr darnm em`jrbatfjh ft fs nmhmraddy hjt pjssfgdm tj emkjupdm ahadysfs. Axfad `jrkms prjeukm gmhefhn efspdakmbmhts ahe gmhefhn `jrkms prjeukm axfad efapdakmbmhts as wfdd gm smmh fh Kcaptmr >. >.
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
Djnfk fh bmkcahfks j` batmrfads 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
76
MSAB_DM 7.1 A gjey uhemr appdfme djaes ahe gjey `jrkms prjeukms tcm `jddjwfhn efspdakmbmht `fmde u 2 ?
7
7
v 2 Ix y ‛ z + Iaxz
w 2 ‛ 7 Ix yz ‛ Iaxy
(M1)
wcmrm u, v, ahe w arm efspdakmbmhts fh tcm x tcm x,, y, y, ahe z ahe z efrmktfjhs, efrmktfjhs, rmspmktfvmdy. @fnurm 7.5 scjws 7.5 scjws tcm fhtmrhad `jrkms ahe bjbmhts (strmss rmsudtahts) jh a krjss smktfjh. Assubfhn a dfhmar, mdastfk, fsjtrjpfk, cjbjnmhmjus batmrfad, emtmrbfhm add tcm fhtmrhad `jrkms ahe bjbmhts fh tmrbs j` bjeudus j` mdastfkfty, _jfssjh‖s ratfj , ahe tcm kjhstahts I kjhstahts I ahe ahe a.
P y
a y
B y
a
x B z
z
7a
H
U
a
P z
a
7.1. @fnurm 7.5 Nmjbmtry fh Mxabpdm 7.1. =
AH[. H 2 ?
5 MI a B z 2 ? 9 P z 2 ? 9 P z 2 ? 9 U 2 ‛ -------------------;1 +
B y 2 ?
MSAB_DM 7.7 Ucm hjrbad strmss jh a krjss smktfjh fs nfvmh gy xx a gy kz , wcmrm a, g, ahe k arm kjhstahts. Emtmrbfhm tcm kjheftfjhs hmkmssary tj jgtafh tcm `jrbuda gmdjw.
xx
B z B y H 2 ---- ‛ ------- y ‛ ------- z F yy A F zz
(M1)
wcmrm, A fs tcm krjss smktfjhad arma, F arma, F yy ahe F ahe F zz arm tcm smkjhe arma bjbmhts j` fhmrtfa agjut tcm y ahe z axfs. AH[.
k
2
A
ye A 2 ?
z k 2
A
ze A
F yz 2
A
yzeA 2 ?
MSAB_DM 7.; Ucm krjss-smktfjh scjwh fh @fnurm 7.= cas 7.= cas a uhf`jrb tcfkihmss t . Assubm t « a . Ucm scmar strmssms fh tcm krjss smktfjh wmrm `juhe tj gm as nfvmh. (a (a) Wmpdakm tcm scmar strmssms gy mqufvadmht scmar `jrkms P y , P z ahe tjrqum U aktfhn aktfhn at tcm kmhtrjfe K . (g (g) Emtmrbfhm tcm kjjrefhatms j` tcm scmar kmhtmr. [jdvm tcm prjgdmb fh tmrbs varfagdms I varfagdms I , a, ahe t . y 7a/;
a z
s
K
7a
xy
2 ?
xz
Is 2 -----t
xy
I 7 7 2 ‛ -------- ‛ 5 a + 0 as ‛ s 7 at
xz
2 ?
xy
2 ?
xz
I s ‛ 0 a 2 ----------------------t
? s 7a 7a s 5a 5a s 0a
7a
@fnurm 7.=
AH[.
7.7
P y 2 ‛ 15 Ia
7
; 9 P z 2
? 9 U 2 05 Ia
;
6 KK^
;7 a 71 9 m y 2 ? 9 m z 2 ;7a
DJNFK FH BMKCAHFK[ J@ BAUMWFAD[
Efspdakmbmhts, strafhs, strmssms, fhtmrhad `jrkms ahe bjbmhts, ahe mxtmrhad `jrkms ahe bjbmhts arm tcm `uheabmhtad varfagdms fh bmkcahfks j` batmrfads. Ucmsm `uheabmhtad varfagdms arm djnfka ddy rmdatme as sybgjdfkaddy scjwh fh@fnurm fh @fnurm 7.0. 7.0. Ucfs djnfk fs usme `jr emvmdjpfhn gjtc tcm sfbpdm tcmjrfms efskussme fh tcm fhtrjeuktjry kjursm jh tcm bmkcahfks j` batmrfads ahe tcm aevahkme tcmjrfms mhkjuhtmrme fh nraeuatm kjursms. Ft fs pjssfgdm tj start at ahy pjfht fh tcm djnfk ahe bjvm mftcmr kdjkiwfsm (sjdfe arrjws) jr
;?
7
Jhm-Efbmhsfjhad [trukturad Bmbgmrs
kjuhtmrkdjkiwfsm (cjddjw arrjws). Ft fs hjt pjssfgdm tj rmdatm efspdakmbmht efrmktdy tj mxtmrhad `jrkms wftcjut fbpjsfhn dfbftatfjhs ahe baifhn assubptfjhs rmnarefhn tcm nmjbmtry j` tcm gjey, batmrfad gmcavfjr, ahe mxtmrhad djaefhn. Ucfs fs mbpcasfzme gy tcm agsmhkm j` arrjws efrmktdy dfhifhn efspdakmbmhts ahe mxt mrhad `jrkms fh @fnurm 7.0. 7.0.
1 7 ; 5 = Efspdakmbmhts 0 > 3 I f h m 6 b a t f k 1? s 11 Mxtmrhad `jrkms 17 [trafhs ahe bjbmhts 1; 15 ds j e m M M 1= h m q 10 u a d B r n y f df f 1> B g r f t mr u 13 m t b a c j B 16 e s 7? 71 [tatfk Fhtmrhad `jrkms 77 [trmssms ahe bjbmhts 7; Mqufvadmhky 75 7= @fnurm 7.0 Djnfk fh strukturad ahadysfs. 70 Ucm djnfk scjwh fh @fnurm 7.0 fs 7.0 fs fhtrfhsfkaddy vmry bjeudar9 tcat fs, tcm mquatfjhs (ifhmbatfks, batmrfad bjemds, statfk 7> mqufvadmhky, ahe mqufdfgrfub) dfhifhn ahy twj `uheabmhtad varfagdms arm butuaddy fhempmhemht . Ucus, f` kjbpdmxfty fs aeeme fh 73 76 jhm part, tcmh tcm tcmjry kah gm rm-emrfvme wftc bfhfbub kcahnm. Ucfs wfdd gm embjhstratme bahy tfbms fh tcfs gjji tcrjunc ;? mxabpdms, fh pjst-tmxt prjgdmbs, ahe fh Kcaptmr = jh = jh tcfh pdatms. ]hemrstahefhn tcm bjeudar haturm j` tcfs djnfk ahe dmarhfhn ;1 cjw kjbpdmxftfms kah gm fhkjrpjratme tj rm`fhm gasfk tcmjrfms fs ah fbpjrtaht dmarhfhn jglmktfvm j` tcfs gjji. Fh Kcaptmr > wm > wm wfdd smm tcat mhmrny bmtcjes arm ah adtmrhatfvm tj Hmwtjhfah bmtcjes j` m qufdfgrfub mquatfjhs ahe kah ;7 ;; rmpdakm tcm dast dfhi fh tcm djnfk. Ucm bjeudar kcaraktmr j` tcm djnfk cjwmvmr rmbafhs uhkcahnme ahe kjbpdmxftfms kah as masfdy ;5 fhkjrpjratme wftc mhmrny bmtcjes as wftc mqufdfgrfub mquatfjhs. ;= ;0 7.; KDA[[FKAD UCMJWFM[ J@ 1-E [UW]KU]WAD BMBGMW[ ;> ;3 Ucfs smktfjh prmsmhts tcm kdassfkad tcmjry j` axfad bmbgmrs, sybbmtrfk gmhefhn j` gmabs, ahe tjrsfjh j` kfrkudar sca`ts fh tagudar ;6 `jrb tj mbpcasfzm tcm kjbbjhadftfms tcm kjbbjhadftfms ahe ef``mrmhkms fh tcm emrfvatfjh j` varfjus `jrbudas. ^m wfdd start gy rmvfmwfhn tcm sfb5? pdmst pjssfgdm tcmjrfms, tcat fs, tcm kdassfkad tcmjrfms efskussme fh tcm fhtrjeukt jry kjursm jh tcm bmkcahfks j` batmrfads. Ucm dfb51 ftatfjhs ahe assubptfjhs tcat dmae tj sfbpdf`fkatfjhs wfdd gm femhtf`fme wftc tcm fhtmht j` erjppfhn tcmb tj fhkjrpjratm 57 kjbpdmxftfms as efskussme fh [mktfjh 7.;.;. 7.;.;. 5; Uagdm 7.1 fs 7.1 fs a syhjpsfs j` tcm emrfvatfjh tcm mdmbmhtary tcmjrfms. Assubptfjhs ahe mquatfjhs wftc su``fxms -A, -G, ahe 55 U rm`mr tj axfad, gmhefhn, ahe tj rsfjh, rmspmktfvmdy. Ah assubptfjh wftcjut ahy su``fx fs appdfkagdm tj add tcrmm tcmjrfms. Ucfs hjta5= tfjh fs usme tj `akfdftatm tcm embjhstratfjh tcat kjbpdmxftfms mhtmr tcm emrfvatfjh at tcm sabm dmvmd ahe arm akkjuhtme `jr fh hmardy 50 a sfbfdar bahhmr `jr add tcrmm strukturad bmbgmrs. 5> 53 7.;.1 Dfbftatfjhs 56 Ucm `jddjwfhn dfbftatfjhs arm kjbbjh tj tcmjrfms agjut axfad bmbgmrs, tjrsfjh j` kfrkudar sca`ts, ahe sybbmtrfk gmhefhn j` gmabs. =? 1. Ucm dmhntc j` tcm bmbgmr fs sfnhf`fkahtdy nrmatmr nrmatmr (apprjxfbatmdy 1? tfbms) tfbms) tcah tcm nrmatmst nrmatmst efbmhsfjh fh tcm krjss krjss =1 =7 smktfjh. Apprjxfbatfjh akrjss tcm krjss smktfjh fs hjw pjssfgdm gmkausm tcm rmnfjh j` apprjxfbatfjh fs sbadd. 7. ^m arm away `rjb rmnfjhs j` strmss kjhkmhtratfjh, wcmrm efspdakmbmhts ahe strmssms kah gm tcrmm-efbmhsfjhad. =; Ucm rmsudts `rjb tcm sfbpdf`fme tcmjrfms kah gm mxtrapjdatme fhtj rmnfjhs j` strmss kjhkmhtratfjh usfhn strmss =5 kjhkmhtratfjh `aktjr. Fh a sfbfdar bahhmr, rmsudts kah gm mxtrapjdatme fhtj rmnfjhs hmar krakis gy bmahs j` strmss == =0 fhtmhsfty `aktjrs. => ;. Ucm varfatfjh j` mxtmrhad djaes jr kcahnms fh tcm krjss-smktfjhad arma fs nraeuad mxkmpt fh rmnfjhs j` strmss kjh=3 kmhtratfjh. Ucm tcmjry j` mdastfkfty scjws tcat tcfs dfbftatfjh fs hmkmssary9 jtcmrwfsm apprjxfbatfjhs akrjss tcm =6 krjss smktfjh wjude gm uhtmhagdm. 0? 01
Kdassfkad tcmjrfms j` 1-E strukturad bmbgmrs 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
;1
5. Ucm mxtmrhad djaes arm sukc tcat tcm axfad, tjrsfjh, ahe gmhefhn prjgdmbs kah gm stuefme fhefvfeuaddy. fhefvfeuaddy. Ucfs rmqufrms hjt jhdy tcat tcm appdfme djaes gm fh a nfvmh efrmktfjh, gut adsj tcat tcm djaes pass tcrjunc spmkf`fk pjfhts assjkfatme wftc tcm krjss smktfjh as efskussme fh [mktfjh 7.1. 7.1. @jr tjrsfjh ahe gmhefhn, wm fbpjsm tcm `jddjwfhn aeeftfjhad dfbftatfjhs. • Ujrsfjh< ^m dfbft jursmdvms tj kfrkudar krjss smktfjhs. Ucfs pmrbfts us tj usm arnubmhts j` axfsybbmtry fh emeukfhn em`jrbatfjh. Ucfs dfbftatfjh fs erjppme fh [mktfjh 0.> tj 0.> tj emvmdjp tcmjry j` tjrsfjh j` hjh-kfrkudar sca`ts. • Gmhefhn< ^m dfbft jursmdvms tj gmab krjss smktfjhs tcat cavm a pdahm j` sybbmtry tj wcfkc tcm djaefhn fs rmstrfktme. Ucfs dfbftatfjh mhsurms sybbmtrfk gmhefhn. Ft fs assubme tcat tcm x tcm x-- y pdahm y pdahm fs tcm pdahm j` sybbmtry ahe tcat gmhefhn fs jkkurrfhn agjut tcm z tcm z axfs axfs (f.m., krjss smktfjhs rjtatm agjut tcm z tcm z axfs). axfs).
7.;.7
Em`fhftfjh j` varfagdms ahe sfnh kjhvmhtfjhs
@fnurm 7.> scjws 7.> scjws a sdmhemr bmbgmr wftc tcm x tcm x axfs axfs adjhn tcm axfad efrmktfjh ahe y ahe y ahe ahe z z fh fh tcm pdahm j` tcm krjss smktfjh. ^m wfdd usm tcfs kjjrefhatm systmb `jr axfad ahe gmhefhn prjgdmbs. @jr tjrsfjh prjgdmbs wm wfdd usm a pjdar kjjrefhatm systmb (, , x). x). Ucm `fnurm fs sybgjdfk ahe tcm tcmjrfms `jr axfad ahe gmhefhn bmbgmrs arm hjt arm hjt dfbftme dfbftme tj kfrkudar krjss smktfjhs. ( () y (v) y (v)
z (w (w)
x
( u )
@fnurm 7.> Kjjrefhatm systmbs ahe em`jrbatfjh varfagdms.
Uagdm 7.1 fs 7.1 fs a syhjpsfs j` tcm emrfvatfjh j` tcm mdmbmhtary kdassfkad tcmjrfms stuefme fh tcm fhtrjeuktjry kjursm jh tcm bmkcahfks j` batmrfads. [mm Pagdm Pagdm Y7?1;R `jr aeeftfjhad emtafds jh tcm emvmdjpbmht j` tcmsm tcmjrfms. A ssubptfjhs arm baem wftc tcm bjtfvatfjh tj emvmdjp tcm sfbpdmst tcm sfbpdmst pjssfgdm tcmjrfms. tcmjrfms. Ucm varfagdms usme fh tcm tagdm arm em`fhme as `jddjws.
1. Ucm efspd efspdakmb akmbmhts mhts u, v, ahe w wfdd gm kjhsfemrme tj gm pjsftfvm fh tcm pjsftfvm x pjsftfvm x,, y, y, ahe z ahe z efrmktfjh, efrmktfjh, rmspmktfvmdy. 7. Ucm rjtatf rjtatfjh jh j` tcm krjss smktfjh wfdd gm kjhsfemrme tj gm pjsftfvm kjuhtmr kdjkiwfsm wftc rmspmkt tj tcm x tcm x axfs. axfs. ;. Ucm mxtmrhad mxtmrhad efstrfgutme efstrfgutme tjrqum tjrqum pmr uhft dmhntc dmhntc t ( x) x) fs pjsftfvm kjuhtmrkdjkiwfsm wftc rmspmkt tj tcm x tcm x axfs. axfs. 5. Ucm mxtmrhad mxtmrhad efstrfgutme efstrfgutme `jrkms `jrkms pmr uhft dmhntc dmhntc p p x( x) x) ahe p ahe p y( x) x) arm kjhsfemrme tj gm pjsftfvm fh tcm pjsftfvm x pjsftfvm x ahe ahe y efrmktfjh, y efrmktfjh, r mspmktfvmdy. =. M fs fs tcm bjeudus j` mdastfkfty ahe N fs scmar bjeudus j` mdastfkfty. 0. A fs tcm krjss smktfjhad arma9 F arma9 F agjut z 9 L 2 pjdar bjbmht j` tcm arma agjut x agjut x.. zz 2 smkjhe arma bjbmht j` fhmrtfa agjut z >. MA 2 MA 2 axfad rfnfefty9 MF rfnfefty9 MF zz 2 gmhefhn rfnfefty9 NL 2 tjrsfjhad rfnfefty. 3. Jh a `rmm gjey efanrab, efanrab, tcm pjsftfvm pjsftfvm fhtmrhad fhtmrhad axfad `jrkm `jrkm H fs fs scjwh as tmhsfdm. 6. Jh a `rmm gjey efanrab, efanrab, tcm pjsftfvm pjsftfvm fhtmrhad fhtmrhad scmar `jrkm `jrkm P y fs scjwh fh tcm sabm efrmktfjh as pjsftfvm xy. 1?.Jh a `rmm gjey efanrab, tcm pjsftfvm fhtmrhad gmhefhn bjbmht B pjsftfvm y `akm `akm fh kjbprmssfjh. z fs scjwh tj put tcm pjsftfvm y 11.Jh a `rmm gjey efanrab, pjsftfvm fhtmrhad tjrqum U fs fs scjwh kjuhtmrkdjkiwfsm wftc rmspmkt tj t cm jutware hjrbad. F` tcm sfnh kjhvmhtfjhs `jr fhtmrhad `jrkms ahe bjbmhts arm `jddjwme fh erawfhn `rmm gjey efanrabs, tcm `jrbudas fh Uagdm 7.1 wfdd 7.1 wfdd nfvm tcm rfnct sfnh `jr efspdakmbmhts ahe strmss kjbpjhmhts. F` tcm fhtmrhad `jrkms ahe bjbmhts arm erawh tj mqufdfgratm tcm mxtmrhad `jrkms ahe bjbmhts, tcm `jrbudas fh Uagdm 7.1 scjude 7.1 scjude gm usme jhdy tj emtmrbfhm tcm banhftuem j` tcm efspdakmbmht ahe strmss kjbpjhmhts9 tcm efrmktfjh j` tcmsm kjbpjhmhts bust gm ejhm gy fhspmktfjh.
7.;.;
Kjbpdmxfty bap
Uwj jgsmrvatfjhs rmnarefhn strmss `jrbudas ahe ef``mrmhtfad mquatfjh jh em`jrbatfjhs arm rmkjreme gmdjw gm`jrm kjhsfemrfhn tcm varfjus kjbpdmxftfms fh mxabpdms, prjgdmbs, ahe datmr kcaptmrs. • Ucm `jrbudas `jrbudas `jr `jr strmssms strmssms nfvmh nfvmh gy Mquatfjhs (7.1;-A), (7.1;-A), (7.1;-G), (7.1;-G), ahe (7.1;-U) jr (7.1;-U) jr tcmfr mqufvadmht `jrb arm jgtafhme gy sugstftutfhn Mquatfjhs (7.17-A), (7.17-A), (7.17-G), (7.17-G), ahe (7.17-U) jr (7.17-U) jr tcmfr mqufvadmht `jrbs fhtj Mquatfjhs (7.>A), A), (7.>-G), (7.>-G), ahe (7.>-U) jr (7.>-U) jr tcmfr mqufvadmht `jrbs, frrmspmktfvm j` tcm kjbpdmxfty fhkdueme.
;7
7
Jhm-Efbmhsfjhad [trukturad Bmbgmrs
Uagdm 7.1 [yhjpsfs j` Mdmbmhtary Ucmjrfms j` Jhm-Efbmhsfjhad [trukturad Bmbgmrs Axfad
Gmhefhn
Ujrsfjh
Em`jrbatfjhs Assubptfjh 1 Assubptfjh 7
Em`jrbatfjhs arm hjt `uhktfjhs j` tfbm. 7-A< _dahm smktfjhs rmbafh pdahm ahe 7a-G< [quascfhn em`jrbatfjh fs sfnhf`fkahtdy paraddmd. sbaddmr tcah em`jrbatfjh eum tj gmhefhn. 7g-G< _dahm smktfjhs gm`jrm em`jrbatfjh rmbafh pdahm a`tmr em`jrbatfjh. 7k-G< A pdahm pmrpmhefkudar tj tcm gmab axfs rmbafhs hmardy pmrpmhefkudar hmardy pmrpmhefkudar a`tmr em`jrbatfjh. u 2 u? x
(7.=-A)
e v u 2 ‛ y -----ex
v 2 vx
7a-U< _dahm smktfjhs pmrpmhefkudar tj tcm axfs rmbafh pdahm eurfhn em`jrbatfjh. 7g-U< Add raefad dfhms rjtatm gy mquad ahndm eurfhn em`jrbatfjh jh a krjss smktfjh. 7k-U< Waefad dfhms rmbafh strafnct eurfhn em`jrbatfjh.
(7.=a-G)
2
x
(7.=-U)
(7.0-G)
x
2
e ------ x
(7.0-U)
[trafhs Assubptfjh ;
Ucm strafhs arm sbadd. 7
eu ? 2 -------- x ex
xx
(7.0-A)
e v 2 ‛ y -------- x 7 ex
xx
ex
[trmssms Assubptfjh Assubptfjh Assubptfjh Assubptfjh
5 = 0 >
Ucm batmrfad fs fsjtrjpfk. Ucmrm arm hj tcmrbad jr jtcmr hjh-bmkcahfkad strafhs. Ucm batmrfad fs mdastfk. [trmss ahe strafh arm dfhmardy rmdatme.
]sfhn Cjjim‖s daw
eu ? 2 M -------- x ex
xx
7
(7.>-A)
e v 2 ‛ My -------- x 7 ex
xx
x
(7.>-G)
e 2 N ------ x ex
(7.>-U)
Fhtmrhad @jrkms ahe Bjbmhts [tatfk mqufvadmhky
H 2
A
y 2 ‛ z
B z 2 ‛ B y
xx e A
A
A
(7.3a-A)
H 2
A
xx e A
2 ?
xx e A
2 ?
(7.3g-A)
xx e A
2
2 ?
U 2
(7.3a-G)
A
B z 2 ‛ y xx e A
x e A
(7.3-U)
(7.3g-G)
A
(7.3k-A)
P y
A
xy e A
[fnh kjhvmhtfjh
(7.3k-G)
Kjbprmssfvm pjsftfvm y pjsftfvm y `akm `akm
xx
xx
xy
+U
x
+ H +P y
+ B z 7
7
[ugs [ugstft tftut utfh fhn n strm strmss ssms ms fhtj fhtj fhtmr fhtmrha hadd `jrk `jrkms ms ahe ahe bjbm bjbmht htss ahe ahe hjtfh hjtfhn n eu ? ex , e v ex , ahe ahe e ex arm arm `uhk `uhktfj tfjhs hs j` x j` x jhdy, jhdy, wcfdm tcm fhtmnratfjh fs wftc rmspmkt tj y tj y ahe ahe z z Jrfnfh djkatfjh
yM e A 2 ?
(7.6-A)
A
eu ? H 2 -------ex Assubptfjh 3 Ucm jrfnfh fs at tcm kmhtrjfe j` tcm krjss smktfjh.
yM e A 2 ?
(7.6-G)
A 7
M e A
(7.1?-A)
A
e v B z 2 -------7 ex
7
My e A
(7.1?-G)
A
e U 2 -----ex
7
N e A
(7.1?-U)
A
Ucm batmrfad fs cjbjnmhmjus akrjss tcm krjss smktfjh.
y e A 2 ?
(7.11-A)
eu H --------? 2 ------ex MA
(7.17-A)
A
A 2
A
y e A 2 ?
(7.11-G)
7 B e v -------- 2 ------- z --7 MF zz ex
(7.17-G)
A
e A
F zz 2
A
7
y e A
e U ------ 2 ------ex NL L 2
A
(7.17-U)
7 e A
[trmss @jrbudas
[ugstftutfhn Mquatfjhs (7.17-A), (7.17-A), (7.17-G), (7.17-G), ahe (7.17-U) fhtj (7.17-U) fhtj Mquatfjhs (7.>-A), (7.>-A), (7.>-G), (7.>-G), ahe (7.>-U)
xx
H 2 --- A
(7.1;-A)
xx
B z y 2 ‛ ---------- F zz
(7.1;-G)
[mm [mktfjh ;.7.; `jr scmar strmssms fh gmhefhn.
x
U 2 ------ L
(7.1;-U)
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
Kdassfkad tcmjrfms j` 1-E strukturad bmbgmrs 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
;;
Bmbgmrs Uagdm 7.1 [yhjpsfs j` Mdmbmhtary Ucmjrfms j` Jhm-Efbmhsfjhad [trukturad Bmbgmrs Axfad
Gmhefhn
Ujrsfjh
Em`jrbatfjh @jrbudas Assubptfjh 6 Assubptfjh 1? Assubptfjh 11
Ucm batmrfad fs cjbjnmhmjus gmtwmmh x gmtwmmh x1 ahe x ahe x7. Ucm strukturad bmbgmr fs hjt tapmrme gmtwmmh x gmtwmmh x1 ahe x ahe x7 Ucm mxtmrhad djaes ej hjt kcahnm wftc x wftc x gmtwmmh gmtwmmh x x1 ahe x ahe x7. Fhtmnratfhn Mquatfjhs (7.17-A) ahe (7.17-A) ahe (7.17-U) (7.17-U) [mm [mktfjh ;.5.; `jr gmab em`dmktfjh. H x 7 ‛ x 1 u 7 ‛ u 1 2 ------------------------(7.15-A) MA
7 ‛ 1
U x 7 ‛ x 1 2 ------------------------NL
(7.15-U)
Mqufdfgrfub Mquatfjhs H
p x( x) x) x
U
H + H
x t ( x) x) x B z + B z
B z
x
P y
p y( x) x) x
P y + P y
x U+U
e H ------- 2 ‛ p x x ex
(7.1=-A)
e P P y --------2 ‛ p y x ex
(7.1=a-G)
e B z ---------2 ‛ P y ex
(7.1=g-G)
e U U ------- 2 ‛ t x ex
(7.1=-U)
Ef``mrmhtfad Mquatfjhs
[ugstftutfhn Mquatfjhs (7.17-A), (7.17-A), (7.17-G), (7.17-G) , ahe (7.17-U) fhtj (7.17-U) fhtj Mquatfjhs (7.1=-A), (7.1=-A), (7.1=a-G), (7.1=a-G) , (7.1=g-G), (7.1=g-G), ahe (7.1=-U) eu e ------ MA --------? 2 ‛ p x x ex ex
(7.10-A)
e e v -------- MF zz -------- 2 p y x 7 7 ex ex 7
7
(7.10-G)
e e ------ NL ------ 2 ‛ t x ex ex
(7.10-U)
• Ucm ef``mrmhtf ef``mrmhtfad ad mquatfjhs mquatfjhs nfvmh nfvmh gy Mquatfjhs (7.10-A), (7.10-A), (7.10-G), (7.10-G), ahe (7.10-U) jr (7.10-U) jr tcmfr mqufvadmht `jrb arm jgtafhme `rjb gy sugstftutfhn Mquatfjhs (7.17-A), (7.17-A), (7.17-G), (7.17-G), ahe (7.17-U) jr (7.17-U) jr tcmfr mqufvadmht `jrb fhtj Mquatfjhs (7.1=-A), (7.1=-A), (7.1=a-G), (7.1=a-G), (7.1=g-G), (7.1=g-G), ahe (7.1=-U) jr (7.1=-U) jr tcmfr mqufvadmht `jrbs, frrmspmktfvm j` tcm kjbpdmxfty fhkdueme. 1. Erjp Erjppf pfhn hn j` Assubptfjh 1 cas 1 cas twj fbpakts. @frst, tfbm gmkjbms ah fhempmhemht varfagdm dfim x dfim x,, cmhkm add jrefhary emrfvatfvms gmkjbm partfad emrfvatfvms j` x j` x.. [mkjhe, tcm mqufdfgrfub mquatfjhs fh wcfkc `jrkms ahe bjbmhts wmrm mquatme tj zmrj arm hjw mquatme tj tcm fhmrtfad `jrkms. Fh Mxabpdm 7.0 wm 7.0 wm kjhsfemr vfgratfjh j` gmabs. Fh prjgdmb Fh prjgdmb 7.71 wm 7.71 wm kjhsfemr tcm wavm mquatfjh fh axfad rjes. Fh prjgdmb Fh prjgdmb 7.75 wm 7.75 wm kjhsfemr vfgratfjh j` kfrkudar sca`ts. 7. Erjp Erjppf pfhn hn j` j` ahy ahy part part j` j` Assubptfjh 7 rmsudts 7 rmsudts fh kcahnfhn tcm `jrb j` em`jrbatfjh mquatfjhs. @jrb j` em`jrbatfjh mquatfjhs bay adsj kcahnm f` sjbm j` tcm dfbftatfjhs arm erjppme. Ucm `djw j` emvmdjpbmht j` tcm hmw mquatfjhs rmbafhs tcm sabm. Fh Mxabpdm 7.0, 7.0, Assubptfjh 7k-G 7k-G fs erjppme ahe scmar fs fhkjrpjratme fhtj gmab tcmjry. Fh prjgdmb Fh prjgdmb 7.>, 7.> , tcm axfad ahe gmhefhn em`jrbatfjh arm kjhsfemrme tjnmtcmr. Fh _rjgdmbs 7.3, 7.3, 7.6, 7.6, ahe 7.1? tcm 7.1? tcm dfbftatfjh jh gmab gmhefhn tcat djaefhn gm fh pdahm j` sybbmtry fs erjppme ahe mquatfjhs `jr uhsybbmtrfk gmhefhn arm emvmdjpme. ;. Erjp Erjppf pfhn hn j` Assubptfjh ; ; fhkjrpjratms nmjbmtrfk hjhdfhmarfty fhtj tcm tcmjry. _rjgdmb 7.70 7.70 kjhsfemrs hjh-dfhmarfty fh kjhtmxt j` axfad rjes. Mxabpdm >.= kjhsfemrs >.= kjhsfemrs darnm strafh fh sybbmtrfk gmab gmhefhn. 5. Assubptfjh = fs = fs erjppme fh prjgdmb fh prjgdmb 7.1; tj 7.1; tj fhkjrpjratm m``mkt j` tmbpmraturm jh gmab gmhefhn. =. Assubptfjh 0 fs 0 fs erjppme fh prjgdmbs 7.1> ahe 7.1> ahe 7.13 tj 7.13 tj fhkjrpjratm mdastfk-pmr`mktdy pdastfk batmrfad bjemd fh tjrsfjh j` kfrkudar sca`ts ahe sybbmtrfk gmhefhn j` gmabs. 0. Assubptfjh > jh > jh batmrfad dfhmarfty fs erjppme ahe `jrbuda `jr strmss fs emvmdjpme fh Mxabpdm 7.= ahe 7.= ahe fh prjgdmb fh prjgdmb 7.76. 7.76. Fh prjgdmb 7.7> strmss 7.7> strmss ahe em`jrbatfjh `jrbudas arm emvmdjpme `jr tjrsfjh j` kfrkudar sca`ts. Fh Mxabpdm >.0 pjwmr >.0 pjwmr daw fs fhkjrpjratme fhtj sybbmtrfk gmab gmhefhn tj emvmdjp gjuheary vadum prjgdmb statmbmht. >. Assubptfjh 3 jh 3 jh batmrfad jh batmrfad cjbjnmhmfty akrjss tcm krjss smktfjh fs erjppme fh Mxabpdm 7.5 `jr 7.5 `jr gmabs. @jrbudas arm emvmdjpme `jr emtmrbfhfhn hmutrad axfs, gmhefhn hjrbad strmss, ahe tcm ef``mrmhtfad mquatfjh. Fh Mxabpdm 7.> tcm 7.> tcm `jrbudas `jr scmar strmss arm emvmdjpme. Fh prjgdmbs 7.11 ahe 7.11 ahe 7.17 strmss 7.17 strmss `jrbudas ahe ef``mrmhtfagdm mquatfjhs arm emvmdjpme `jr dabfhatme axfad rjes ahe tjrsfjh j` dabfhatme kfrkudar sca`ts. 3. Assubptfjh 1? 1? j` uhtapmrme bmbgmr fs erjppme fh prjgdmb fh prjgdmb 7.; 7.; `jr axfad rje ahe fh prjgdmb fh prjgdmb 7.5 7.5 `jr tjrsfjh j` kfrkudar sca`ts. 6. Assubptfjh 11 fs 11 fs erjppme fh prjgdmb fh prjgdmb 7.;; tj 7.;; tj fhkjrpjratm efstrfgutme axfad `jrkm ahe fh prjgdmb fh prjgdmb 7.;5 tj 7.;5 tj fhkjrpjratm efstrfgutme tjrqum.
;5
7
Jhm-Efbmhsfjhad [trukturad Bmbgmrs
Fh prjgdmb Fh prjgdmb 7.16 tcm 7.16 tcm ef``mrmhtfad mquatfjh `jr a gmab jh mdastfk `juheatfjh fs kjhsfemrme. [jdutfjhs `jr gmabs jh mdastfk `juheatfjhs arm efskussme fh emtafds fh Kcaptmr ;. ;. Bjrm tcah jhm kjbpdmxfty bay gm aeeme sfbudtahmjusdy as embjhstratme fh Mxabpdm 7.0 fh 7.0 fh wcfkc vfgratfjh j` Ufbjscmhij gmabs fs kjhsfemrme. Fh prjgdmbs 7.15 tcrjunc 7.15 tcrjunc 7.10 tmbpmraturm 7.10 tmbpmraturm m``mkts fh dabfhatme bmbgmrs fs kjhsfemrme. Fh prjgdmbs 7.;1 ahe 7.;1 ahe 7.;7 twj 7.;7 twj hjh-dfhmar batmrfads baim up tcm strukturad bmbgmrs. Fh a sfbfdar bahhmr wm kjude krmatm vfgratfjhs j` dabfhatme axfad rjes, gmabs, ahe sca`ts. Kjbgfhatfjhs j` kjbpdmxftfms fs vmry darnm, gut cjpm`uddy tcm rmaemr gy mhe j` tcfs kcaptmr wfdd emvmdjp a smhsm j` cjw tcmsm kjbpdmxftfms kah gm aeeme tj kdassfkad tcmjrfms j` axfad rjes, gmhefhn j` sybbmtrfk gmabs, ahe tjrsfjh j` kfrkudar sca`ts. Ah fbpjrtaht kcmki jh tcmsm kjbpdmx tcmjrfms fs tcat wm scjude gm ag dm tj rmprjeukm rmsudts rmsudts j` kdassfkad tcmjrfms gy apprjprfatm sugstftutfjh j` varfagdms. varfagdms. Ucfs kcmki wfdd gm pmr`jrbme fh tcm mxabpdms prmsmhtme hmxt.
1 7 ; 5 = 0 > 3 6 1? 11 17 MSAB_DM 7.5 1; @fnurm 7.3 scjws 7.3 scjws a dabfhatme kjbpjsftm krjss smktfjh. ^m assubm tcat add batmrfad krjss smktfjhs arm sybbmtrfk agjut tcm y tcm y 15 axfs ahe tcat tcm djaefhn fh stfdd fh tcm pdahm j` sybbmtry, tcus mhsurfhn sybbmtrfk gmhefhn agjut tcm z tcm z axfs. axfs. Add assubptfjhs 1= mxkmpt Assubptfjh 3 fh 3 fh Uagdm 7.1 arm 7.1 arm vadfe. (a) Assubfhn hj axfa d `jrkm, emtmrbfhm tcm djkatfjh j` tcm jrfnfh, tcat fs djkatfjh j` 10 hmutrad axfs.(g) Jgtafh a `jrbuda rmdatfhn hjrbad strmss xx ahe fhtmrhad gmhefhn bjbmht B z . (k) Jgtafh tcm `jurtc jremr ef``mr1> mhtfad mquatfjh njvmrhfhn tcm em`dmktfjh v( x) x) j` tcm gmab. (e) [cjw tcat tcm rmsudts `jr parts a, g ahe k nfvm tcm sabm rmsudts as 13 kdassfkad gmab tcmjry `jr cjbjnmhjus batmrfad. 16 y 7? 71 M h 77 7; 75 M f 7= z 70 7> 73 76 M 1 ;? ;1 7.5. @fnurm 7.3 Dabfhatme krjss smktfjh fh Mxabpdm 7.5. ;7 kf M A 7 eA f f M y e7 v h A f e f1 f B z -------AH[. k ------------------------------9 wcmrm M l F zz l 2 p y ;; kf 2 ------------------ 9 xx f 2 ‛ -----------------------------h h 7 7 l 1 A f ;5 M A M l F zz l e x e x f f f 1 l 1 ;= ;0 ;> MSAB_DM 7.= ;3 h Ucm strmss‛strafh kurvm `jr a batmrfad fs nfvmh gy 2 M . Assubm tcm batmrfad gmcavfjr fs sabm fh tmhsfjh ahe kjbprmssfjh. ;6 Add assubptfjhs mxkmpt Assubptfjh > fh > fh Uagdm 7.1 arm 7.1 arm vadfe. (a) @jr tcm rmktahnudar krjss smktfjh scjwh fh @fnurm 7.6, 7.6, jgtafh ah 5? mquatfjh rmdatfhn gmhefhn hjrbad strmss xx ahe fhtmrhad gmhefhn bjbmht B z . (g) Jgtafh tcm ef``mrmhtfad mquatfjh njvmrhfhn tcm 51 57 em`dmktfjh v( x) x) j` tcm gmab (k) [cjw tcat `jr h 2 1 wm jgtafh tcm rmsudt j` kdassfkad gmab tcmjry. 5; y 55 5= 50 z 5> c. 53 56 =? =1 g =7 7.=. @fnurm 7.6 Nmjbmtry fh Mxabpdm 7.=. =; h+1 =5 7 ‛ y h --------------y? == h + 7 7 c B z h+7 7 h 7 gc c 7 e v e 7 gM ----------------------- --------7 2 p y x =0 AH[. xx 2 9 7 e x h+1 h h+7 => e x 7 y ----------y? ‛ h + 7 7 c B z =3 gc =6 0? 01
Kdassfkad tcmjrfms j` 1-E strukturad bmbgmrs
;=
1 MSAB_DM 7.0 7 Fh Ufbjscmhij gmabsa tcm assubptfjh j` pdahms rmbafhfhn pmrpmhefkudar tj tcm axfs j` tcm gmab fs erjppme tj akkjuht `jr ; scmar gy pmrbfttfhn tcm krjss smktfjh tj rjtatm gy ah ahndm `rjb tcm vmrtfkad sybbmtrfk axfs y. y. (a) Jgtafh tcm ef``mrmhtfad 5 mquatfjhs jh em`dmktfjh v( x, x , t ) ahe ahndm x t `jr vfgratfjh j` sybbmtrfk Ufbjscmhij gmab gy startfhn wftc tcm efspdakm = bmht `fmde nfvmh gmdjw. ]sm tcm `jddjwfhn varfagdms< bass emhsfty b9 scmar bjeudus N9 bjeudus j` mdastfkfty M mdastfkfty M 9 efstrfgutme 0 djae y x t 9 krjss smktfjhad arma A9 ahe smkjhe arma bjbmht j` fhmrtfa F fhmrtfa F zz . (g) [cjw tcat `jr e v/ v/ ex ahe wftc hj eyhabfk > m``mkts wm jgtafh tcm rmsudt j` kdassfkad gmab tcmjry. 3 u 2 ‛ y x t v 2 v x t 6 7 7 1? v v v AH[. NA ‛ x + p y x t 2 b A 7 MF zz + NA ‛ x 2 b F zz 7 11 x x x x x t t 17 1; a. [mm @uhn @uhn Y160=R Y160=R jr Wmeey Wmeey Y166;R. Y166;R. 15 7.;.5 [cmar strmss fh tcfh sybbmtrfk gmabs 1= 10 Assubptfjh 7k-G fbpdfms tcat tcm scmar strafh xy bust gm sj sbadd tcat wm kah hmndmkt ft fh ifhmbatfk mquatfjhs. Gy Cjjim‖s daw 1> tcfs fbpdfms tcat xy scjude gm sbadd. Fh gmab gmhefhn, a kcmki jh tcm vadfefty j` tcm ahadysfs fs tj kjbparm tcm baxfbub scmar 13 strmss xy wftc tcm baxfbub hjrbad strmss xx `jr tcm mhtfrm gmab. mhtfrm gmab. F` tcm twj strmss kjbpjhmhts arm kjbparagdm, tcmh tcm scmar 16 strafh kahhjt gm hmndmktme fh ifhmbatfk kjhsfemratfjhs ahe wm scjude usm Ufbjscmhij gmab tcmjry j` Mxabpdm 7.0. 7.0. Ucm baxf7? bub gmhefhn hjrbad strmss xx fh tcm gmab scjude gm hmardy ah jremr j` banhftuem nrmatmr tcah tcm baxfbub gmhefhn scmar 71 strmss xy as scjwh gmdjw. 77 (7.1>-G) xx ba x ¹ xy ba x 7; Kjhsfemr a kfrkudar krjss smktfjh tcat fs ndume tjnmtcmr `rjb sfx wjjemh strfps as scjwh fh @fnurm 7.1?. 7.1?. Ucm gmhefhn 75 djae _ , kausms tcm bjbmht B (7.1;-G) wm ihjw tcat tcm banhftuem j` 7= djae _ z tj vary adjhn tcm dmhntc j` tcm gmab. Ucus `rjb Mquatfjh (7.1;-G) wm 70 tcm hjrbad strmss xx wfdd kcahnm adjhn tcm dmhntc j` tcm gmab. Ucm scmar strmss at tcm ndum ljfhts bust gadahkm tcm varfatfjh j` 7> hjrbad strmss xx adjhn tcm dmhntc j` tcm gmab. 73 76 t @rmm [ur`akm ;? @rmm [ur`akm ;1 H ;7 y ;; ;5 ;= @rmm [ur`akm P 2 xs(tex) tex) 1 ;0 ;> s ;3 z 7 ex H +e +e H ;6 1 0 5? 7 = ; 5 51 s 57 (a) (g) x 5; _ 55 @fnurm 7.1? [cmar strmss fh a kfrkudar krjss smktfjh. 5= Ucm jutware hjrbad j` tcm sur`akm wfdd gm fh a ef``mrmht efrmktfjh `jr makc ndum sur`akm jh wcfkc wm kjhsfemr tcm scmar 50 kjjrefhatm s fh tcm tahnmht efrmktfjh tj tcm kmhtmrdfhm j` tcm krjss smktfjh. Ucm jutware hjrbad tj 5> strmss. ^m em`fhm a tahnmhtfad kjjrefhatm s tcm s efrmktfjh efrmktfjh ahe tcm scmar strmss wfdd gm sx. Jhkm bjrm, gy sybbmtry j` scmar strmssms, xs 2 sx. At 53 tcm ndume sur`akm wfdd gm fh tcm s tcm s efrmktfjh efrmktfjh ahe tcm y tcm y efrmktfjh efrmktfjh arm tcm sabm, tcmh xs wfdd mquad pdus jr bfhus xy. tcm s efrmktfjh efrmktfjh ahe tcm z tcm z efrmktfjh efrmktfjh 56 a pjfht, f` tcm s xy. F` tcm s arm tcm sabm at a pjfht, tcmh wfdd mquad pdus jr bfhus . xs xz =? @fnurm 7.1?g 7.1?g scjws a ef``mrmhtfad mdmbmht j` tcm gmab. Ucm scmar `jrkm tcat gadahkms tcm kcahnm fh axfad `jrkm H s fs =1 =7 scjwh jh jhdy jhm sur`akm. Ucm sur`akm jh tcm jtcmr mhe j` tcm `rmm gjey efanrab fs assubme tj gm a `rmm sur`akm9 tcat fs, tcm =; scmar strmss fs zmrj jh tcmsm jtcmr sur`akms. Ucm efrmktfjh j` tcm s kjjrefhatm fs `rjb tcm `rmm sur`akm tjware tcm pjfht at wcfkc tcm =5 scmar strmss fs gmfhn kadkudatme. Fh a gmab krjss smktfjh, tcm tjp, gjttjb, ahe sfem sur`akms arm adways assubme tj gm sur`akms jh wcfkc tcm scmar strmss == smktfjh (f.m. t fs fs sbadd). Ucm assubp=0 fs zmrj. ^m `urtcmr assubm tcat tcm gmab fs tcfh pmrpmhefkudar tj tcm kmhtmrdfhm j` tcm krjss smktfjh (f.m. => tfjhs fbpdy =3 • Ucm scmar strmss hjrbad tj tcm kmhtmrdfhm fs sbadd gmkausm tcm sur`akms jh mftcmr sfem j` tcm kmhtmrdfhm arm `rmm sursur=6 `akms, ahe cmhkm kah gm hmndmktme. 0? • Ucm scmar scmar strmss strmss sx fs uhf`jrb fh tcm efrmktfjh hjrbad tj tcm kmhtmrdfhm. 01
;0
7
Jhm-Efbmhsfjhad [trukturad Bmbgmrs
Gy mqufdfgrfub j` `jrkms fh tcm x tcm x efrmktfjh efrmktfjh fh @fnurm 7.1?g 7.1?g, wm jgtafh ( H ex ?, jr H H s e s) H s sx t ex e H
e 2 ‛ -----e x e x
sx t 2 ‛ --------- s
xx e A
(7.13-G)
A s
Arma A s fs tcm arma jh tcm krjss smktfjh tcat fs gmtwmmh tcm `rmm sur`akm ahe tcm pjfht at wcfkc tcm scmar strmss fs gmfhn mvaduatme. [ugstftutfhn Mquatfjh (7.1;-G) fhtj (7.1;-G) fhtj Mquatfjh (7.13-G), (7.13-G), ahe hjtfhn tcat tcm bjbmht B z ahe tcm arma bjbmht j` fhmrtfa F zz ej hjt vary jvmr tcm krjss smktfjh, wm jgtafh e B z Q z 2 ------ ------------e x F zz
e B ------ ------ z - y e A e x F zz A s
sx t 2
(7.16-G)
wcmrm Q z fs rm`mrrme tj as tcm `frst bjbmht j` tcm arma A arma A s ahe fs em`fhme as Q z 2
y e A
(7.7?-G)
A s
^m assubm tcm assubm tcm gmab fs hjt tapmrme . Ucfs fbpdfms tcat F tcat F zz ahe Q z arm hjt `uhktfjhs j` x j` x,, ahe tcmsm quahtftfms kah gm taimh jutsfem tcm emrfvatfvm sfnh, wm jgtafh sx t 2 Q z F zz e B (7.1=g-G), tcm rmdatfjhscfp gmtwmmh scmar z e x . [ugstftutfhn Mquatfjh (7.1=g-G), `jrkm P y ahe bjbmht B z , wm jgtafh Q z P y sx t 2 ‛ ----------- F zz
sx
jr
2
xs
P y Q z 2 ‛ ------------ F zz t
(7.71-G)
@rjb Mquatfjh (7.7?-G), (7.7?-G), wm hjtm tcat Q z fs tcm `frst bjbmht j` tcm arma A arma A s agjut tcm z tcm z axfs. axfs. @fnurm 7.11 scjws 7.11 scjws tcm arma A s gmtwmmh tcm tjp `rmm sur`akm ahe tcm pjfht at wcfkc scmar strmss fs gmfhn `juhe (tcm [ ‛ [ [ dfhm). dfhm). Ucm fhtmnrad fh Mquatfjh (7.7?-G) fs tcm hubmratjr fh tcm em`fhftfjh j` tcm kmhtrjfe j` tcm arma A arma A s. Ahadjnjus tj tcm bjbmht eum tj a `jrkm, tcm `frst bjbmht j` ah arma kah gm `juhe gy pdakfhn tcm arma A arma A s at fts kmhtrjfe ahe `fhefhn tcm bjbmht agjut tcm hmutrad axfs. Ucat f s,Q s, Q z fs tcm prjeukt j` prjeukt j` A s
y
Dfhm adjhn wcfkc scmar strmss fs gmfhn `juhe. `juhe. [
Kmhtrjfe j` A j` A s y s Q z 2 A s y s [ Hmutrad Axfs Axfs
z
A s*
ys*
Q 2 A y z s s
Kmhtrjfe j` A j` A s*
@fnurm 7.11 Kadkudatfjh j` Q z .
arma A s ahe tcm efstahkm j` tcm kmhtrjfe j` tcm arma A s `rjb tcm hmutrad axfs, axfs, as scjwh fh @fnurm 7.11. 7.11. Adtmrhatfvmdy, Q z kah gm `juhe gy usfhn tcm gjttjb sur`akm as tcm `rmm sur`akm, scjwh as Q z * fh @fnurm 7.11. 7.11. Fh tcm fhtrjeuktjry kjursm jh tcm bmkcahfks j` batmrfads ft was scjwh tcat Q z fs baxfbub at tcm hmutrad axfs9 axfs9 cmhkm gmhefhn scmar strmss fs baxfbub at tcm hmutrad axfs j` a krjss smktfjh. ^m hjtm tcat sfhkm y sfhkm y fs fs bmasurme `rjb tcm kmhtrjfe, tcm `frst bjbmht j` tcm mhtfrm krjss-smktfjhad arma fs zmrj9 cmhkm Q z Q z ? . Ucm mquatfjh fbpdfms tcat Q z ahe Q z * wfdd cavm tcm sabm banhftuem gut jppjsftm sfnhs. Ucus, wm wfdd nmt tcm sabm banhftuem j` tcm scmar strmss wcmtcmr wm usm Q z jr Q z * fh Mquatfjh (7.71-G). (7.71-G). Gut wcfkc nfvms tcm kjrrmkt sfnh (efrmktfjh): Ucm ahswmr fs tcat gjtc wfdd nfvm tcm kjrrmkt sfnh, prjvfeme tcm `jddjwfhn pjfht fs rmbmbgmrme. Ucm s Ucm s efrmktfjh efrmktfjh fh Mquatfjh (7.71-G) fs (7.71-G) fs bmasurme `rjb tcm `rmm sur`akm usme fh tcm kadkudatfjh j` Q Q z.. Ucm prjeukt j` tcm scmar strmss ahe tcm tcfkihmss fs kaddme tcm scmar tcm scmar `djw, `djw, emsfnhatme gy tcm sybgjd q, as scjwh gmdjw. q 2
xs t
(7.77-G)
Ucm uhfts j` scmar `djw q arm `jrkm arm `jrkm pmr uhft dmhntc. dmhntc. Ucm tmrbfhjdjny fs `rjb `dufe `djw fh kcahhmds, gut tcm tmrb fs usme mxtmhsfvmdy `jr efskussfhn scmar strmssms fh tcfh krjss smktfjhs, prjgagdy gmkausm j` tcm vfsuad fbanm j` a `djw ft kjhvmys fh tcm efskussfjh j` scmar strmss efrmktfjh.
MSAB_DM 7.> (a) @jr tcm kjbpjsftm dabfhatme gmab fh Mxabpdm 7.5, 7.5, jgtafh a `jrbuda rmdatfhn gmhefhn scmar strmss xs ahe tcm fhtmrhad scmar `jrkm P y. Assubm tcm gmab fs hjt tapmrme. (g) [cjw tcat tcm rmsudts `jr part a nfvm tcm sabm rmsudts as kdassfkad gmab tcmjry `jr cjbjnmhjus batmrfad. Qkjbp P y AH[. sx 2 xs 2 ‛ ---------------------------------h t M l F zz l
2
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
Gjuheary vadum prjgdmbs 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
7.5
;>
GJ]HEAWV PAD]M _WJGDMB[
A gjuheary vadum prjgdmb fs a batcmbatfkad dfstfhn j` tcm ef``mrmhtfad mquatfjhs, tcm gjuheary kjheftfjhs, ahe ahy jtcmr kjheftfjhs hmkmssary tj sjdvm tcm ef``mrmhtfad mquatfjhs. Ucm ef``mrmhtfad mquatfjhs `jr tcm axfad, gmhefhn, ahe tjrsfjh kasms arm nfvmh gy Mquatfjhs (7.10-A), (7.10-A), (7.10-G), (7.10-G), ahe (7.10U), U), rmspmktfvmdy fh Uagdm 7.1. 7.1. Ucm efstrfgutme djaes p djaes p x( x), x), p p y( x), x), ahe t ( x) x) fh tcm ef``mrmhtfad mquatfjhs arm ihjwh `uhktfjhs. Ucm fhtmnratfjh j` tcmsm efstrfgutme djaes rmsudts fh tcm partfkudar sjdutfjh. Ucm sub j` tcm partfkudar sjdutfjh ahe cjbjnmhjus sjdutfjh rmprmsmhts tcm kjbpdmtm sjdutfjh j` tcm ef``mrmhtfad mquatfjh. Ucm cjbjnmhjus sjdutfjh kjrrmspjhes tj tcm rfnct cahe sfem j` tcm ef``mrmhtfad mquatfjh smt tj zmrj, tcat fs, wm smt p smt p x( x), x), p p y( x), x), ahe t ( x) x) tj zmrj fh tcm ef``mrmhtfad mquatfjh tj jgtafh tcm cjbjnmhjus sjdutfjh gy fhtmnratfjh. Ucm cjbjnmhjus sjdutfjh kjhtafhs uhihjwh kjhstahts tcat arm emtmrbfhme `rjb tcm gjuheary kjheftfjhs. Ucm gjuheary kjheftfjhs arm kjheftfjhs jh tcm fhtmrhad `jrkms/bjbmhts jr efspdakmbmhts/rjtatfjhs at tcm twj mhes (gjuheary) j` tcm bmbgmr. Uj uhemrstahe tcm prfhkfpdm j` cjw gjuheary kjheftfjhs arm emtmrbfhme, wm nmhmradfzm a prfhkfpdm efskussme fh statfks `jr emtmrbfhfhn tcm rmaktfjh `jrkm ahe/jr bjbmhts. Wmkadd tcat fh erawfhn `rmm gjey efanrabs, wm usme tcm `jddjwfhn prfhkfpdms `jr emtmrbfhfhn rmaktfjh `jrkms ahe bjbmht at tcm suppjrts.
1. F` a pjfht kahhjt bjvm fh a nfvmh efrmktfjh, tcmh tcmh a rmaktfjh `jrkm jppjsftm tj tj tcm efrmktfjh akts at tcm suppjrt pjfht. 7. F` a dfhm kahhjt rjtatm rjtatm agjut ah axfs fh a nfvmh efrmktfjh, tcmh a rmaktfjh bjbmht bjbmht jppjsftm tj tcm efrmktfjh efrmktfjh akts at tcm suppjrt. suppjrt.
F` wm wmrm tj baim ah f banfhary kut vmry kdjsm tj tcm suppjrt ahe tcmh eraw a `rmm gjey efanrab, wm wjude `fhe tcat tcm fhtmrhad `jrkm wfdd mquad tj tcm rmaktfjh `jrkm ahe tcm fhtmrhad bjbmht wfdd mquad tcm rmaktfjh bjbmht. Ucus, tcm `frst jgsmrvatfj h fbpdfms tcat f` a pjfht kahhjt bjvm (efspdakmbmht fs zmrj), tcmh tcm fhtmrhad `jrkm fs hjt ihjwh gmkausm tcm rmaktfjh `jrkm fs hjt ihjwh. [fbfdardy, tcm smkjhe jgsmrvatfjh fbpdfms tcat f` a dfhm kahhjt rjtatm (rjtatfjh fs zmrj) arjuhe ah axfs at tcm suppjrt, tcmh tcm fhtmrhad bjbmht fs hjt ihjwh gmkausm tcm rmaktfjh bjbmht fs hjt ihjwh. Ucm rmvmrsm fs mquaddy trum. F` strukturad bmbgmr‖s mhe fs `rmm, tcmh tcm fhtmrhad `jrkm ahe fhtmrhad bjbmht wfdd gm zmrj9 gut tcm `rmm mhe kah efspdakm ahe rjtatm gy ahy abjuht tcat fs efktatme gy tcm djaefhn. Ucus, wcmh wm spmkf`y a vadum `jr fhtmrhad `jrkm, tcmh wm kahhjt spmkf`y efspdakmbmht9 ahe wcmh wm spmkf`y tcm vadum `jr fhtmrhad bjbmht, tcmh wm kahhjt spmkf`y rjtatfjh.
7.5.1
Axfad efspdakmbmht
Ucm axfad efspdakmbmht u?( x) x) kah gm `juhe `rjb tcm ef``mrmhtfad mquatfjh (7.10-A) wcfkc (7.10-A) wcfkc wm rmwrftm gmdjw `jr kjhvmhfmhkm. • Ef``mrmhtfa Ef``mrmhtfadd mquatfjh< mquatfjh< eu ? e MA 2 ‛ p x x e x e x Fhtmnratfhn tcm ef``mrmhtfad mquatfjh twfkm wm jgtafh<
(7.7;-A)
eu ? 2 F 1 x + K 1 MA u? 2 F 7 x + K 1 x + K 7 wc m rm F 1 x 2 ‛ p x x e x F 7 x 2 F 1 x e x (7.75-A) e x wcmrm, tcm fhem`fhftm fhtmnrads kah gm `juhe as p as p x( x) x) fs ihjwh9 K 1 ahe K 7 arm kjhstahts tj gm emtmrbfhme `rjb tcm gjuheary kjhef F x MA tfjhs. Ucm partfkudar sjdutfjh fs 7 ahe tcm cjbjnmhjus sjdutfjh fs K 1 x + K 7 MA . Fh tcm agjvm fhtmnratfjh, wm assubme MA assubme MA fs fs a kjhstaht, gut f` ft fs hjt, tcmh ft kah gm masfdy akkjuhtme `jr gy efvfefhn Mquatfjh (7.75-A) gy (7.75-A) gy MA MA gm`jrm gm`jrm fhtmnratfhn. ^m hjtm tcat tcm fhtmrhad axfad `jrkm fs< H 2 MA e u ? e x . As efskussme mardfmr, ft fs tcm haturm j` gjuheary kjheftfjhs tcat at makc mhe (gjuheary) j` tcm axfad bmbgmr, mftcmr tcm axfad efspdakmbmhtu efspdakmbmht u? jr tcm fhtmrhad axfad `jrkm H bust bust gm spmkf`fme as scjwh gmdjw.
MA
• Gjuhea Gjuheary ry kjheftfjh kjheftfjhs< s< At makc mhe spmkf`y u?
jr
H
(7.7=-A) p x x A
H A
@ mxt
@fnurm 7.17 Gjuheary kjheftfjh jh tcm fhtmrhad axfad `jrkm.
Uj emtmrbfhm tcm gjuheary kjheftfjh jh tcm fhtmrhad axfad `jrkm H , wm baim ah fbanfhary kut ah fh`fhftmsfbad efstahkm () `rjb tcm mhe (say pjfht A pjfht A)) ahe eraw a `rmm gjey efanrab j` tcm fh`fhftmsfbad mdmbmht as fh @fnurm 7.17. 7.17. Ucm fhtmrhad axfad `jrkm fs erawh fh tmhsfjh fh akkjreahkm wftc jur sfnh kjhvmhtfjh (smm Uagdm 7.1). 7.1). Gy mqufdfgrfub j` `jrkms ahe gy taifhn tcm dfbft ? , wm jgtafh tcm `jddjwfhn gjuheary kjheftfjh< dfb @ mxt ‛ H A ‛ p x x A 2 ?
?
jr
H A 2 @ mxt
Ucm agjvm mquatfjh scjws tcat tcm efstrfgutme axfad `jrkm ejms hjt a``mkt tcm gjuheary kjheftfjh jh tcm fhtmrhad axfad `jrkm. Ucm banhftuem j` tcm fhtmrhad axfad `jrkm H at at tcm mhe j` ah axfad gar fs mquad tj tcm kjhkmhtratme mxtmrhad axfad `jrkm
;3
7
Jhm-Efbmhsfjhad [trukturad Bmbgmrs
appdfme at tcm mhe, gut nmttfhn tcm rfnct tcm rfnct sfnh rmqufrms sfnh rmqufrms erawfhn tcm `rmm gjey efanrab wftc tcm fhtmrhad `jrkm H erawh erawh fh akkjreahkm wftc tcm sfnh kjhvmhtfjh. F` @ F` @ mxt fs zmrj, wm kah wrftm tcm gjuheary kjheftfjh j` zmrj vadum j` H gy gy fhspmktfjh.
7.5.7
Ujrsfjhad rjtatfjh
Ucm rjtatfjh ( x) x) kah gm `juhe `rjb tcm ef``mrmhtfad mquatfjh (7.10-U) wcfkc (7.10-U) wcfkc wm rmwrftm gmdjw `jr kjhvmhfmhkm. • Ef``mrmhtfa Ef``mrmhtfadd mquatfjh< mquatfjh< e e NL 2 ‛ t x e x e x
(7.70-U)
Fhtmnratfhn tcm ef``mrmhtfad mquatfjh twfkm wm jgtafh< e (7.7>-A) NL 2 F 1 x + K 1 NL 2 F 7 x + K 1 x + K 7 wcmrm F 1 x 2 ‛ t x e x F 7 x 2 F 1 x e x e x wcmrm, tcm fhem`fhftm fhtmnrads kah gm `juhe as t ( x) x) fs ihjwh9 K 1 ahe K 7 arm kjhstahts tj gm emtmrbfhme `rjb tcm gjuheary kjheftfjhs. Ucm partfkudar sjdutfjh fs F 7 x NL ahe tcm cjbjnmhjus sjdutfjh fs K 1 x + K 7 NL . Fh tcm agjvm fhtmnratfjh, wm assubme NL fs fs a kjhstaht, gut f` ft fs hjt, tcmh ft kah gm masfdy akkjuhtme `jr gy efvfefhn Mquatfjh (7.7>-A) gy (7.7>-A) gy NL gm`jrm gm`jrm fhtmnratfhn. ^m hjtm tcat fhtmrhad tjrqum fs< U 2 NL e /e x /e x . As efskussme mardfmr, ft fs tcm haturm j` gjuheary kjheftfjhs tcat at makc mhe (gjuheary) j` tcm kfrkudar sca`t, mftcmr tcm rjtatfjh jr tcm fhtmrhad tjrqum U bust bust gm spmkf`fme as scjwh gmdjw.
• Gjuheary Gjuheary kjheftfjhs< kjheftfjhs< At makc makc mhe spmkf`y spmkf`y
jr
U
(7.73-U)
Uj emtmrbfhm tcm gjuheary kjheftfjh jh fhtmrhad tjrqum U , wm baim ah fbanfhary kut ah fh`fhftmsfbad efstahkm () `rjb tcm mhe (say pjfht A pjfht A)) ahe eraw a `rmm gjey efanrab j` tcm fh`fhftmsfbad mdmbmht as scjwh fh @fnurm 7.1;. 7.1;. Ucm fhtmrhad tjrqum U fs fs erawh fh akkjreahkm wftc jur sfnh kjhvmhtfjh (smm Uagdm 7.1).. 7.1).. U A
t x A
U mxt
@fnurm 7.1; Gjuheary kjheftfjh jh tcm fhtmrhad tjrqum.
Gy mqufdfgrfub j` bjbmhts ahe gy taifhn tcm dfbft ? , wm jgtafh tcm gjuheary kjheftfjh dfb U mxt ‛ U A ‛ t x A 2 ?
?
jr
U A 2 U mxt
Ucm agjvm mquatfjh scjws tcat tcm efstrfgutme tjrqum ejms hjt a``mkt tcm gjuheary kjheftfjh jh tcm fhtmrhad tjrqum. Ucm banhftuem j` tcm fhtmrhad tjrqum U at at tcm mhe j` a sca`t fs mquad tj tcm kjhkmhtratme mxtmrhad tjrqum appdfme at tcm mhe, gut nmttfhn tcm rfnct tcm rfnct sfnh rmqufrms sfnh rmqufrms erawfhn tcm `rmm gjey efanrab wftc tcm fhtmrhad tjrqum tjrqum U erawh fh akkjreahkm wftc tcm sfnh kjhvmhtfjh. F` U mxt fs zmrj, wm kah wrftm tcm gjuheary kjheftfjh j` zmrj vadum j` U gy gy fhspmktfjh.
7.5.;
Gmab em`dmktfjh
Ucm gmab em`dmktfjh v( x) x) kah gm `juhe `rjb tcm ef``mrmhtfad mquatfjh (7.10-G) wcfkc (7.10-G) wcfkc fs wrfttmh gmdjw `jr kjhvmhfmhkm. • Ef``mrmhtfa Ef``mrmhtfadd mquatfjh< mquatfjh< 7
7
e e v -------- MF zz -------- 2 p y x 7 7 e x e x
(7.76a-G)
Fhtmnratfhn tcm ef``mrmhtfad mquatfjh `jur tfbms wm jgtafh< 7
e e v ------ MF zz -------- 2 F 1 x + K 1 7 e x e x
F 1 x 2
7
e v MF zz --------7 2 F 7 x + K 1 x + K 7 e x
F 7 x 2
ev 7 MF zz ------ 2 F ; x + K 1 x 7 + K 7 x + K ; e x MF zz v 2 F 5 x + K 1 x
;
0 + K 7 x 7 7 + K ; x + K 5
p x e x
(7.76g-G)
y
F x e x
(7.76k-G)
1
F x e x F x 2 F x e x
F ; x 2 5
7
;
(7.76e-G) (7.76m-G)
wcmrm, tcm fhem`fhftm fhtmnrads kah gm `juhe as p as p y( x) x) fs ihjwh9 K 1, K 7, K ;, K 5 arm kjhstahts tj gm emtmrbfhme `rjb tcm gjuheary ; 7 kjheftfjhs. Ucm partfkudar sjdutfjh fs F 5 x MF zz ahe tcm cjbjnmhjus sjdutfjh fs K 1 x 0 + K 7 x 7 + K ; x + K 5 MF zz .
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
Gjuheary vadum prjgdmbs
;6
1 Fh tcm agjvm fhtmnratfjh, wm assubme MF zz fs a kjhstaht, gut f` ft fs hjt, tcmh ft kah gm masfdy akkjuhtme `jr gy efvfefhnMquatfjh efvfefhnMquatfjh 7 (7.76k-G) gy MF zz gm`jrm fhtmnratfhn. ; ^m wrftm tcm `jddjwfhn mxprmssfjhs `jr tcm fhtmrhad bjbmht ahe scmar `jrkm `rjb Mquatfjhs (7.17-G) ahe (7.17-G) ahe (7.1=g-G) 5 7 7 e v e e v e B z 2 ‛ ----= -------x 2 ‛ MF B z x 2 MF zz --------7 P (7.76`-G) y e x e x zz e x7 e x 0 B z , jr P Ucm fhtmnratfjh j` tcm ef``mrmhtfad mquatfjh `jur tfbms rmsudtme fh v, ev e x , B > y. Ucm `jur quahtftfms `jrb twj 3 nrjups `jr tcm purpjsm j` emtmrbfhfhn tcm gjuheary kjheftfjhs. 6 • Nrjup 1< 1< At a gjuheary pjfht mftcmr tcm em`dmktfjh v jr tcm fhtmrhad scmar `jrkm P y kah gm spmkf`fme, gut hjt gjtc. 1? • Nrjup 7< 7< At a gjuheary pjfht mftcmr tcm sdjpm ev e x jr tcm fhtmrhad gmhefhn bjbmht B z kah gm spmkf`fme, gut 11 hjt gjtc. 17 1; Uj nmhmratm `jur gjuheary kjheftfjhs, twj kjheftfjhs arm spmkf`fme at makc mhe j` tcm gmab. ^m kcjjsm jhm kjheftfjh 15 `rjb nrjup 1 ahe jhm `rjb nrjup 79 adtmrhatfvmdy wm kah baim tcm `jddjwfhn statmbmht. 1= • Gjuheary Gjuheary kjheftfjhs< kjheftfjhs< At makc makc mhe spmkf`y spmkf`y 10 1> -ev ----P y B z (7.76n-G) jr ah e jr v e x 13 16 Uj emtmrbfhm tcm gjuheary kjheftfjh jh tcm fhtmrhad bjbmht B z , ahe tcm scmar `jrkm P y, wm baim ah fbanfhary kut ah 7? fh`fhftmsfbad efstahkm () `rjb tcm mhe (say pjfht A pjfht A)) ahe eraw a `rmm gjey efanrab j` tcm fh`fhftmsfbad mdmbmht as scjwh fh @fn71 urm 7.17. 7.17. Ucm fhtmrhad scmar `jrkm ahe fhtmrhad gmhefhn bjbmht arm erawh fh akkjreahkm wftc jur sfnh kjhvmhtfjh (smm Uagdm 7.1) 7.1) 77 7; P A p y x A 75 y 7= B @ mxt A 70 z 7> x 73 76 B mxt ;? ;1 ;7 ;; ;5 @fnurm 7.15 Gjuheary kjheftfjh jh tcm fhtmrhad scmar `jrkm ahe gmhefhn bjbmht. ;= Gy mqufdfgrfub j` `jrkms ahe bjbmht ahe gy taifhn tcm dfbft ? , wm jgtafh tcm gjuheary kjheftfjhs gmdjw. ;0 P A 2 ‛ @ mxt (7.;?a-G) dfb @ mxt + P A ‛ p y x A 2 ? jr ;> ? ;3 7 ;6 B A 2 B mxt (7.;?g-G) dfb B A ‛ B mxt + @ mx t + p y x A ---- 2 ? jr 7 ? 5? Ucm agjvm mquatfjhs scjw tcat tcm efstrfgutme `jrkm ejms hjt a``mkt tcm gjuheary kjheftfjh jh mftcmr tcm fhtmrhad scmar 51 y ahe tcm gmhefhn bjbmht B z fs mquad 57 `jrkm jr tcm gmhefhn bjbmht. Jh a gjuheary pjfht, tcm vadum j` tcm fhtmrhad scmar `jrkm P tcm rfnct rfnct sfnh 5; fh banhftuem tj tcm kjhkmhtratme mxtmrhad `jrkm ahe tcm kjhkmhtratme mxtmrhad bjbmht, rmspmktfvmdy9 gut nmttfhn tcm P B rmqufrms erawfhn tcm `rmm gjey efanrab wftc tcm fhtmrhad scmar `jrkm ahe tcm fhtmrhad gmhefhn bjbmht erawh fh akkjreahkm y z 55 F` @ mxt ahe B ahe B mxt arm zmrj, wm kah wrftm tcm gjuheary kjheftf jh j` zmrj vadum `jr scmar `jrkm ahe bjbmht 5= wftc tcm sfnh kjhvmhtfjh. F` @ gy fhspmktfjh. 50 A syhjpsfs j` tcm gjuheary vadum prjgdmbs `jr axfad, tjrsfjh, ahe sybbmtrfk gmhefhn fs prmsmhtme fh Uagdm 7.7. 7.7. 5> 53 Padum _rjgdmbs Uagdm 7.7 [yhjpsfs j` Gjuheary Padum 56 =? Axfad Ujrsfjh [ybbmtrfk Gmhefhn =1 7 Ef``mrmhtfad mquatfjh e e e 7 7 ------ MA e u ? e x 2 ‛ p x x ------ NL e e x 2 ‛ t x -- MF zz e v e x 2 p y x =7 e x e x 7 e x =; =5 Gjuheary kjheftfjhs at u? jr jr H U -ev ----ahe v jr P y makc mhe e x jr B z == =0 Fhtmrhad `jrkms/ e NL e e x 7 7 7 7 H MA e u ? e x U NL e => B z MF zz e v e x P y ‛ ------ MF zz e v e x bjbmhts e x =3 =6 0? 01
5?
7
Jhm-Efbmhsfjhad [trukturad Bmbgmrs
MSAB_DM 7.3 A dfnct pjdm fs suglmktme tj wfhe prmssurm tcat varfms quaeratfkaddy as scjwh fh @fnurm 7.1=. 7.1=. Fh tmrbs j` M j` M , F , w, D, D, ahe x ahe x,, emtmrbfhm (a) tcm em`dmktfjh at tcm `rmm mhe ahe (g) tcm nrjuhe rmaktfjhs. w
G
7
x w -----7 D
D
x y A 7.3. @fnurm 7.1= Gmab ahe djaefhn `jr Mxabpdm 7.3.
AH[. v D
5 1; wD 13? MF zz ‛ 1;wD
9 W A
wD ;
7
B A
wD
5
W A
; wD 3
MSAB_DM 7.6 Fh tmrbs j` M j` M , F , w, D, D, ahe x ahe x,, emtmrbfhm (a) tcm mdastfk kurvm ahe (g) tcm rmaktfjh `jrkm at A at A fh @fnurm 7.10. 7.10. y x
A
G D 7.6. @fnurm 7.10 Gmab ahe djaefhn `jr Mxabpdm 7.6.
AH[. v x
7.=
‛ w 53 MF zz 7 x 5 ‛ = Dx ; ; D 7 x 7 9
EF[KJHUFH]FUV @]HKUFJH[
A strukturad bmbgmr usuaddy cas pjfht djaes appdfme tj ft. Fhtmrhad `jrkms ahe bjbmhts lubp gy tcm vadums j` mxtmrhad pjfht `jrkms ahe bjbmhts as jhm krjssms tcm pjfht djaes `rjb jhm sfem tj tcm jtcmr. Fh tcm fhtrjeuktjry kjursm jh bmkcahfks j` batmrfads, tcm efspdakmbmhts ahe tcm rjtatfjh kadkudatfjhs cae tj gm kjheuktme `jr makc smnbmht smparatmdy, ahe tcm kjhtfhufty kjheftfjhs jh efspdakmbmht ahe rjtatfjhs cae tj gm fbpjsme mxpdfkftdy tj mxpdfkftdy tj sjdvm tcm prjgdmb. Ucm efskjhtfhufty `uhktfjhs pmrbft us tj bjemd tcm efskjhtfhuftfms fh djaes sukc tcat a sfhndm mxprmssfjh `jr efstrfgutme djae kah gm usme jvmr tcm mhtfrm dmhntc j` tcm strukturad bmbgmr. Ucm sfhndm mxprmssfjh `jr efstrfgutme djae kah gm sugstftutme fhtj tcm ef``mrmhtfad mquatfjhs ahe tcm gjuheary vadum prjgdmb sjdvme as wm efe fh [mktfjh 7.5. 7.5. Ucm kjhtfhufty kjheftfjhs arm fbpdfkftdy satfs`fme fh tcm sjdutfjh j` gjuheary vadum prjgdmb. Ucm sjdutfjh fs fhempmhemht j` wcmtcmr tcm prjgdmb fs statfkaddy emtmrbfhatm jr fhemtmrbfhatm. Jhkm tcm efspdakmbmht `uhktfjh fs jgtafhme, tcmh strafhs, strmssms, fhtmrhad `jrkms ahe bjbmhts kah gm `juhe.
7.=.1
Em`fhftfjhs
Kjhsfemr a efstrfgutme djae p djae p ahe ahe ah mqufvadmht djae _ djae _ p as scjwh fh @fnurm 7.1>. 7.1>. [uppjsm wm hjw dmt tcm fhtmhsfty j` tcm efstrfgutme djae kjhtfhujusdy fhkrmasm tj fh`fhfty, wcfdm wm emkrmasm tcm dmhntc jvmr wcfkc tcm efstrfgutme `jrkm fs appdfme tj zmrj fh sukc a bahhmr tcat tcm arma p arma p rmbafhs a `fhftm quahtfty9 tcmh wm jgtafh a kjhkmhtratme `jrkm _ `jrkm _ appdfme appdfme at x at x a. Batcmbatfkaddy, tcfs fs statme as `jddjws< ‛ 1
_ 2 dfb dfb p 2 _ x ‛ a
(7.;1)
p ?
‛ 1
Ucm `uhktfjh x ‛ a fs kaddme tcm Efrak emdta `uhktfjh, jr lust tcm emdta `uhktfjh. Ucm emdta `uhktfjh fs zmrj mxkmpt fh ah fh`fhftmsfbad rmnfjh hmar a. As x As x tmhes tmhes tjware a, tcm emdta `uhktfjh tmhes tj fh`fhfty, gut tcm arma uhemr t cm `uhktfjh fs mquad tj jhm. Ucm emdta `uhktfjh fs em`fhme as `jddjws<
x ‛ a ‛ 1
2
?
xa xa
a +
x ‛ a
‛ 1
e x
1
(7.;7a)
a
Kjhsfemr tcm fhtmnrad j` tcm emdta `uhktfjh. Ucm djwmr dfbft j` bfhus fh`fhfty mbpcasfzms tcat tcm pjfht fs gm`jrm a. ^m wrftm tcm fhtmnrad as tcm sub j` tcrmm fhtmnrads< x
‛
x ‛ a ‛ 1 e x
a
2
‛
x ‛ a ‛ 1 e x +
a
a
x ‛ a ‛ 1 e x +
x
a+
x ‛ a ‛ 1 e x
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
Efskjhtfhufty `uhktfjhs
51
‛ 1 1 x ‛ a 7 x 2 a ; 7 7 5 x x x 2 a = 0 x > _ 2 p x 2 a p 3 7 7 (a) 6 (g) 1? 11 @fnurm 7.1> Ucm emdta `uhktfjh. 17 Ucm `frst ahe tcm tcfre fhtmnrads arm zmrj gmkausm tcm emdta `uhktfjh fs zmrj at add pjfhts fh tcm fhtmrvad j` fhtmnratfjh, 1; wcfdm tcm smkjhe fhtmnrad fs mquad tj 1, pmr Mquatfjh (7.;7a). (7.;7a). Ucus, tcm fhtmnrad fs zmrj gm`jrm a ahe mquad tj jhm a`tmr a. Ucm fhtm15 (7.;7g) ahe scjwh fh @fnurm 7.13a 7.13a. 1= nrad fs kaddme tcm stmp `uhktfjh as em`fhme fh Mquatfjh (7.;7g) ahe 10 x x a ? (7.;7g) x ‛ a ? 2 x ‛ a ‛ 1 e x 2 1> ‛ 1 x a 13 16 7 x ‛ a 1 x ‛ a ? x ‛ a 7? Wabp `uhktfjh [tmp `uhktfjh 71 77 7; x x x 75 x 2 a x2a x2a 7= 70 (a) (g) (k) 7> @fnurm 7.13 Efskjhtfhufty `uhktfjhs. 73 76 Hjw kjhsfemr tcm fhtmnrad j` tcm stmp `uhktfjh. ^m kah wrftm tcm fhtmnrad as sub j` twj fhtmnrads. x a x ;? ? ? x ‛ a e x 2 x ‛ a e x + x ‛ a ? e x ;1 ‛ ‛ a ;7 Ucm `frst fhtmnrad fs zmrj gmkausm tcm stmp `uhktfjh fs zmrj at add pjfhts fh tcm fhtmrvad j` fhtmnratfjh, wcfdm tcm smkjhe fhtm;; nrad vadum fs ( x ‛ x ‛ a). Ucm smkjhe fhtmnrad fs kaddme tcm rabp `uhktfjh scjwh fh @fnurm 7.13g 7.13g . _rjkmmefhn fh tcfs bahhmr9 wm kah ;5 batcmbatfkaddy em`fhm ah mhtfrm kdass j` `uhktfjhs scjwh fh Mquatfjh (7.;;). (7.;;). ;= x a ? ;0 h? (7.;;) x ‛ a h 2 h ;> x ‛ a x a ;3 ^m kah adsj nmhmratm tcm fhtmnrad `jrbuda j` Mquatfjh (7.;5) `rjb (7.;5) `rjb Mquatfjh (7.;;)< (7.;;)< ;6 x x ‛ a h + 1 5? h? (7.;5) x ‛ a h e x 2 ------------------------h+1 51 ‛ 57 Jhm bjrm `uhktfjh, kaddme tcm ejugdmt `uhktfjh fs em`fhme fh Mquatfjh (7.;=). (7.;=). 5; x xa ? 55 (7.;=) ahe x ‛ a ‛ 7 2 x ‛ a ‛ 7 e x 2 x ‛ a ‛ 1 5= xa ‛ 50 Ucm emdta `uhktfjh ahe tcm ejugdmt `uhktfjh gmkjbm fh`fhftm at x at x a. Adtmrhatfvmdy statme, tcmsm `uhktfjhs arm sfhnudar at 5> h x a ahe arm rm`mrrme tj as sfhnudarfty `uhktfjhs. Ucm mhtfrm kdass j` `uhktfjhs x ‛ a `jr pjsftfvm ahe hmnatfvm h arm kaddme tcm 53 efskjhtfhufty `uhktfjhs. 56 Hjtm tcat tcm efskjhtfhufty `uhktfjhs arm zmrj f` tcm arnubmht fs hmnatfvm. hmnatfvm. Gy ef``mrmhtfatfhn Mquatfjhs (7.;7g), (7.;7g), (7.;5), (7.;5), =? ahe (7.;=) wm (7.;=) wm jgtafh tcm `jrbudas `jr ef``mrmhtfatfjh. Ucm fhtmnrad `jrbudas arm fhvmrsm j` tcm ef``mrmhtfatfjh `jrbudas. =1 h ‛ 1 ? h ‛ 1 e x ‛ a =7 ‛ 7 ‛ 1 e x ‛ a e x ‛ a 2 h x ‛ a `jr h 1 2 x ‛ a 2 x ‛ a e x =; e x e x (7.;0) h =5 ‛ 7 ‛ 1 ‛ 1 ? x ‛ a h ‛ 1 x ‛ a e x 2 x ‛ a x ‛ a e x 2 x ‛ a x ‛ a e x 2 ------------------- `jr h 1 == h =0 => 7.=.7 Axfad efspdakmbmht =3 @fnurm 7.16a 7.16a scjws a tmbpdatm j` ah axfad gar fh wcfkc a kjhkmhtratme `jrkm fs appdfme at x at x 2 2 a. F` wm baim ah fbanfhary kut fh =6 tcm rmnfjh x rmnfjh x 4 4 a, tcm fhtmrhad axfad `jrkm H 2 2 ?. F` wm baim ah fbanfhary kut fh tcm rmnfjh x rmnfjh x 8 8 a, tcm fhtmrhad axfad `jrkm H 2 2 ‛ @ . 0? ^m kah usm tcm efskjhtfhufty `uhktfjh tj wrftm tcm fhtmrhad `jrkm ahe sugstftutm ft fhtj Mquatfjh (7.1=-A) tj (7.1=-A) tj jgtafh tcm twj mqua01
57
7
Jhm-Efbmhsfjhad [trukturad Bmbgmrs
tfjhs scjwh as tmbpdatm mquatfjhs fh @fnurm 7.16a 7.16a. F` tcm pjfht `jrkm jh tcm aktuad bmbgmr fs fh tcm efrmktfjh scjwh fh @fnurm 7.16, 7.16, tcm fhtmrhad `jrkm ahe efstrfgutme `jrkm arm as nfvmh. F` tcm pjfht `jrkm fs jppjsftm tj tcat scjwh fh@fnurm fh @fnurm 7.16a 7.16a, tcm sfnhs arm rmvmrsme fh tcm tmbpdatm mquatfjhs. @jr statfkaddy emtmrbfhatm prjgdmbs ahe `jr prjgdmbs `jr wcfkc x fs a sfbpdm `uhktfjh, ft bay gm pjssfgdm tj emtmrbfhm tcm fhtmrhad axfad `jrkm H ahe ahe fhtmnratm Mquatfjh (7.1=-A) tj (7.1=-A) tj nmt jgtafh tcm axfad efspdakmbmhts. Gut startfhn wftc tcm ef``mrmhtfad mquatfjhs (7.10-A) ahe (7.10-A) ahe sjdvfhn tcm gjuheary vadum prjgdmb as wm efe fh [mktfjh 7.5, 7.5, fs a nmhmrad apprjakc tcat kjvmrs gjtc tcm statfkaddy emtmrbfhatm ahe fhemtmrbfhatm prjgdmbs. Umbpdatm mquatfjhs
Umbpdatm mquatfjhs
H 2 ‛ @ x ‛ a @
a
U mxt
?
p x 2 @ x ‛ a
H
U
a
‛ 1
x
U 2 ‛ U mx t x ‛ a
?
‛ 1
t 2 U mx t x ‛ a
x (a)
(g)
( a) Axfad bmbgmrs. (g (g) Ujrsfjh j` sca`ts. @fnurm 7.16 Umbpdatm ahe tmbpdatm mquatfjhs `jr usm j` efskjhtfhufty `uhktfjhs fh (a
7.=.;
Ujrsfjhad rjtatfjh
@fnurm 7.16g 7.16g scjws a tmbpdatm j` a sca`t jh wcfkc a kjhkmhtratme tjrqum fs appdfme at x at x 2 2 a. F` wm baim ah fbanfhary kut fh tcm rmnfjh x rmnfjh x 4 4 a, tcm fhtmrhad tjrqum U 2 2 ?. F` wm baim ah fbanfhary kut fh tcm rmnfjh x rmnfjh x 8 8 a, tcm fhtmrhad tjrqum fs U 2 2 ‛Umxt. ^m kah usm tcm efskjhtfhufty `uhktfjh tj wrftm tcm fhtmrhad tjrqum ahe sugstftutm ft fhtj Mquatfjh (7.1=-U) tj (7.1=-U) tj jgtafh tcm twj mquatfjhs scjwh as tmbpdatm mquatfjhs fh @fnurm 7.16g 7.16g. F` tcm kjhkmhtratme tjrqum U mxt jh tcm aktuad sca`t fs fh tcm efrmktfjh scj wh fh @fnurm 7.16g 7.16g, tcm fhtmrhad tjrqum ahe tcm efstrfgutme tjrqum arm as nfvmh. F` tcm kjhkmhtratme tjrqum fs jppjsftm tj tcat scjwh fh @fnurm 7.16g 7.16g, tcm sfnhs arm rmvmrsme fh tcm tmbpdatm mquatfjhs Ucm usm j` efskjhtfhufty `uhktfjhs fh tjrsfjh fs sfbfdar tj tcat `jr axfad efspdakmbmht ahe wfdd hjt gm efskussme `urtcmr. y
y
x
a
y
a
z
x
a x
_
w
z
z
B mxt B z
B z
P y
(a)
B z 2
x a
Umbpdatm mquatfjhs B z 2 ‛ B x ‛ a
?
‛ 7
p y 2 ‛ B x ‛ a
B z
(g)
x a
? ‛ B
P y
B z 2
? ‛ _ x ‛ a
(k) x a x a
Umbpdatm mquatfjhs B z 2 ‛ _ x ‛ a
P y
1
‛ 1
p y 2 ‛ _ x ‛ a
B z 2
? w x ‛ a 7 ‛ ----------------------7
x a x a
Umbpdatm mquatfjhs 7
x ‛ -a----- B z 2 ‛ w --------------7 p 2 ‛ w x ‛ a
?
@fnurm 7.7? Umbpdatms `jr tcm usm j` tcm efskjhtfhufty `uhktfjhs fh gmabs.
7.=.5
Gmab em`dmktfjh
@fnurm 7.7? scjws 7.7? scjws a `rmm gjey efanrab j` a gmab fh wcfkc a kjhkmhtratme `jrkm, bjbmht, jr tcm start j` efstrfgutme `jrkm jkkurs at x at x 2 2 a. F` wm baim ah fbanfhary kut fh tcm rmnfjh x rmnfjh x 4 4 a, tcmh tcm fhtmrhad gmhefhn bjbmht B z 2 ?. F` wm baim ah fbanfhary kut fh rmnfjh x rmnfjh x 8 8 a, tcmh tcm fhtmrhad gmhefhn bjbmht B z kah gm `juhe gy bmahs j` tcm bjbmht mqufdfgrfub, as scjwh fh @fnurm 7.7?. 7.7?. Ucm bjbmht kah tcmh gm wrfttmh gy usfhn tcm em`fhftfjh j` efskjhtfhufty `uhktfjhs. Gy sugstftutfhn Mquatfjh (7.1=a-G) fhtj (7.1=a-G) fhtj Mquatfjh (7.1=g-G), (7.1=g-G), wm jgtafh Mquatfjh (7.;>-G). (7.;>-G). 7
e B z -----------2 p y x 7 e x
(7.;>-G)
Gy sugstftutfhn tcm bjbmht mxprmssfjhs `jr makc kasm jgtafhme fh @fnurm 7.7? fhtj 7.7? fhtj Mquatfjh (7.;>-G), (7.;>-G), wm jgtafh ah mxprmssfjh `jr tcm efstrfgutme `jrkm `jr makc kasm as scjwh fh @fnurm 7.7?. 7.7?. Ucm tmbpdatm mquatfjhs arm assjkfatme wftc tcm kjjrefhatm systmb scjwh fh @fnurm 7.7?. 7.7?. F` tcm kjjrefhatm systmb fs kcahnme, kjrrmspjhefhn kcahnms bust gm baem fh t cm tmbpdatm mquatfjhs. F` tcm mxtmrhad `jrkms ahe bjbmhts arm fh tcm efrmktfjhs scjwh jh tcm tmbpdatm `rmm gjey efanrab, tcmh tcm mxprmssfjhs `jr B ahe p y wfdd gm fh akkjreahkm wftc tcm tmbpdatm mquatfjhs. z ahe p F` tcm mxtmrhad `jrkms ahe bjbmhts arm jppjsftm, tcmh tcm sfnhs wfdd cavm tj gm rmvmrsme fh tcm tmbpdatm mquatfjhs.
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
Efskjhtfhufty `uhktfjhs 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
5;
MSAB_DM 7.1? Ucm kjdubh scjwh fh @fnurm 7.71 cas 7.71 cas a spmkf`fk wmfnct j` 2 ?.1 dg/fh;, bjeudus j` mdastfkfty j` M j` M 2 2 5??? isf, ahe krjss-smk7 tfjhad arma A arma A 2 2 1?? fh . Emtmrbfhm (a) tcm bjvmbmht j` tcm rfnfe pdatm at K ahe (g) tcm rmaktfjh `jrkm at A. at A. A 7= fh. fh. 7 ifps
x
7 ifps
G
0? fh. fh. 5 ifps
5 ifps K
7? fh. fh. E
7.1?. @fnurm 7.71 Kjdubh `jr Mxabpdm 7.1?. ‛ ;
AH[. u k 2 ?.;6; 1?
f h 9 W A
=?60 dg upware
MSAB_DM 7.11 ]sm tcm efskjhtfhufty `uhktfjhs `jr tcm tcrmm tmbpdatms scjwh fh @fnurm 7.77 tj 7.77 tj wrftm tcm bjbmht ahe efstrfgutme `jrkm mxprmssfjhs. _
w
y B mxt
y
x
x
x
x 2 a
x 2 a
x 2 a
x 2 g
y Kasm F
Kasm FFF
Kasm FF
7.11. @fnurm 7.77 Ucm tcrmm kasms j` Mxabpdm 7.11.
AH[. Kasm F< B z 2 B mx t x ‛ a
?
‛ 7
p y 2 B mx t x ‛ a
9 Kasm FF< B z 2 _ x ‛ a
?
?
‛ 1
1
AH[. Kasm FFF< p y 2 ‛ w x ‛ a + w x ‛ g 9 B z 2 ‛ w x ‛ a
p y 2 _ x ‛ a 7
7 + w x ‛ g 7 7
MSAB_DM 7.17 Emtmrbfhm tcm mquatfjh j` tcm mdastfk kurvm fh @fnurm 7.7; fh 7.7; fh tmrbs j` M j` M , F , D, D, _ , ahe x ahe x.. _
y A
7 _D
G
x
D
K
D
7.17. @fnurm 7.7; Gmab ahe djaefhn `jr Mxabpdm 7.17.
AH[.
_ ; ; 7 v x 2 ‛ ---------------- 7 x ‛ D + x ‛ = D x 17 MF zz
MSAB_DM 7.1; A gmab wftc a gmhefhn rfnfefty j` MF j` MF 2 2 57,??? iH ¾ b7 fs scjwh fh @fnurm 7.75. 7.75. Emtmrbfhm (a) tcm em`dmktfjh at pjfht G pjfht G ahe ahe (g) tcm baxfbub fhtmrhad gmhefhn bjbmht. y
= iH 5 iH/b
A
x
= iH/b 7.? 7.? b
G
17 iH-b K
1.? b
E
;.? b
@fnurm 7.75 Gmab ahe djaefhn `jr Mxabpdm 7.1;
AH[. v 7 2 ‛ ?.;= bb 9 B ba x 2 0.17= iH-b
55
7
7.0
Jhm-Efbmhsfjhad [trukturad Bmbgmrs
KDJ[]WM
Ucm djnfk scjwh fh @fnurm 7.0 mbpcasfzms 7.0 mbpcasfzms tcat tcm ifhmbatfk mquatfjhs gmtwmmh mquatfjhs gmtwmmh efspdakmbmhts ahe strafhs, tcm kjhstftutfvm mquatfjhs gmtwmmh tfjhs gmtwmmh strafhs ahe strmssms, tcm statfk tcm statfk mqufvadmhky mquatfjhs mqufvadmhky mquatfjhs gmtwmmh strmssms ahe fhtmrhad `jrkms/bjbmhts, ahe tcm mqufdfgrfub mquatfjhs gmtwmmh mquatfjhs gmtwmmh fhtmrhad `jrkms/bjbmhts ahe mxtmrhad `jrkms/b jbmhts kah gm trmat as fhempmhemht bjeudms, pmrbfttfhn jhm tj aee kjbpdmxftfms tj jhm bjeudm wftcjut kcahn fhn tcm `uheabmhtad mquatfjhs fh tcm jtcmr bjeudms. Ucus, eyhabfks m ``mkts ahe mdastfk `juheatfjh m``mkts kcahnm tcm mqufdfgrfub mquatfjhs wftcjut kcahnfhn tcm ifhmbatfks mquatfjhs, kjhstftutfvm mquatfjhs, ahe mqufvadmhky mquatfjhs. Fh tcm Ufbjscmhij gmab, wm kcahnme tcm ifhmbatfk mquatfjhs gut hjt tcm mquatfjhs fh jtcmr bjeudms. Fh kjbpjsftm dabfhatme strukturad bmbgmrs, wm erjp tcm assubptfjh j` batmrfad cjbjnmhmfty akrjss tcm krjss smktfjh tcat wm baim fh tcm bjeudm j` statfk mqufvad mhky, gut tcmrm arm hj jtcmr kcahnms. Uj fhkjrpjratm batmrfad hjh-dfhmarfty jr fhmdastfk strafhs, wm erjp tcm assubptfjhs rmdatme tj Cjjim‖s daw, tcus kcahnfhn tcm kjhstftutfvm mquatfjhs gut hjtcfhn mdsm. Fh uhsybbmtrfk gmhefhn, wm erjp tcm rmstrfktfjhs jh tcm nmjbmtry j` tcm krjss smktfjhs j` sybbmtrfk gmabs, tcus kcahnfhn tcm ifhmbatfk mquatfjhs wftcjut kcahnfhn tcm `uheabmhtad mquatfjhs fh jtcmr bjeudms. Fh Kcaptmr = jh = jh Ucfh _datms, wm wfdd `jddjw a sfbfdar patc j` emvmdjpfhn `frst tcm sfbpdmst pjssfgdm tcmjry usfhn djnfk scjwh fh @fnurm 7.0 ahe 7.0 ahe tcmh aee kjbpdmxftfms jhm at a tfbm. Fh Kcaptmr > jh > jh Mhmrny Bmtcjes, wm rmpdakm tcm mhtfrm bjeudm j` mqufdfgrfub mquatfjhs wftc mhmrny bmtcje mquatfjhs gut ej hjt kcahnm tcm mquatfjhs j` tcm jtcmr bjeudms. Ah uhemrstahefhn j` tcm bjeudar kcaraktmr j` tcm djnfk, ahe cjw kjbpdmxftfms arm fhkjrpjratme hjt jhdy wfdd cmdp yju wftc tcm rmbafhfhn kcaptmrs fh tcfs gjji gut wfdd adsj nfvm yju tcm kapagfdfty j` aeefhn kjbpdmxftfms tj strukturms hjt kjvmrme cmrm. Fh tcfs kcaptmr wm adsj saw tcm efskjhtfhufty `uhktfjhs ahe tcmfr appdfkatfjhs fh sjdvfhn gjuheary vadum prjgdmbs. Fh Kcaptmr ; jh ; jh Fh`dumhkm Fh`dumhkm @uhktfjhs wm @uhktfjhs wm wfdd usm tcmsm efskjhtfhufty `uhktfjhs `jr sjdvfhn tcm gjuheary vadum prjgdmbs wftc bjrm kjbpdmx djaefhns ahe gmabs jh mdastfk `juheatfjhs.
7.>
[VHJ_[F[ J@ MQ]AUFJH[ Axfad
[ybbmtrfk Gmhefhn
Efspdakmbmhts u 2 u j x [trafh
[trmss
Fhtmrhad `jrkms ahe bjbmhts Jrfnfh
2
xx
eu j 2 M x e x
H 2
e x
x
B y 2 ?
ev e x
2 ‛ y
xx
2 ‛ My
e x
7
x
e v e x
7
x
B z 2 ‛ y xx e A
B z 2 ?
P y 2
A
A
2
x
2 N
e x
A
U 2
xy e A
A
e x e x
x e A
y e A 2 ?
H 2 ------ MA
e U 2 ------e x NL
7
B ev 2 ------- z --7 MF zz ex B z y 2 ‛ ---------- F zz
7 ‛ 1
xx
H 2 --- A
xx
Mqufdfgrfub
e H 2 ‛ p x x e x
eP y
Ef``mrmhtfad mquatfjhs
eu j e MA 2 ‛ p x x e x e x
e v MF zz 7 2 p y x 7 e x e x jr U
u?
e x
jr
A
e
H
2 ‛ p y x
xs e B z e x
x
2 ‛ P y
eU 2 ‛ t x e x
7
7
e x ‛ a e x
‛ 7
2 x ‛ a
‛ 7 x ‛ a e x
2
x ‛ a ‛ 1
e x ‛ a e x
?
U 2 ------ L
e e NL 2 ‛ t x e x e x
v jr
P y
h
‛ 1
2 x ‛ a
‛ 1 x ‛ a e x
2
x ‛ a ?
U x 7 ‛ x 1 2 ------------------------NL
P y Q z 2 ‛ ------------- F zz t
y xz ‛ z xy e A ‛ 1
Efskjhtfhufty `uhktfjhs
e x
x
A
H x 7 ‛ x 1 u 7 ‛ u 1 2 ------------------------ MA
U 2
x
2
2
[trmss `jrbudas
Gjuheary kjheftfjhs at makc mhe Ujrsfjh hjhkfrkudar gars
7
y e A 2 ?
e x
e v
xx
xx e A
A
u 2 ‛ y 7
xx
eu j Em`jrbatfjh `jrbudas
eu j
v 2 vx
Ujrsfjh
h ‛ 1 e x ‛ a 2 h x ‛ a `jr h 1 e x h ‛ 1 x ‛ a e x
h
x ‛ a 2 -------------------- `jr h 1 h
ahe
v e ----- ex jr B z
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
;
Fh`dumhkm @uhktfjhs
DMAWHFHN JGLMKUFPM 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5;
1. ]hemrstahe tcm kjhkmpt j` fh`dumhkm `uhktfjhs ahe fts appdfkatfjhs tj kdassfkad gmabs ahe gmabs jh mdastfk `juheatfjhs. Nrmmh‖s `uhktfjh was `uhktfjh was tcm habm nfvmh tj tcm `uhktfjh Nmjrnm Nrmmh fhtrjeukme fh 1373 fh tcm sjdutfjh j` _jfssjh‖s mquatfjh njvmrhfhn tcm pjtmhtfad j` ah mdmktrfk `fmde. As tcm appdfkatfjhs j` Nrmmh‖s kjhkmpt nrmw, tcm habm Nrmmh‖s `uhktfjh was `uhktfjh was rmpdakme gy fh`dumhkm `uhktfjh. `uhktfjh. @uheabmhtad sjdutfjh fs sjdutfjh fs tcm habm usme `jr fh`dumhkm `uhktfjh wcmh tcm rmnfjh j` fhtmrmst fs fh`fhftm. @uheabmhtad sjdutfjhs arm usme fh tcm `jrbudatfjh j` gjuheary fhtmnrad mquatfjhs as mquatfjhs as a sjdutfjh tj gjuheary vadum prjgdmbs. Ucm hubmrfkad sjdutfjh j` gjuheary fhtmnrad mquatfjhs fs kaddme tcm Gjuheary tcm Gjuheary Mdmbmht Bmtcje , jr GMB jr GMB . Fh tcfs kcaptmr wm emvmdjp tcm kjhkmpt j` fh`dumhkm `uhktfjhs ahe `uheabmhtad sjdutfjhs ahe appdy ft tj kdassfkad gmabs ahe gmabs jh mdastfk `juheatfjhs wftc argftrary djaefhn.
;.1
GA[FK KJHKM_U[
Em`fhftfjhs ahe tcm batcmbatfkad mquatfjhs `jr sjdvfhn prjgdmbs j` kdassfkad gmabs ahe gmabs jh mdastfk `juheatfjhs arm fhtrjeukme fh tcfs smktfjh.
;.1.1
Em`fhftfjhs
A sjurkm pjfht () fs a pjfht fh tcm batmrfad at wcfkc a efsturgahkm fs pdakme. A `fmde pjfht ( x) x) fs a pjfht fh tcm batmrfad wcmrm tcm fbpakt j` efsturgahkm fs mvaduatme. Fh`dumhkm `uhktfjh N x rmdatms a vadum j` a varfagdm at tcm `fmde pjfht x) (x) tj tcm uhft vadum j` tcm efsturgahkm pdakme at tcm sjurkm pjfht (). Fh gmab gmhefhn appdfkatfjhs, tcm pjfhts at wcfkc wm pdakm kjhkmhtratme `jrkm jr bjbmht arm tcm sjurkm pjfhts, ahe pjfhts at wcfkc wm mvaduatm gmab em`dmktfjh, sdjpm, scmar `jrkm, ahe gmhefhn bjbmht arm tcm `fmde pjfhts. Fh`dumhkm `uhktfjhs assjkfatme wftc fh`fhftm gjefms arm kaddme `uheabmhtad sjdutfjhs. Ucm fh`dumhkm `uhktfjh fs safe tj gm sfhnudar f` ft gmkjbms fh`fhftm at tcm sjurkm pjfht. Ucm efsturgahkm assjkfatme wftc a sfhnudar fh`dumhkm `uhktfjh fs kaddme a sfhnudarfty. Ucm bjst kjbbjh rmprmsmhtatfjh j` sfhnudarfty fs tcm emdta `uhktfjh. @fnurm 7.1> scjws 7.1> scjws sfhnudarfty fh tcm emdta `uhktfjh. ^cmh tcm sfhnudarfty fs rmprmsmhtme gy tcm emdta `uhktfjh ft fs kaddme tcm sjurkm sfhnudarfty. A ejugdmt sfhnudarfty fs krmatme gy pdakfhn twj sjurkm sfhnudarftfms j` mquad ahe jppjsftm banhftuem at a efstahkm tcat scrfhis tj zmrj wcfdm tcm sjurkm strmhntc tmhes tj fh`fhfty sukc tcat tcm prjeukt j` sjurkm strmhntc ahe efstahkm rmsudts fh a `fhftm vadum. _jfht `jrkm fs a sjurkm sfhnudarfty ahe pjfht bjbmht fs a ejugdmt sfhnudarfty. Fh tcm hmxt smktfjh wm emvmdjp mquatfjhs ahe hjtatfjhs tcat arm appdfkagdm tj kdassfkad gmabs, fh`fhftm ahe `fhftm gmabs jh mdastfk `juheatfjhs wcfkc wm wfdd stuey datmr.
;.1.7
@jrkm (sjurkm) sfhnudarfty fh`dumhkm `uhktfjhs fh gmabs
Ucm `jrkfhn `uhktfjh jh tcm rfnct cahe sfem j` tcm ef``mrmhtfad mquatfjh fs rmpdakme gy a emdta `uhktfjh ahe tcm gjuheary vadum prjgdmb fs sjdvme tj jgtafh tcm fh`dumhkm `uhktfjh fh ah appdfkatfjh. Ucm ef``mrmhtfad mquatfjh njvmrhfhn sybbmtrfk gmab em`dmk p e y
e @fnurm ;.1 @jrkm aktfhn jh a ef``mrmhtfad dmhntc j` a gmab. 5=
50
;
Fh`dumhkm @uhktfjhs
tfjh fs a `jurtc jremr jrefhary ef``mrmhtfad mquatfjh. Ucm `jrkfhn `uhktfjh fs tcm efstrfgutme djae p djae p y fh tcm y tcm y-efrmktfjh. -efrmktfjh. ^m rmpdakm p y gy tcm emdta `uhktfjh ahe sjdvm tcm gjuheary vadum prjgdmb tj jgtafh tcm fh`dumhkm `uhktfjh N x tcat rmprmsmhts tcm em`dmktfjh at `fmde pjfht x pjfht x eum eum tj a uhft vadum j` kjhkmhtratme `j rkm pdakme at sjurkm pjfht . Budtfpdyfhn tcm fh`dumhkm `uhktfjh gy tcm banhftuem j` `jrkm _ `jrkm _ nfvms nfvms us tcm efspdakmbmht v(x) eum tj `jrkm _ `jrkm _ as as v x 2 N x _
(;.1)
Fh gmab gmhefhn, p y e fs tcm `jrkm aktfhn jh tcm ef``mrmhtfad dmhntc at sjurkm pjfht as scjwh fh @fnurm ;.1. ;.1. Gy efstrfgutfhn tcm `jrkm jvmr tcm dmhntc j` tcm gmab wm jgtafh tcm em`dmktfjh at ahy pjfht x as x as vx 2
Dm hn tc
N x p y e
(;.7a)
^m jgtafh tcm sdjpm, bjbmht, ahe scmar `jrkm gy sukkmssfvm ef``mrmhtfatfjh j` Mquatfjh (;.7a) as (;.7a) as scjwh gmdjw. ev 2 e x
N x
N 1 x p y e
wc m rm
N 1 x 2
ev B z x 2 M F 7 2 e x
N 7 x p y e
wc m rm
N 7 x 2 MF
e B z e x 2
N ; x p y e
wcmrm
N ; x 2 ‛
x
2
Dm hn tc
7
Dm hn tc
P y x 2 ‛
Dm hn tc
(;.7g)
N 1 x
(;.7k)
N 7 x
(;.7e)
Ucm fh`dumhkm `uhktfjhs fhtrjeukme agjvm arm tcm< tcm sdjpm N 1 x , tcm fhtmrhad gmhefhn bjbmht N 7 x , ahe tcm fhtmrhad scmar `jrkm N ; x at `fmde pjfht x pjfht x eum eum tj a uhft vadum j` `jrkm pdakme at sjurkm pjfht . Fh sjdvfhn tcm `jurtc jremr ef``mrmhtfad mquatfjh tj jgtafh N x , wm cavm tj usm tcm gjuheary kjheftfjhs tj mvaduatm tcm fhtmnratfjh kjhstahts. Ucm fh`dumhkm `uhktfjh fs vadfe `jr a nfvmh smt j` gjuheary kjheftfjhs. kjheftfjhs . Fh [mktfjh ;.7.1 wm ;.7.1 wm wfdd jgtafh N x `jr a sfbpdy suppjrtme gmab. ^m kah tcmh usm Mquatfjh (;.7a) `jr (;.7a) `jr a sfbpdy suppjrtme gmab wftc ahy djaefhn p y. ^m mvaduatm tcm agjvm fhtmnrads ahadytfkaddy wcmh p y fs a sfbpdm `uhktfjh. ^cmh p ^cmh p y fs jgtafhme `rjb mxpmrfbmhtad eata jr rmprmsmhtme gy a kjbpdfkatme `uhktfjh tcmh wm wfdd mvaduatm tcm fhtmnrad hubmrfkaddy.
;.1.;
Bjbmht (ejugdmt) sfhnudarfty fh`dumhkm `uhktfjhs fh gmabs
A pjfht bjbmht fs a kjupdm j` twj mquad ahe jppjsftm `jrkms pdakme at ahe + as scjwh fh @fnurm ;.7. ;.7. Gy supmrpjsftfjh wm kah wrftm Mquatfjh (;.1) as< (;.1) as< v x 2 N x _ 1 + N x + _ 7 2 ‛ N x _ + N x + _ _ 12- _ _
y z
x
_ 72 _
@fnurm ;.7 Bjbmht sfhnudarfty
Mxpahefhn tcm smkjhe tmrb gy Uaydjr smrfms wm jgtafh 7
v x 2 ‛ N x _ +
^m hjw dmt
1 N 7 N + ---N x + + 7! 7
? ahe _ sukc tcat
7
_ 2
1 N N + ---_ + 7! 7
dfb dfb _ 2 B tj jgtafh
? _
v x 2 B
N
2 C x B
C x 2
N
(;.;a)
Mquatfjh (;.;a) fs (;.;a) fs fhempmhemht j` tcm suppjrt kjheftfjhs ahe nfvms tcm fh`dumhkm `uhktfjh assjkfatme wftc a pjfht bjbmht. Hjtm tcat B tcat B fs pjsftfvm fs pjsftfvm kjuhtmrkdjkiwfsm wftc kjuhtmrkdjkiwfsm wftc rmspmkt tj tcm z tcm z -axfs. -axfs. Jhkm bjrm, gy sukkmssfvm ef``mrmhtfatfjh j` Mquatfjh (;.;a), (;.;a), wm jgtafh tcm sdjpm, tcm fhtmrhad gmhefhn bjbmht, ahe tcm fhtmrhad scmar `jrkm as nfvmh gmdjw
x
2
ev 2 C 1 x B e x
C 1 x 2
wc m rm
C 7 x 2 MF
7
ev B z x 2 M F 7 2 C 7 x B e x
C x
wc m rm
C 1 x
(;.;g)
(;.;k)
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
Gasfk kjhkmpts
5>
1 e B z C 7 C ; x 2 ‛ P y x 2 ‛ (;.;e) w c m rm 7 e x 2 C ; x B x ; Ucm fh`dumhkm `uhktfjhs fhtrjeukme agjvm arm< tcm efspdakmbmht C x , tcm sdjpm C 1 x , tcm fhtmrhad gmhefhn 5 C bjbmht pjfht x eum eum tj a uhft vadum j` bjbmht pdakme at sjurkm pjfht . 7 x , ahe tcm fhtmrhad scmar `jrkm C ; x at `fmde pjfht x = 0 ;.1.5 Hubmrfkad fhtmnratfjh > 3 Ucm fhtmnratfjh fh Mquatfjh (;.7a) tcrjunc (;.7a) tcrjunc (;.;e) kah (;.;e) kah gm ejhm ahadytfkaddy wcmh tcm djaefhn p djaefhn p y fs sfbpdm, jr hubmrfkaddy f` tcm 6 djaefhn fs em`fhme gy mxpmrfbmhtad eata jr rmprmsmhtme gy a kjbpdfkatme `uhktfjh. 1? Kjhsfemr tcm fhtmnrad 11 g (;.5a) F 2 ` e 17 a 1; 15 Fh hubmrfkad fhtmnratfjh, tcm agjvm fhtmnrad fs apprjxfbatme as H 1= (;.5g) F w ` f f 10 f 1 1> wcmrm, arm kaddme tcm gasm pjfhts9 w arm kaddme tcm wmfncts, ahe H ahe H fs fs tcm hubgmr j` smnbmhts fhtj wcfkc tcm fhtmnrad fhtmrvad f f 13 (g-a) fs efvfeme fhtj. Adtmrhatfvmdy statme, tcm fhtmrvad (g (g-a) cas H cas H +1 +1 gasm pjfhts ahe tcm `uhktfjh ` `uhktfjh ` () fs mvaduatme at tcmsm gasm 16 pjfhts. Ucmrm arm bahy hubmrfkad fhtmnratfjh skcmbms tcat gmmh cavm emvmdjpme tj rmeukm tcm mrrjr j` apprjxfbatfjh. Ucmsm fhtm7? nratfjh skcmbms ef``mr `rjb jhm ahjtcmr fh tcm djkatfjh j` tcm gasm pjfhts ahe tcm vadum j` tcm wmfncts w . Nauss-quaeraturm f f 71 fs tcm bjst pjpudar fhtmnratfjh skcmbm gut wm wfdd usm tcm sfbpdmst hubmrfkad fhtmnratfjh skcmbm kaddme tcm Urapmzjfead Urapmzjfead rudm r udm tj 77 embjhstratm tcm kjhkmpts. 7; 75 `( ) 7= `( ) f 70 `( f-1 ) 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 x ;> f ?2 a 1 f-1 H 2 H 2 g ;3 2 ‛ ;6 f f f ‛ 1 5? @fnurm ;.; Fhtmnratfjh gy trapmzjfead rudm. 51 Ucm fhtmnrad fh Mquatfjh (;.5a) rmprmsmhts (;.5a) rmprmsmhts tcm arma uhemr tcm kurvm em`fhme gy tcm `uhktfjh ` `uhktfjh ` (). ^m kah apprjxfbatm 57 ;.;. Ucm strafnct dfhm apprjxfbatfjh j` 5; tcm `uhktfjh gy a strafnct dfhm gmtwmmh twj sukkmssfvm gasm pjfhts as scjwh fh @fnurm ;.;. tcm `uhktfjh krmatms ah arma j` a trapmzjfe, cmhkm tcm habm j` tcm rud m. Ucm sub j` tcm armas j` tcm trapmzjfes fs tcm vadum j` t cm 55 5= fhtmnrad as scjwh gmdjw. 50 H H f H ‛ 1 f + f + 1 -------- ` f + ` f ‛ 1 2 ---------1 ` ? + ---------------------------------- ` f + -------- F ` H (;.5k) 5> f 1 7 f1 7 7 7 53 Kjbparfhn Mquatfjhs (;.5g) ahe (;.5g) ahe (;.5k) wm smm tcat w ? 2 1 7 , w H 2 H 7 , ahe w f 2 f + f + 1 7 `jr f 2 1 56 tj H tj H -1. - 1. @jr @jr `urtc urtcmr mr sfbp sfbpdf df`f `fka katf tfjh jh,, wm wm ass assub ubmm tcm tcm gasm gasm pjfh pjfhts ts arm arm mqu mquad addy dy spa spakme― kme― f 2 `jr `jr add addff‖s. Mquatfjh (;.5k) sfbpdf(;.5k) sfbpdf=? `fms as =1 H ‛ 1 ` ? + ` H =7 (;.5e) F 2 ------------------------------+ ` f f1 7 =; =5 == ;.1.= Hjh-efbmhsfjhad varfagdms =0 Cfncmr akkurakfms kah gm jgtafhme fh hubmrfkad kadkudatfjhs gy usm j` hjh-efbmhsfjhad varfagdms. Hjh-efbmhsfjhadfzfhn adsj => sfbpdf`fms mquatfjhs `jr prjnrabbfhn ahe baims add varfagdms uhft dmss. =3 Parfagdms wftc kurvme gars wf dd rm`mr tj tcm hjh-efbmhsfjhad varfagdm. ^m start wftc hjh-ef bmhsfjhadfzfhn kjjref=6 hatms wftc rmspmkt tj sjbm kcaraktmrfstfk dmhntc D dmhntc Dj ahe efstrfgutme djae pj as scjwh fh Mquatfjh (;.=a), (;.=a), (;.=g), (;.=g), ahe (;.=k). (;.=k). ^m 0? 7 ; 5 hjtm< p j Dj cas tcm efbmhsfjh j` `jrkm9 j Dj cas tcm efbmhsfjh j` bjbmht9 pj D j MF cas tcm efbmhsfjh j` sdjpm9 p j D j MF 01
53
Fh`dumhkm @uhktfjhs
;
cas tcm efbmhsfjh j` em`dmktfjh, wcmrm MF wcmrm MF fs fs tcm gmhefhn rfnfefty j` tcm gmab. Mquatfjhs (;.=m) tcrjunc (;.=m) tcrjunc (;.=f) arm (;.=f) arm jgtafhme usfhn tcm agjvm jgsmrvatfjhs. Mquatfjh (;.1) kah (;.1) kah gm wrfttmh as
v )
x
p j D j 5 MF
)
)
2 N x _ p j Dj
x
)
v
)
)
)
x
)
jr
v
2 N x )
)
)
2 N x MF Dj
;
)
_
jr
)
_
)
wcmrm, N x fh tcm agjvm mquatfjh fs nfvmh gy Mquatfjh (;.0a). (;.0a). Fh a sfbfdar bahhmr tcm rmbafhfhn mquatfjhs fh Uagdm ;.1 kah gm emrfvme (smm prjgdmbs ;.1;.1-;.;). ;.;). )
Uagdm ;.1 Dfst j` hjh-efbmhsfjhad varfagdms Jrfn Jrfnfh fhad ad varf varfag agdm dm
HjhHjh-ef efbm bmhs hsfj fjha hadf dfzm zme e varf varfag agdm dm x 2 x D j
x
)
)
(;.=g)
p y 2 p y pj
N
N 2 N MF Dj
)
N1 2
(;.=k)
)
_ 2 _ p j D j
_
N x
;
(;.0a)
)
7
)
N 1 x
N 1 2 N 1 MF D j
)
N x
2
)
(;.0g)
)
2 e v ex
)
2
7
7
7 7 e P y 2 ‛ MF e v ex e x
5
N;
N ; 2 N; 2
C
C 2 C MF D j
(;.=n)
C 1 2 C x
C 1 2 C 1 MF D j 2 C x
(;.=c)
C 7
C 7 2 C 7 2 C 1 x
(;.=f)
C ;
C ; 2 C ; D j 2 e C 7 e x
(;.=m)
MF p j D j
)
B z 2 MF e v ex
N 7 2 N 7 D j 2
;
B 2 B p j D j
B
)
7
)
N7
)
(;.0k)
(;.=e)
v 2 v MF p j D j
v
;.7
D j
HjhHjh-ef efbm bmhs hsfj fjha hadf dfzm zme e varf varfag agdm dm
)
p y
2
(;.=a)
Jrfn Jrfnfh fhad ad varf varfag agdm dm
(;.=`)
B z 2 B z p j D j
7
)
)
)
N 7 x 7
)
(;.0e)
(;.0m)
)
)
)
)
)
)
(;.0`) (;.0n)
)
P y 2 P y pj D j
)
)
)
(;.0c)
FH@D]MHKM @]HKUFJH[ @JW KDA[[FKAD GMAB[
Ucm ef``mrmhtfad mquatfjh njvmrhfhn em`dmktfjh j` a kdassfkad gmab wftc kjhstaht gmhefhn rfnfefty fs nfvmh gy 5
ev MF 5 2 p y e x
(;.>) ‛ 1
Uj emtmrbfhm tcm fh`dumhkm `uhktfjh wm rmpdakm p rmpdakm p y( x) x) gy tcm emdta `uhktfjh x ‛
ahe v( x) x) gy N x tj jgtafh
5
eN ‛ 1 MF 5 2 x ‛ e x
(;.3)
Ucm cjbjnmhmjus N c x ahe tcm partfkudar sjdutfjh N p x tj tcm agjvm mquatfjh arm ;
N c x
7
x x 2 k 1 ---- + k 7 ---- + k ; x + k 5 0 7
; 1 N p x 2 --------- x ‛ 0 MF
(;.6)
Ucm sjdutfjh tj Mquatfjh (;.3) fs (;.3) fs ;
7
; 1 x x N x 2 N c x + N p x 2 --------- x ‛ + k 1 ---- + k7 ---- + k ; x + k5 0 MF 0 7
(;.6a)
Ucm kjhstahts k‖s wfdd gm emtmrbfhme `rjb tcm gjuheary kjheftfjhs `jr a spmkf`fk gmab. Jhkm tcm `jur kjhstahts arm emtmrbfhme, tcm fh`dumhkm `uhktfjh N x wfdd gm em`fhme.
;.7.1
Fh`dumhkm `uhktfjhs `jr sfbpdy suppjrtme gmab
^m wfdd emvmdjp tcm sjdutfjh `jr a sfbpdy suppjrtme gmab scjwh fh @fnurm ;.5a ;.5a tj mdagjratm tcm emtmrbfhatfjh j` kjhstahts k‖s fh Mquatfjh (;.6a) `rjb (;.6a) `rjb tcm gjuheary kjheftfjhs. y
y
D x
y
D x
z
x
z
_ (a)
D
z
_ 12-_
_ 72_
B
(g)
(a) [jurkm sfhnudarfty jr pjfht `jrkm. (g ( g) Ejugdmt sfhnudarfty jr pjfht bjbmht. @fnurm ;.5 [fbpdy [uppjrtme Gmab. (a
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
Fh`dumhkm `uhktfjhs `jr kdassfkad gmabs 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
56
Hjtfhn tcat em`dmktfjh ahe bjbmht arm zmrj at makc mhe j` tcm gmab scjwh fh @fnurm ;.5a ;.5a, tcm gjuheary kjheftfjhs jh N x arm< 7
eN MF 7 ? 2 ? e x
N ? 2 ?
7
eN MF 7 D 2 ? e x
N D 2 ?
(;.1?)
]sfhn tcm `jur gjuheary kjheftfjhs fh Mquatfjh (;.1?) wm (;.1?) wm jgtafh tcm fhtmnratfjh kjhstahts as ;
D ‛ k 1 2 ‛ ---------------- MF D
D ‛ D ‛ D k ; 2 ‛ ------------------- + -------------------0 MF D 0 MF
k7 2 ?
k5 2 ?
(;.11)
[ugstftutfhn Mquatfjh (;.11) fhtj (;.11) fhtj Mquatfjh (;.6a) wm (;.6a) wm jgtafh< ;
1 D ‛ ; D ‛ x + D ‛ Dx ; N x 2 --------- x ‛ ‛ ----------------- x ‛ ------------------- x 0 MF D D
(;.17a)
@fnurm ;.5g ;.5g scjws a ejugdmt sfhnudarfty jgtafhme gy pdakfhn twj mquad ahe jppjsftm `jrkms ahe rmpdakfhn tcm kjupdm gy tcm bjbmht. Ucm fh`dumhkm `uhktfjh assjkfatme wftc tcm pjfht bjbmht kah gm `juhe `rjb Mquatfjh (;.;a) as (;.;a) as scjwh gmdjw<
N 2 C x 2 Ucm agjvm fh`dumhkm `uhktfjhs N x
7
1 ; D ‛ 7 x --------- ‛ ; x ‛ + ---+ ---------------------- x ‛ Dx 0 MF D D ;
(;.17g)
ahe C x fhkjrpjratms tcm gjuheary kjheftfjhs j` sfbpdm suppjrt. Ucmsm `uhktfjhs kah hjw gm usme `jr `fhefhn sjdutfjhs j` sfbpdy suppjrtme gmabs wftc ahy djaefhn. Uagdm ;.7 scjws ;.7 scjws add tcm fh`dumhkm `uhktfjhs tcat arm appdfkagdm tj sfbpdy suppjrtme gmab ahe arm jgtafhme gy apprjprfatm ef``mrmhtfatfjh j` Mquatfjhs (;.17a) ahe (;.17a) ahe (;.17g) ahe (;.17g) ahe gy apprjprfatm hjh-efbmhsfjahdfzfhn tcm fh`dumhkm `uhktfjhs. Uagdm ;.7 Fh`dumhkm `uhktfjhs `jr a sfbpdy suppjrtme gmab Fh`dumhkm `uhktfjhs
Hjh-efbmhsfjhadfzme `jrb j` fh`dumhkm `uhktfjhs
;
D ‛ ; D ‛ ‛ ----------------- x ‛ ------------------- x + D ‛ Dx D D
(;.1;a)
; ; 1 N 2 -- x ‛ ‛ 1 ‛ x ‛ 1 ‛ 0
(;.1;g)
7 7 1 N 1 2 -- ; x ‛ ‛ ; 1 ‛ x ‛ 1 ‛ 0
D ‛ ‛ ----------------- x D
(;.1;k)
N7 2
? D ‛ N ; 2 ‛ x ‛ ‛ ---------------- D
(;.1;e)
N ; 2 ‛ x ‛ ‛ 1 ‛
(;.1;m)
7 1 C 2 -- ‛ ; x ‛ + x 0
1 x 1 ; D ‛ C 1 2 --------- ‛ 0 x ‛ + ; ----- + ------------------------ ‛ D 0 MF D D
(;.1;`)
1 1 C 1 2 -- ‛ ; x ‛ + ; x 0
? x C 7 2 ‛ x ‛ + - D
(;.1;n)
C 7 2 ‛ x ‛ + x
1 ‛ - D
(;.1;c)
C ; 2
1 N 2 --------0 MF
x ‛
;
)
;
1 7 D ‛ 7 D ‛ N 1 2 --------- ; x ‛ ‛ ; ----------------- x ‛ ------------------- + D ‛ D 0 MF D D N7 2
x ‛
)
1
7
;
1 ; D ‛ 7 x ---------------- x ‛ Dx C 2 --------- ‛ ; x ‛ + ----- + --------0 MF D D 7
C ; 2
7
)
)
)
1
)
)
)
)
?
)
)
)
)
)
?
)
;
)
)
x + 1 ‛ x )
)
;
+ 1 ‛
)
)
)
;
7
)
7
+ ; 1 ‛
)
x ‛ x
)
)
7
‛ 1
)
(;.15m)
(;.15`)
(;.15n)
‛ 1
x ‛ ‛ 1
)
(;.15g)
(;.15e)
+ ; 1 ‛
)
(;.15a)
(;.15k)
)
)
)
)
)
)
x ‛ ‛ 1 ‛ x
)
)
)
)
)
)
‛ 1
x ‛
)
)
)
(;.15c)
MSAB_DM ;.1 Jgtafh tcm mdastfk kurvm v(x) `jr tcm gmab ahe djaefhn scjwh fh @fnurm ;.= usfhn ;.= usfhn tcm fh`dumhkm `uhktfjhs fh Uagdm ;.7. ;.7. y
w x D
;.1. @fnurm ;.= Gmab fh Mxabpdm ;.1.
AH[.
v x 2
‛ w 5 ; ; ----------- 75 MF x ‛ 7 Dx + D x
=?
;
Fh`dumhkm @uhktfjhs
MSAB_DM ;.7 Jgtafh tcm mdastfk kurvm v( x) x) `jr tcm gmab scjwh fh @fnurm ;.0a ;.0a usfhn tcm fh`dumhkm `uhktfjhs fh Uagdm ;.7. ;.7. y
y
w x
w
B w
Fbpjsm zmrj sdjpm
x
D
D
(a)
(g)
( g) Mqufvadmht gmab @fnurm ;.0 (a) Jrfnfhad gmab. (g 7
wx 7 7 v x 2 ‛ ------------ 7 x ‛ = xD + ; D 53 MF
AH[.
MSAB_DM ;.; A 1? `t. sfbpdy suppjrtme gmab scjwh fh @fnurm ;.> fs ;.> fs djaeme wftc a efstrfgutme `jrkm wcjsm vadums vary as scjwh fh Uagdm ;.;. ;.;. Ucm gmab cas a rmktahnudar krjss smktfjh w ftc emptc j` 3 fh. fh tcm y tcm y efrmktfjh efrmktfjh ahe wfetc j` 7 fh. fh tcm z tcm z efrmktfjh. efrmktfjh. Ucm bjeudus j` mdastfkfty `jr j` tcm gmab batmrfad fs 3,1?? isf. Emtmrbfhm< (a) Ucm em`dmktfjh ahe gmhefhn bjbmht at tcm bfe smktfjh j` tcm gmab. (g) Ucm baxfbub em`dmktfjh ahe gmhefhn hjrbad st rmss fh tcm gmab. p y
y
Uagdm ;.; Efstrfgutme djae vadums fh Mxabpdm ;.;. ;.;. x (`t.)
x
py( x) x) (dg/`t)
1?`t
?
1
7
;
5
=
-;?? -;1? -;?0 -733 -7=0 -71?
0
>
3
6
1? 1?
?
?
?
?
?
;.;. @fnurm ;.> Gmab ahe djaefhn fh Mxabpdm ;.;.
AH[.
v x 2 = 2 ‛ ?.?567 fh. 9 B z x 2 = 2 7=7>7 fh.-dg 9 v ba x 2 ‛ ?.?567 fh. 9 xx ba x 2 113= psf (K) jr (U)
MSAB_DM ;.5 A 1? `t. sfbpdy suppjrtme gmab scjwh fh @fnurm ;.3 cas ;.3 cas a efstrfgutme `jrkm tcat varfms as scjwh. Ucm gmab cas a rmktahnudar krjss smktfjh wftc emptc j` 3 fhkc fh tcm y tcm y efrmktfjh efrmktfjh ahe wfetc j` 7 fhkc fh tcm z tcm z -efrmktfjh. -efrmktfjh. Ucm bjeudus j` mdastfkfty `jr j` tcm gmab batmrfad fs 3,1?? isf. Emtmrbfhm< (a) Ucm em`dmktfjh ahe gmhefhn bjbmht at tcm bfe smktfjh j` tcm gmab. (g) Ucm baxfbub em`dmktfjh ahe gmhefhn hjrbad strmss fh tcm gmab. ‛ x 1?
y
p y 2 ;??m ;?? m
sfh x 1? dg/`t.
x 1? `t ;.5. @fnurm ;.3 Gmab ahe djaefhn fh Mxabpdm ;.5.
AH[.
;.;
v x 2 = 2 ‛ ?.?5>= fh. 9 B z x 2 = 2 77 07= 07= fh.fh.-dg dg 9 v ba x 2 ‛ ?.?5>= fh 9
xx ba x
2 1?0?.0 psf (K) jr (U)
GMAB[ JH MDA[UFK @J]HEAUFJH[
Wafdrjae trakis rmst jh mdastfk suppjrt baem j` krjss tfms ahe gaddast j` kruscme rjkis, djhn stmmd pfpms rmst jh martc jr a smrfms j` pmrfjefk suppjrts, djae gmarfhn wadds fh gufdefhns cavm gmabs suppjrtme pmrfjefkaddy gy stues tcat rmst jh `juheatfjh gmabs, arm sjbm mxabpdms j` gmabs jh mdastfk `juheatfjhs. Ucm sfbpdmst bjemd `jr fhkjrpjratfhn tcm mdastfk `juheatfjh m``mkt fs tcm ^fhidmr bjemd bjemd fh wcfkc tcm `juheatfjh rmsfstahkm fs assubme prjpjrtfjhad tj tcm gmab em`dmktfjh. Ucfs dfhmar bjemd wjris wmdd `jr sbadd em`dmktfjh―tcm gasfk assubpt fjh fh jur tcmjry. ^m rmprmsmht tcm bjeudus j` `juheatfjh gy i , wcfkc fs sprfhn fs sprfhn kjhstaht pmr uhft dmhntc ahe dmhntc ahe cas tcm efbmhsfjh j` `jrkm j` `jrkm pmr dmhntc squarme . Ucm `juheatfjh mxmrts a `jrkm pmr uhft dmhntc j` i v ejwhwares tj rmsfst tcm upware pjsftfvm em`dmktfjh. ^m taim a sbadd ef``mrmhtfad mdmbmht j` a gmab rmstfhn jh tcm mdastfk `juheatfjh as scjwh fh @fnurm ;.6a ;.6a. Fh tcm fh`fhftmsfbad dmhntc wm kah assubm p assubm p y fs a kjhstaht ahe rmpdakm ft gy ah mqufvadmht `jrkm as scjwh fh @fnurm ;.6g ;.6g. Gy `jrkm mqufdfgrfub fh tcm y tcm y-efrmktfjh -efrmktfjh ahe bjbmht mqufdfgrfub agjut tcm bfepjfht fh @fnurm ;.6g ;.6g wm jgtafh
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
@uheabmhtad sjdutfjhs `jr gmabs jh mdastfk 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
pyex ex/7 ex/7
p y B z
B z + e B z
ex
B z
P y +eP y
P y
ex/7
B z + e B z
ex
P y +eP y
P y
(iex)v iex)v
(iex)v iex)v
(a)
(g)
@fnurm ;.6 Ef``mrmhtfad mdmbmht j` a gmab jh mdastfk `juheatfjh.
P y + p y e x ‛ i e x v 2 ? e P
P y e P -------- ‛ iv 2 ‛ p y ex
jr
ex ex B z + e B z + P y + e P P y ------ + P y ------ ‛ B z 2 ? 7 7
jr
(;.1=a)
e B P y 2 ‛ ---------- z ex 7
(;.1=g) 7
Fh tcm agjvm mquatfjh wm hmndmktme tcm smkjhe jremr tmrb (eP yex). [ugstftutfhn B z x 2 MF e v e x ahe Mquatfjh (;.1=g) fhtj Mquatfjh (;.1=a) wm (;.1=a) wm jgtafh tcm ef``mrmhtfad mquatfjh njvmrhfhn em`dmktfjh j` tcm gmab rmstfhn jh mdastfk `juheatfjh. 7
e v MF zz 7 7 e x e x e
;.5
7
+ i v 2 p y
(;.10)
@]HEABMHUAD [JD]UFJH[ @JW GMAB[ JH MDA[UFK @J]HEAUFJH[
Ucm `uheabmhtad sjdutfjh fs tcm fh`dumhkm `uhktfjh assjkfatme wftc ah fh`fhftm gmab jh mdastfk `juheatfjh. Uj sfbpdf`y kadkudatfjhs, wm wfdd pdakm tcm kjhkmhtratme `jrkm at tcm jrfnfh as scjwh fh @fnurm ;.1? ahe ;.1? ahe datmr nmhmradfzm tcm sjdutfjh gy trahsdatfhn tcm jrfnfh tj ah argftrary djkatfjh . Uj immp `jkus jh tcm kjhkmpts fh tcfs smktfjh, tcm b atcmbatfkad emtafds j` tcm emrfvatfjh j` tcm ‛ 1 `uheabmhtad sjdutfjhs arm smparatme ahe nfvmh fh [mktfjh ;.3. ;.3. ^m sugstftutm y 2 _ x fhtj Mquatfjhs (;.10) tj (;.10) tj jgtafh 7
e v MF 7 7 e x e x e
7
‛ 1
+ i v 2 _ x
-
(;.1>a)
+
x
-
+ y
@fnurm ;.1? Fh`fhftm gmab jh mdastfk `juheatfjh wftc a kjhkmhtratme `jrkm.
Ucm efsturgahkm fhtrjeukme gy tcm kjhkmhtratme `jrkm bust efm jut at fh`fhfty. Ucus, scmar `jrkm ahe gmhefhn bjbmht bust nj tj zmrj at fh`fhfty as scjwh gmdjw 7
e e v dfb MF 7 x e x e x
7
ev
?
dfb MF zz 7 e x
x
?
(;.1>g)
Fh aeeftfjh, efspdakmbmhts ahe sdjpm bust rmbafh gjuheme at fh`fhfty as scjwh gmdjw. dfb v
x
ev e x
dfb
x
(;.1>k)
Mquatfjhs (;.1>a) tcrjunc (;.1>a) tcrjunc (;.1>k) rmprmsmht (;.1>k) rmprmsmht tcm gjuheary vadum prjgdmb wm hmme tj sjdvm. Fhtmnratfhn Mquatfjh (;.1>a) `rjb bfhus fh`fhfty tj pdus fh`fhfty, hjtfhn tcm fhtmnrad j` emdta `uhktfjh fs jhm ahe usfhn Mquatfjh (;.1>g) wm (;.1>g) wm jgtafh 7
e v e MF zz 7 e x e x
+ ‛
‛
i v e x 2 _
‛
x ‛ 1 e x
jr
i ve x 2 _
(;.1>e)
‛
Mquatfjh (;.1>e) rmprmsmhts (;.1>e) rmprmsmhts statfk mqufdfgrfub as tcm dm`t cahe sfem fs tcm tjtad `juheatfjh rmsfstahkm wcfkc scjude mquad tj tcm appdfme `jrkm _ `jrkm _ . ^m usm tcm sybbmtry fh tcm prjgdmb tj rmeukm tcm adnmgra gy kjhsfemrfhn jhdy x ? ahe datmr nmhmradfzfhn
=7
;
Fh`dumhkm @uhktfjhs
tcm sjdutfjh tj fhkduem x ? . Gy sybbmtry wm jgtafh tcat tcm sdjpm at tcm jrfnfh fs zmrj, wcfkc rmpdakms tcm kjheftfjh jh bjbmht at fh`fhfty. Ucm efspdakmbmht wfdd gm sybbmtrfk ahe Mquatfjh (;.1>e) kah (;.1>e) kah gm wrfttmh as< 7
?
i v e x 2 _
v ? x
(;.1>m)
2 ?
(;.13a)
Fh tcm emvmdjpbmht tcat `jddjws wm scadd assubm tcat tcm gmhefhn rfnfefty MF rfnfefty MF zz 2 MF 2 MF fs fs a kjhstaht. @jr x ? , tcm sjdutfjh tj tcm agjvm emskrfgme gjuheary vadum prjgdmb as scjwh fh Mquatfjh (;.51) fs (;.51) fs v x 2
‛ x
_ ------ 7 i m
kjs x + sfh x
x?
(;.13g)
wcmrm tcm parabmtmr fs
2
i -------- 5 MF
1 5
(;.13k)
^m nmhmradfzm tcm sjdutfjh gy taifhn tcm agsjdutm vadum j` x j` x ahe ahe mxtmhe jur sjdutfjh tj hmnatfvm vadums j` x j` x.. Uj `urtcmr nmhmradfzm jur sjdutfjh, wm kjhsfemr tcm kjhkmhtratme `jrkm at x at x 2 2 ratcmr tcah tcm jrfnfh. Ucfs fs a sfbpdm trahsdatfjh j` tcm jrfnfh fh ah fh`fhftm gmab ahe tcm fbpakt fs tcat wm wrftm x wrftm x - fh pdakm j` x j` x.. Hjtfhn tcm em`fhftfjh j` `uheabmhtad sjdutfjh wm rmwrftm Mquatfjh (;.13g) as (;.13g) as v x 2 N x _
N x 2
‛ x ‛
- ---- 7 i m
kjs x ‛
+ sfh x ‛
(;.16)
Kjhsfemr tcm `jddjwfhn emrfvatfvm j` ah agsjdutm quahtfty x ‛ 2
x ‛ ‛ x ‛
x
x ‛ x
x
2
1 ‛ 1
x x
(;.7?)
^m fhtrjeukm tcm em`fhftfjh j` snh j` snh `uhktfjh gmdjw `uhktfjh gmdjw wcfkc baims tcm emrfvatfvm j` agsjdutm `uhktfj hs kjbpakt. snh x ‛ 2
x
1 ‛ 1
x
(;.71)
^m hjtm tcm `jddjwfhn femhtftfms usfhn Mquatfjh (;.71)
x ‛ x
2 ‛
x ‛
N x
2 snh x ‛
N
2 ‛
(;.77)
(;.7;)
Mquatfjh (;.7;) pmrbft (;.7;) pmrbft us tj kcahnm tcm emrfvatfvm `rjb x `rjb x tj tj . Ucm agjvm femhtfty cjdes jhdy `jr `uheabmhtad sjdutfjhs. Ucm agjvm femhtfty fs hjt vadfe `jr fh`dumhkm `uhktfjh fh Mquatfjh (;.17a), (;.17a), wcfkc fs assjkfatme wftc a gmab j` `fhftm dmhntc. Gy taifhn tcm apprjprfatm emrfvatfvms j` N x wftc rmspmkt tj x tj x wm wm jgtafh mxprmssfjhs `jr `uheabmhtad sjdutfjhs assjkfatme wftc tcm sdjpm, gmhefhn bjbmht, ahe scmar `jrkm. Ucmsm mxprmssfjhs arm nfvmh gy Mquatfjhs (;.7=g), (;.7=g), (;.7=k), (;.7=k), ahe (;.7=e) fh (;.7=e) fh Uagdm ;.5. ;.5. Ucm emtafds j` tcm emrfvatfjh j` tcmsm mquatfjhs arm nfvmh fh [mktfjh ;.3. ;.3. Hjw tcm `uheabmhtad sjdutfjh assjkfatme wftc kjhkmhtratme bjb mht fs nfvmh gyMquatfjh gy Mquatfjh (;.;a). (;.;a). ]sfhn tcm femhtfty j` Mquatfjh (;.7;), (;.7;), wm kjhkduem tcat C x C x 2 ‛ N 1 x
(;.75)
C x fs scjwh fh Mquatfjh (;.7=m). Ucm mxprmssfjh `jr C x (;.7=m). ^m bay tcmh taim tcm emrfvatfvms j` C x wftc rmspmkt tj x tj x tj jgtafh mxprmssfjhs `jr mxprmssfjhs `jr `uheabmhtad sjdutfjhs assjkfatme wftc tcm sdjpm, gmhefhn bjbmht, ahe scmar `jrkm. Ucmsm mxprmssfjhs arm nfvmh gy Mquatfjhs (;.7=`), (;.7=`), (;.7=n), (;.7=n), ahe (;.7=c). (;.7=c). Ucm emtafds j` tcm emrfvatfjh j` tcmsm mquatfjhs arm nfvmh fh [mktfjh ;.3. ;.3.
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
@uheabmhtad sjdutfjhs `jr gmabs jh mdastfk Uagdm ;.5 @uheabmhtad sjdutfjhs `jr ah fh`fhftm gmab jh mdastfk `juheatfjh..F 1 7 @uheabmhtad [jdutfjhs Hjh-efbmhsfjhadfzme `jrb j` @uheabmhtad [jdutfjhs ; 1 ‛ x ‛ 5 ‛ x ‛ kjs x ‛ + sfh x ‛ N 2 ----------- m kjs x ‛ + sfh x ‛ (;.70a) N 2 ------ m (;.7=a) 7 i 3 ; = 0 1 ‛ x ‛ ‛ x ‛ 7 > N 1 2 ‛ ----------- snh x ‛ m sfh x ‛ (;.70g) N 1 2 ‛ ----- snh x ‛ m sfh x ‛ (;.7=g) 7 5 i 3 6 1 ‛ x ‛ 1 ‛ x ‛ N 7 2 --------- m (;.70k) sfh x ‛ ‛ kjs x ‛ sfh x ‛ ‛ kjs x ‛ (;.7=k) 1? N 7 2 -5----- m 5 11 ‛ x ‛ 1 17 N ; 2 ‛ 1-- snh x ‛ m ‛ x ‛ kjs x ‛ (;.7=e) N ; 2 ‛ -- snh x ‛ m kjs x ‛ (;.70e) 7 7 1; 1 15 7 ‛ x ‛ C 2 ----------- snh x ‛ m sfh x ‛ (;.70m) ----- m ‛ x ‛ sfh x ‛ C 2 snh x ‛ (;.7=m) 7 1= 5 i 10 ; 1 ‛ x ‛ 1> C 2 ‛ ---- m ‛ x ‛ sfh x ‛ ‛ kjs x ‛ C 1 2 ‛ --------- m (;.70`) sfh x ‛ ‛ kjs x ‛ (;.7=`) 1 i 5 13 16 ‛ x ‛ 1 ‛ x ‛ 1 C 7 2 ‛ -- snh x ‛ m kjs x ‛ (;.7=n) C 7 2 ‛ -- snh x ‛ m kjs x ‛ (;.70n) 7? 7 7 71 ‛ x ‛ kjs x ‛ + sfh x ‛ ‛ x ‛ kjs x ‛ + sfh x ‛ (;.7=c) 77 C ; 2 ‛ -7- m C ; 2 ‛ ------ m (;.70c) 7 7; 75 Ucm hjh-efbmhsfjhadfzme `jrbs j` `uheabmhtad sjdutfjhs fh Uagdm ;.5 arm ;.5 arm jgtafhme usfhn varfagdms em`fhme fh Uagdm ;.1 7= ahe tcm `jddjwfhn. 70 7> (;.7>) i 2 iD j pj 2 D j 73 Ft scjude gm hjtme tcat tcm snh tcm snh `uhktfjh fs a kjhstaht, cmhkm fts emrfvatfvm fs zmrj `jr x ahe x gut hmftcmr tcm 76 `uhktfjh hjr ft‖s emrfvatfvm mxfsts at x 2 . Ucm mquatfjhs fh Uagdm ;.5 arm vadfe `jr x ahe x gut hjt at x 2 , uhdmss tcm ;? appdfkatfjh j` tcm `jrbuda scjw tcat tcm varfagdm fs kjhtfhujus at x 2 . Ft adsj `jddjws tcateurfhn tcat eurfhn fhtmnratfjh ft wfdd gm hmkms;1 sary tj kjhsfemr tcm rmnfjhs x ahe x smparatmdy. smparatmdy. ;7 ;; ;.5.1 [jbm prjpmrtfms j` `uheabmhtad sjdutfjhs assjkfatme wftc `jrkm sfhnudarfty ;5 (;.7=a) tcrjunc (;.7=e) wm (;.7=e) wm hjtm ;= @rjb (;.7=a) tcrjunc ;0 • N x ahe N 7 x arm mvmh `uhktfjhs agjut tcm sjurkm pjfht ;> • N 1 x ahe N ; x arm ah jee `uhktfjh agjut tcm sjurkm pjfht ;3 ;6 Ucus, tcm efspdakmbmht fs ah mvmh `uhktfjh j` x ‛ . Ucm jee emrfvatfvms j` efspdakmbmht wftc rmspmkt tj x tj x arm arm jee 5? `uhktfjhs ahe mvmh emrfvatfvms arm mvmh `uhktfjhs, f.m., 51 ev 57 v ‛ x + 2 v x ‛ B z ‛ x + 2 B z x ‛ P y ‛ x + 2 ‛ P y x ‛ (;.73) ‛ x + 2 ‛ ev x ‛ e x e x 5; 55 ^m `urtcmr hjtm tcat abpdftuem j` add varfagdms emkrmasms mxpjhmhtfaddy as wm bjvm away `rjb tcm pjfht j` appdfkatfjh 5= j` tcm `jrkm. Ucus, tcm banhftuem j` tcm baxfbub vadums j` tcm varfagdms fs lust gm`jrm jr lust a`tmr tcm pjfht j` appdfkatfjh j` tcm 50 `jrkm. [ugstftutfhn x [ugstftutfhn x 2 2 + ahe dmttfhn nj tj zmrj wm jgtafh tcm banhftuem j` tcm baxfbub vadums as scjwh gmdjw< 5> v ba x 2 _ 7 i (;.76) e v ex ba x 2 _ 7 i B z ba x 2 _ 5 P y ba x 2 _ 7 53 56 @fnurm ;.11 scjws ;.11 scjws tcm varfatfjh j` tcm `jur varfagdms hjh-efbmhsfjhadfzme wftc rmspmkt tj tcmfr baxfbub vadums `jr tcm =? rahnm j` ‛ 7 x 7 . =1 @fnurm ;.11 scjws ;.11 scjws tcm `jddjwfhn< =7 1. Efspdakmbmht Efspdakmbmht ahe bjbmht bjbmht arm mvmh mvmh `uhktfjh `uhktfjhss j` x j` x ahe ahe sdjpm ahe scmar `jrkm arm jee `uhktfjhs agjut tcm =; sjurkm, kjhsfstmht wftc Mquatfjh (;.73). (;.73). =5 7. Ucm efsturgahkm efsturgahkm fhtrjeukme fhtrjeukme gy tcm kjhkmhtrat kjhkmhtratme me `jrkm at tcm jrfnfh jrfnfh efms jut rapfedy as jhm bjvm bjvm away `rjb == tcm sjurkm. Ucfs fs mbpcasfzme gy add `jur varfagdms. =0 ;. Efspdakmbmht, Efspdakmbmht, sdjpm sdjpm ahe bjbmht arm arm kjhtfhujus kjhtfhujus at tcm jrfnfh gut tcm tcm scmar `jrkm `jrkm lubps gy tcm vadum j` => appdfme djae _ djae _ . Ucfs fs as mxpmktme as tcm gmab ejms hjt grmai jr `jrb ifhis, cmhkm tcm kjhtfhufty j` tcmsm =3 varfagdms. Ucmrm fs hj mxtmrhad bjbmht at tcm jrfnfh, cmhkm hj lubp fh tcm fhtmrhad bjbmht gut tcmrm fs a =6 kjhkmhtratme `jrkm _ `jrkm _ , cmhkm tcm lubp fh fhtmrhad scmar `jrkm. 0? ;.1>). 01 Fh a sfbfdar bahhmr wm kah jgtafh prjpmrtfms assjkfatme wftc bjbmht sfhnudarfty ([mm prjgdmb ;.1>). )
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=5
;
Fh`dumhkm @uhktfjhs 1.?
ev ev e x e x bax
1.?
v/|vbax| bax|
?.=
?.=
x -0
-5
-7
7
5
x -0
0
-5
-7
7
-?.=
-?.=
-1.? 1.? B B z z bax
-1.? 1.?
?.=
5
0
P y P y bax
?.=
x -0
-5
-7
7
5
x
0
-0
-5
-7
7
-?.=
-?.=
-1.?
-1.?
5
0
@fnurm ;.11 _djt j` hjh-efbmhsfjhadfzme varfagdms fh ah fh`fhftm gmab wftc kjhkmhtratme `jrkm.
MSAB_DM ;.= Jgtafh tcm efspdakmbmht ahe fhtmrhad gmhefhn bjbmht `uhktfjh `jr a pjfht jh ah fh`fhftm gmab jh mdastfk `juheatfjh uhemr a uhf`jrb djae scjwh fh @fnurm ;.17. ;.17. D w -
+ +
x
-
+ + y
;.=. @fnurm ;.17 Gmab jh mdastfk `juheatfjh fh Mxabpdm ;.=.
AH[.
‛ x + D 7
v x 2 w 7 i 7 ‛ snh x + D 7 m
AH[. B z x 2
‛ x ‛ D 7
kjs x + D 7 + snh x ‛ D 7 m
kjs x ‛ D 7
w 5 7 ‛ snh x + D 7 m ‛ x + D 7 sfh x + D 7 + snh x ‛ D 7 m ‛ x ‛ D 7 sfh x ‛ D 7 MSAB_DM ;.0
A vmry djhn rmktahnudar gmab wftc a bjeudus j` mdastfkfty j` ;?,??? isf rmst jh ah mdastfk `juheatfjh j` bjeudus j ` 7 isf. Ucm gmab krjss smktfjh ahe djaefhn arm as scjwh fh @fnurm ;.1;. ;.1;. Emtmrbfhm tcm baxfbub em`dmktfjh, baxfbub gmhefhn hjrbad strmss, ahe tcm baxfbub `juheatfjh `jrkm pmr uhft dmhntc aktfhn jh tcm gmab. 5 fh. fh. 10 `t w 2 17ifps/ 17ifps/ `t. `t. -
0 fh. fh. +
x
-
+ y
z y Krjss smktfjh
;.0. @fnurm ;.1; Gmab jh mdastfk `juheatfjh fh Mxabpdm ;.0.
AH[.
v ba x 2 ?.=;1 fh.
@ ba x 2 17.>5 ifps/`t
xx ba x
2 5.=;0 isf (U) jr (K)
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
@fhftm gmabs 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
==
MSAB_DM ;.> @fnurm ;.15 scjws ;.15 scjws a vmry djhn rmktahnudar gmab wftc a bjeudus j` mdastfkfty j` ;?,??? isf tcat rmsts jh ah mdastfk `juheatfjh j` ‛ x 1? bjeudus j` 7 isf. Ucm gmab fs suglmktme tj a trahsvmrsm djae j` y 2 ;??m ;?? m sfh x 1? dg/`t ahe cas tcm krjss smktfjh scjwh. Emtmrbfhm tcm em`dmktfjh ahe gmhefhn bjbmht at tcm bfe smktfjh j` tcm gmab. 5 fh. fh. p y 0 fh. fh. -
z
+
x
+
y Krjss smktfjh
1? `t y
@fnurm ;.15
AH[.
;.=
^ v x 2 = 2 ?.?=7 fh. 9 B z x 2 = 2 5>0>> fh.-dg
@FHFUM GMAB[
^m kjhsfemr a `fhftm gmab AG gmab AG rmstfhn rmstfhn jh mdastfk `juheatfjh fh @fnurm ;.1=a ;.1=a. Ucmrm fs hj djss j` nmhmradfty tj assubm tcm jrfnfh at tcm kmhtmr j` tcm gmab. Fh @fnurm ;.1=g ;.1=g wm kjhsfemr tcm `fhftm gmab AG gmab AG as part j` ah fh`fhftm gmab wftc a `jrkm ahe a bjbmht appdfme at makc mhe. Ucmsm `jrkms arm fh tcm pjsftfvm y pjsftfvm y-efrmktfjh -efrmktfjh ahe tcm bjbmht fs pjsftfvm fh tcm kjuhtmr-kdjkiwfsm efrmktfjh wftc rmspmkt tj tcm z tcm z axfs. axfs. Ucmsm `jrkms ahe bjbmhts arm uhihjwh quahtftfms tj gm emtmrbfhme as emskrfgme datmr. Ucmsm `jrkms ahe bjbmhts rmprmsmht lubps fh tcm scmar `jrkm ahe gmhefhn bjbmhts as wm bjvm `rjb tcm fhsfem j` tcm `fhftm gmab tj tcm jutsfem fhtj tcm fh`fhftm gmab. W A
p y x x
A x2-D/ 7
-
G
A
x
B A
x2D/ 7
W G
p y x G B G
+
y
y (a)
(g)
@fnurm ;.1= Gmabs jh Mdastfk @juheatfjhs. (a) @fhftm gmab. (g) _art j` ah fh`fhftm gmab.
]sfhn tcm `uheabmhtad sjdutfjhs fh Uagdm ;.5 wm ;.5 wm kah wrftm tcm `jddjwfhn gy supmrpjsftfjh. v x 2 W A N x ‛ D 7 + B A C x C x ‛ D 7 + W G N x D 7 + B G C x D 7 +
v x x
‛ D 7
D 7
B x 2 W N x ‛ D 7 + B C x ‛ D 7 + W N x D 7 + B C x D 7 + P x 2 W N x ‛ D 7 + B C x ‛ D 7 + W N x D 7 + B C x D 7 + 2 W A N 1 x ‛ D 7 + B A C 1 x ‛ D 7 + W G N 1 x D 7 + B G C 1 x D 7 +
z
A
7
A
7
G
7
G
7
y
A
;
A
;
G
;
G
;
N x p y e
‛ D 7
N 1 x p y e
(;.;?g)
N 7 x p y e
(;.;?k)
N ; x p y e
(;.;?e)
‛ D 7
‛ D 7
(;.;?a)
Fh tcm agjvm mquatfjhs, add quahtftfms arm ihjwh mxkmpt `jrkms ahe bjbmhts at A at A ahe ahe G G fh fh @fnurm ;.1=g ;.1=g. Appdyfhn tcm `jur gjuheary kjheftfjhs tcat arm appdfkagdm tj tcm `fhftm gmab wm wfdd nmhmratm 5 mquatfjhs fh 5 uhihjwhs tcat kah gm sjdvme. Ucm mquatfjhs agjvm tcmh nfvms us tcm kjbpdmtm sjdutfjh at ahy pjfht x pjfht x.. Fh jremr tj gm agdm tj mvaduatm tcm snh tcm snh `uhktfjhs, `uhktfjhs, f` tcm dm`t gjuheary at A at A fs kjhsfemrme tcmh x 2 ‛ D 7 + ahe f` tcm rfnct gjuheary at G at G fs kjhsfemrme tcmh x 2 D 7 ‛ . ^m tcmh dmt ? . Uagdm ;.= scjws ;.= scjws tcm vadums j` tcm `uheabmhtad sjdutfjhs at tcm twj gjuhearfms `jr kjhvmhfmhkm. Fh praktfkm, tcm adnmgra hmmeme tj sjdvm `jr tcm `jur uhihjwhs kah gm tmefjus. [j wm wfdd kjhsfemr sfbpdf`fkatfjhs tcat wfdd dmt us jgtafh ahadytfkad sjdutfjhs, ahe fh kasm j` kjbpdfkatme efstrfgutfjh j` p y x wm kjude nmt hubmrfkad sjdutfjhs usfhn sprmaescmmts as wm cavm ejhm fh smvmrad mxabpdms. Fh emvmdjpfhn tcmsm sjdutfjhs wm wfdd assubm tcm `jddjwfhn< Gmab fs kjhtfhujusdy attakcme tj tcm mdastfk `juheatfjh frrmspmktfvm j` tcm djaes ahe gjuheary kjheft fjhs. Ucm agjvm assubptfjh fs hmkmssary tj emvmdjp ahadytfkad sjdutfjhs. F` gmab fs hjt attakcme kjhtfhujusdy tcmh tcm hmnatfvm vadums j` em`dmktfjh v at a pjfht fhefkatms tcm gmab ahe tcm `juheatfjh smparatms at tcat pjfht. Ft fs tcm rmspjhsfgfdfty j` tcm emsfnhmr tj mhsurm tcm batcmbatfkad bjemd aktuaddy sfbudatms tcm rmad df`m kjheftfjh ahe tcus scjude baim a emkfsfjh f` hmnatfvm vadums j` em`dmktfjh v arm akkmptagdm jr hjt.
=0
Fh`dumhkm @uhktfjhs
;
Uagdm ;.= @uheabmhtad sjdutfjhs vadums jh tcm gjuheary.. Dm`t sfem x 2 ‛ D 7 +
2 ‛ D 7
2 D 7
‛ D
Wfnct [fem x 2 D 7 ‛ 2 ‛ D 7
kjs D + sfh D
‛ D
kjs D + sfh D
N 2 -----7 i
N 2 ------ m 7 i
N1 2 ?
N 1 2 ----- m i
1 N 7 2 ‛ ------ 5
1 ‛ D N7 2 ------ m sfh D ‛ kjs D 5
1 ‛ D N 7 2 ------ m sfh D ‛ kjs D 5
1 N 7 2 ‛ ------ 5
1 N ; 2 ‛ -- 7
1 ‛ D N; 2 -- m kjs kjs D 7
1 ‛ D N ; 2 ‛ -- m kjs kjs D 7
1 N ; 2 -7
C 2 ?
C 2 ‛ ----- m i
7
;
7
sfh D
;
‛ D
1 ‛ D C 7 2 -- m kjs kjs D 7
‛ D
C ; 2 ‛ -- m 7
N1 2 ?
sfh D
‛ D
sfh D
C 2 ----- m i
;
sfh D ‛ kjs D
1 C 7 2 ‛ -- 7
;.=.1
7
sfh D
N 2 -----7 i
‛ D
N 1 2 ‛ ----- m i
‛ D
C 1 2 ‛ ----- m i
7
‛ D
C 1 2 ----i
C ; 2 ‛ -- 7
N 2 ------ m 7 i
2 D 7
‛ D
C 1 2 ‛ ----- m i
C 2 ? ;
sfh D ‛ kjs D
C 1 2
1 ‛ D C 7 2 ‛ -- m kjs kjs D 7
kjs D + sfh D
‛ D
C ; 2 ‛ -- m 7
----- i
1 C 7 2 -7
C ; 2 ‛ -- 7
kjs D + sfh D
[ybbmtrfk djaefhn
^m assubm tcat tcm gmab cas sybbmtrfk gjuheary kjheftfjhs ahe djaefhn, tcat fs p y x 2 p y ‛ x 2 p s x .^ftc tcmsm sfbpdf`fkatfjhs wm kjhkduem tcat W G 2 W A 2 W s
ahe
B G 2 ‛ B A 2 ‛ B s
(;.;1a)
[ugstftutfhn Mquatfjh (;.;1a) fhtj (;.;1a) fhtj Mquatfjhs (;.;?a) tcrjunc (;.;?a) tcrjunc (;.;?e) wm (;.;?e) wm jgtafh C x v x 2 W s N x ‛ D 7 + N x D 7 + B s C x ‛ D 7 ‛ C x D 7 +
v x x
D 7
‛ D 7
2 W s N 1 x ‛ D 7 + N 1 x D 7 + B s C 1 x ‛ D 7 ‛ C 1 x D 7 +
N x p s e
D 7
‛ D 7
B z x 2 W s N7 x ‛ D 7 + N 7 x D 7 + B s C 7 x ‛ D 7 ‛ C 7 x D 7 +
N 1 x p s e
(;.;7g)
N 7 x p s e
(;.;7k)
N ; x p s e
(;.;7e)
D 7
‛ D 7
P y x 2 W s N ; x ‛ D 7 + N ; x D 7 + B s C ; x ‛ D 7 ‛ C ; x D 7 +
D 7
‛ D 7
(;.;7a)
^m hjw hmme jhdy twj gjuheary kjheftfjhs at jhm j` tcm mhes tj jgtafh W jgtafh W s ahe B ahe B ;.3. s as embjhstratme fh Mxabpdm ;.3.
;.=.7
Asybbmtrfk djaefhn ahe gjuheary kjheftfjhs
^m assubm tcat tcm gmab cas asybbmtrfk gjuheary kjheftfjhs ahe tcm djaefhn, tcat fs, p y x 2 ‛ p y ‛ x 2 pa x . ^ftc tcmsm sfbpdf`fkatfjhs wm kjhkduem tcat W G 2 ‛ W A 2 ‛ W a
ahe
B G 2 B A 2 B a
(;.;7m)
[ugstftutfhn Mquatfjh (;.;7m) fhtj (;.;7m) fhtj Mquatfjhs (;.;?a) tcrjunc (;.;?a) tcrjunc (;.;?e) wm (;.;?e) wm jgtafh v x 2 W a N x ‛ D 7 ‛ N x D 7 + B a C x ‛ D 7 + C x D 7 +
v x x
D 7
‛ D 7
2 W a N 1 x ‛ D 7 ‛ N 1 x D 7 + B a C 1 x ‛ D 7 + C 1 x D 7 +
B z x 2 W a N 7 x ‛ D 7 ‛ N 7 x D 7 + B a C 7 x ‛ D 7 + C 7 x D 7 + P y x 2 W a N ; x ‛ D 7 ‛ N ; x D 7 + B a C ; x ‛ D 7 + C ; x D 7 +
N x pa x e
D 7
‛ D 7
N 1 x pa x e
(;.;;g)
N 7 x pa x e
(;.;;k)
N ; x pa x e
(;.;;e)
D 7
‛ D 7
D 7
‛ D 7
(;.;;a)
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
@fhftm gmabs 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
;.=.;
=>
Nmhmrad djaefhn ahe gjuheary kjheftfjhs
@fnurm ;.10 scjws ;.10 scjws cjw a gmab uhemr argftrary efstrfgutme djaes ahe gjuheary kjheftfjhs kah gm ahadyzme as a sub j` sybbmtrfk ahe ah asybbmtrfk prjgdmb. Ucm varfagdms `jr tcm sybbmtrfk ahe asybbmtrfk kah gm rmdatme tj tcm argftrary efstrfgutme djaefhn as scjwh gmdjw. p s x 2 p y x + p y ‛ x 7
_ s 2
pa x 2 p y x ‛ p y ‛ x 7
_ a 2
_ 1 + _ 7 7 _ 1 ‛ _ 7 7
B s 2 B 1 + B 7 7
(;.;5a)
B a 2 B 1 ‛ B 7 7
Fh wrftfhn tcm agjvm mquatfjh wm baem usm j` tcm `akt tcat ahy `uhktfjh kah gm wrfttmh as a sub j` a sybbmtrfk `uhktfjh ahe asybbmtrfk `uhktfjh as scjwh `jr tcm `jrkms _ `jrkms _ 1 ahe _ ahe _ 7 gmdjw. Fh a sfbfdar bahhmr jtcmr varfagdms k ah gm wrfttmh. _ 1 2 _ 1 + _ 7 7 + _ 1 ‛ _ 7 7 2 _ s + _ a _ 1
_ s
_ 7
_ s
p s x
p y x B 1
_ 7 2 _ 1 + _ 7 7 ‛ _ 1 ‛ _ 7 7 2 _ s ‛ _ a
ahe
_ a
B s
B 7
B a
B s
Nmhmrad
pa x
(;.;5g) _ a B a
Asybbmtrfk
[ybbmtrfk
@fnurm ;.10 Nmhmrad gjuheary djaes.
Ucm sjdutfjhs j` sybbmtrfk ahe asybbmtrfk prjgdmb arm `juhe as emskrfgme fh [mktfjhs ;.=.1ahe ;.=.1ahe ;.=.7. ;.=.7. Gy supmrpjsftfjh wm jgtafh tcm sjdutfjhs tj tcm nmhmrad prjgdmb. Fh jtcmr wjres W A 2 W s + W a
W G 2 W s ‛ W a
B A 2 B s + B a
B G 2 ‛ B s + B a
(;.;=)
[ugstftutfhn tcm agjvm fhtj Mquatfjhs (;.;?a) tcrjunc (;.;?a) tcrjunc (;.;?e) wm (;.;?e) wm jgtafh tcm kjbpdmtm sjdutfjh j` tcm jrfnfhad prjgdmb, wftc p wftc p y gmfhn tcm jrfnfhad efstrfgutme djae fh tcm mquatfjhs. ^m embjhstratm tcfs apprjakc fh tcm hmxt twj mxabpdms. Ucfs apprjakc wfdd gm j` nrmatmr usm f` wm krmatm a tagdm j` sjdutfjhs `jr sybbmtrfk ahe asybbmtrfk prjgdmbs.
MSAB_DM ;.3 A `fhftm gmab jh mdastfk `juheatfjh fs djaeme as scjwh fh @fnurm ;.1>. ;.1>. (a (a) Fh tmrbs j` M j` M , F , , i , ahe D ahe D emtmrbfhm emtmrbfhm tcm uhihjwh `jrkm W `jrkm W s ahe bjbmht B bjbmht B fh Mquatfjhs (;.;7a) tcrjunc (;.;7a) tcrjunc (;.;7e). (;.;7e) . (g ( g ) Baim a pdjt j` tcm em`dmktfjh, sdjpm, fhtmrhad gmhefhn s bjbmht, ahe fhtmrhad scmar `jrkm akrjss tcm gmab assubfhn i 2 7 isf , _ 2 1.= ifps ahe D 2 1? `t . 5 fh. fh.
_
D
_
A
x
G
0 fh. fh.
z y Krjss smktfjh
y @fnurm ;.1> @fhftm gmab uhemr mhe `jrkms. ‛ D
‛ D
5 1 + m kjs D AH[. W s 2 _ ------------------------------------------------------ ‛ 7 D ‛ D 1 ‛ m + 7 m sfh D
7 _ 1 ‛ m sfh D ‛ kjs D B s 2 ‛ ------- ----------------------------------------------------------- 1 ‛ m ‛ 7 D + 7 m ‛ D sfh D
(M1)
MSAB_DM ;.6 A `fhftm gmab jh mdastfk `juheatfjh fs djaeme as scjwh fh @fnurm ;.1>. ;.1>. (a) Fh tmrbs j` M j` M , F , , i , ahe D ahe D emtmrbfhm emtmrbfhm tcm uhihjwh `jrkms W `jrkms W A, W G ahe bjbmhts B bjbmhts B , B fh Mquatfjhs (;.;7a) tcrjunc (;.;7a) tcrjunc (;.;7e). (;.;7e) . (g) Baim a pdjt j` tcm em`dmktfjh, sdjpm, fhtmrhad A G gmhefhn bjbmht, ahe fhtmrhad scmar `jrkm akrjss tcm gmab assubfhn tcm gmab krjss-smktfjh ahe batmrfad prjpmrtfms j` tcm gmab ahe `juheatfjh arm tcm sabm as fh Mxabpdm ;.3. ;.3. 7 _
D
A
x
G
y ;.6. @fnurm ;.13 @fhftm gmab uhemr mhe `jrkms fh Mxabpdm ;.6.
AH[. W A 2 ‛ 1.01?0 ifps
W G 2 17.1>0; 17.1>0; ifps
B A 2 ;.3>70 `t-ifps
B G 2 7;.13> `t-ifps
=3
;.0
;
Fh`dumhkm @uhktfjhs
KDJ[]WM
Ucm kjhkmpt j` fh`dumhkm `uhktfjh baims ft pjssfgdm tj emvmdjp tmkchfqums gy wcfkc wm kah ahadyzm gmab em`jrbatfjh ahe strmssms `jr kjbpdmx djaefhn jr djaefhn jgtafhme mxpmrfbmhtaddy. Fh`dumhkm `uhktfjhs fhtrfhsfkaddy satfs`y tcm gjuheary kjheftfjhs. Cjwmvmr, gy rmpdakfhn suppjrts wftc apprjprfatm rmaktfjh `jrkms ahe / jr bjbmhts wm kah adsj ahadyzm gmabs wftc ef``mrmht gjuheary kjheftfjhs. ^m emvmdjpme tcm kjhkmpt j` `uheabmhtad sjdutfjhs gy mxtmhefhn tcm kjhkmpt j` fh`dumhkm `uhktfjhs tj fh`fhftm gmabs jh mdastfk `juheatfjhs. Ucm gjuheary kjheftfjhs hjw rmqufrme tcat ahy efsturgahkm eum tj `jrkm jr bjbmht bust efm jut at fh`fhfty. ^m ^m tcmh fhtmrprmtme tcm `f hftm gmab as part j` tcm fh`fhftm gmab wftc uhihjwh `fktftfjus `jrkm ahe bjbmht appdfme at tcm gjuheary j` tcm `fhftm gmab. ^m kjhsfemrme ahy djaefhn ahe gjuheary kjheftfjhs as a sub j` sybbmtrfk ahe asybbmtrfk prjgdmb, wcfkc cmdpme rmeukm tcm adnmgrafk tmefub tj tcm sjdutfjh j` twj uhihjwhs. Ucm uhihjwhs wmrm emtmrbfhme gy satfs`yfhn tcm gjuheary kjheftfjhs at jhm mhe j` tcm `fhftm gmab. ^m wmrm agdm tj `fhe ahadytfkad sjdutfjhs wcmh djaefhn was sfbpdm `jr gjtc tcm kdassfkad gmabs ahe gmabs jh mdastfk `juheatfjh. ^m usme hubmrfkad fhtmnratfjh wcmh djaefhn was kjbpdmx jr fh eata `jrb sukc as tcjsm tcat wfdd gm jgtafhme mxpmrfbmhtaddy. ^m ^m saw tcat `jr hubmrfkad fhtmnratfjh wm scjude hjh efbmhsfjhadfzm tcm varfagdms. ^m usme trapmzjfead rudm tj embjhstratm tcm kjhkmpt. F` cfncmr akkurakfms arm hmmeme tcmh tcm trapmzjfead rudm scjude gm rmpdakme gy Nauss Quaeraturm. Ucfs fs a stahe adjhm kcaptmr ahe ejms hjt fbpakt sugsmqumht kcaptmrs. Cjwmvmr, tcm kjhkmpts j` fh`dumhkm `uhktfjhs ahe `uheabmhtad sjdutfjhs arm usme mxtmhsfvmdy fh skfmhkm ahe mhnfhmmrfhn dftmraturm.
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
[yhjpsfs j` mquatfjhs 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
;.>
=6
[VHJ_[F[ J@ MQ]AUFJH[ Fh`dumhkm @uhktfjhs
vx 2
D
N x p y e
x
_jfht `jrkm e B z e x
P y x 2 ‛
2
D
C 7 x
C x
Hubmrfkad Fhtmnratfjh Fhtmnratfjh ` ? + ` H Mquaddy spakme F 2 ------------------------------- + gasm pjfhts 7 x 2 x D j
)
v 2 v MF p j D j
)
)
N 2 N MF D j
;
)
C 2 C MF D j
)
5
H ‛ 1
f 1
7
)
N 1 2 N 1 MF D j
C ; x 2 ‛
)
7
)
B 2 B p j D j )
)
P y 2 P y p j D j )
N 7 2 N 7 D j 2
)
)
7
)
N x
2
)
2 C ; x B
_ 2 _ pj D j
MF pj D j;
2
P y x 2 ‛
C 7 x
C ; x 2 ‛
)
e B z e x
e v 2 C 7 x B B z x 2 M F 7 e x
p y 2 p y p j )
N7 x
N ; x 2 ‛
D j
2
7
` f
N 7 x p y e 9
D
N 7 x 2 MF
C 7 x
C ; x 2 ‛
N1 x
N x
x 2 ev 2 C 1 x B e x
_jfht bjbmht
Hjh-efbmh Hjh-efbmhsfjhad varfagdms
N x p y e 9 N 1 x 2
v x 2 C x B C 1 x 2
7
ev N 1 x p y e 9 B z x 2 M F 2 7 D e x
ev 2 e x
2
)
C 1 2 C 1 MF D j 2 C x
)
N 1 x )
)
C 7 2 C 7 2 C 1 x
)
)
N 7 x
N ; 2 N; 2
)
)
)
C ; 2 C ; D j 2 e C 7 e x
)
)
Fh`dumhkm `uhktfjhs `jr kdassfkad gmabs ;
7
; 1 x x Nmhmrad sjdutfjh N x 2 --------- x ‛ + k 1 ----- + k 7 ----- + k ; x + k 5 0 MF 0 7
[fbpdy sup pjrtme gmab
;
1 ; D ‛ ; D ‛ N x 2 --------- x ‛ ‛ ----------------- x ‛ ------------------- x + D ‛ Dx 9 C x 2 0 MF D D Gmab jh mdastfk `juheatfjhs 9
N x
N
2 ‛
E 7 x 2
‛ x ‛
m
‛ x ‛ )
)
)
m
)
)
)
x ‛ ‛ kjs
)
)
)
)
x ‛
)
)
;
)
)
x ‛
)
)
sfh x ‛ ‛ kjs x ‛ i
kjs x ‛
+ sfh x ‛
7 )
‛ x ‛
)
sfh
‛ x ‛ )
)
x ‛
)
)
)
)
)
7
)
)
‛ x ‛ )
)
)
)
x ‛
)
)
7
)
C ; 2 ‛
)
)
sfh
sfh
)
)
5
)
7
x ‛
)
7
)
)
kjs
)
x ‛
)
)
x ‛ ‛ kjs
)
)
x ‛
)
)
)
5
)
‛ x ‛ )
)
m
v x 2 W A N x ‛ D 7 + B A C x ‛ D 7 + W G N x D 7 + B G C x D 7 +
)
)
C 1 2 ‛ m
)
kjs
‛ x ‛
)
N ; 2 ‛ snh x ‛ m )
)
5
)
N 1 2 ‛ snh x ‛ m
5
i
kjs x ‛ 7
)
)
3
)
‛ x ‛
C 7 2 ‛ snh x ‛ m @fhftm gmabs
‛ x ‛
sfh x ‛
)
)
sfh
C 2 snh x ‛ m )
N ; 2 ‛ snh x ‛ m
)
)
)
‛ x ‛
‛ x ‛
C ; 2 ‛ m )
7
N 1 2 ‛ snh x ‛ m ; ‛ x ‛
+ sfh x ‛
7
7 i
kjs x ‛ 7 )
kjs x ‛ ‛ sfh x ‛ 7
‛ x ‛
x ‛ + sfh
+ sfh x ‛
‛ x ‛
C 1 2 ‛ m
)
kjs x ‛
sfh x ‛ i
)
+ sfh x ‛
‛ kjs x ‛
7 ‛ x ‛
kjs
‛ x ‛
)
+ sfh x ‛
kjs x ‛
‛ x ‛
)
m
N7 2
‛ x ‛
sfh x ‛ ‛ kjs x ‛ 5
C 7 2 ‛ snh x ‛ m )
Hjh-efbmhsfjh Hjh-efbmhsfjhadfzme `jrb
kjs x ‛
C 2 snh x ‛ m
)
snh x ‛ m
kjs x ‛ e 2 snh x ‛ m
‛ x ‛
N 2
2
‛ x ‛
N7 2 m _jfht bjbmht
snh x ‛ m ‛
sfh x ‛ e 2 snh x ‛ m
7
_jfht `jrkm
x ‛
2
‛ x ‛
m L x 2 m L 1 x 2
N x 2
x
m ‛ x ‛ kjs x ‛ m ‛ x ‛ sfh x ‛
E 1 x 2 Fh`fhftm gmabs
x
1 ‛ 1
9 snh x ‛ 2
7
;
1 ; D ‛ 7 x ---------------- x ‛ Dx 2 --------- ‛ ; x ‛ + ----- + --------0 MF D D
N
kjs
‛ D 7
)
x ‛ + sfh
)
)
x ‛
)
)
7
N x p y e
At ; x 2 ‛ D 7 + N 2 7 i N1 2 ? N 7 2 ‛ 1 5 N ; 2 ‛ 1 7 9 C 2 ? C 1 2 i C 7 2 ‛ 1 7 C ; 2 ‛ 7 2 ‛ D 7 ‛ D 7 ‛ D ‛ D ‛ D At N 2 m N 1 2 m sfh D i N7 2 m N; 2 m kjs kjs D 7 kjs D + sfh D 7 i sfh D ‛ kjs D 5 x 2 ‛ D 7 + 7 ‛ D ; ‛ D ‛ D ‛ D C 2 ‛ m sfh D i C 1 2 ‛ m C 7 2 m kjs kjs D 7 C ; 2 ‛ m sfh D ‛ kjs D i kjs D + sfh D 7 2 D 7 ‛ D
At x 2 D 7 ‛
N 2
2 ‛ D 7 At x 2 D 7 ‛
C 2
N 2
7 i
2 D 7
m
kjs D + sfh D 7 i
7 ‛ D
m
sfh D i N1 2 ?
7 ‛ D
N 1 2 ‛ m ; ‛ D
C 1 2 ‛ m
sfh D i
sfh D ‛ kjs D i
N 7 2 ‛ 1 5
‛ D
N7 2 m
‛ D
C 7 2 ‛ m
N ; 2 1 7 9 C 2 ?
sfh D ‛ kjs D 5 kjs kjs D 7
C 1 2
; i
‛ D
N ; 2 ‛ m ‛ D
C ; 2 ‛ m C 7 2 1 7
kjs kjs D 7
kjs D + sfh D 7 C ; 2 ‛ 7
0?
Fh`dumhkm @uhktfjhs
;
;.3
BAUCMBAUFKAD EMUAFD[ J@ UCM EMWFPAUFJH J@ @]HEABMHUAD [JD]UFJH[
Fh tcfs smktfjh tcm emtafds j` tcm emrfvatfjh j` mquatfjhs fhtrjeukme fh [mktfjh ;.5 arm ;.5 arm prmsmhtme.
;.3.1
@uheabmhtad sjdutfjh `jr efspdakmbmht
@jr x ? , ahe assubfhn MF assubfhn MF fs fs a kjhstaht, Mquatfjh (;.13a) kah (;.13a) kah gm wrftm as 5
i + ------ v 2 ? 5 e x MF ev
(;.;0)
^m sugstftutm v 2 Am x tj jgtafh 5 i x A + ------ m 2 ? MF
i 5 + -----
jr
MF
2 ?
(;.;>a)
Ucm `jur rjjts j` Mquatfjh (;.;>a) arm (;.;>a) arm
1
2
1 + f
7
2 ‛ 1 + f
;
1 ‛ f
2
5
2 ‛ 1 ‛ f
(;.;>g)
wcmrm, f 2 ‛ 1 ahe fs nfvmh gy Mquatfjh (;.13k). (;.13k). Ucm ;re ahe 5tc rjjts arm kjbpdmx kjhlunatm rjjts 1 ahe 7. Ucus, tcm sjdutfjh j` Mquatfjh (;.;0) fs (;.;0) fs v 2 Wm A 1 m
1 + f x
‛ 1 + f x
+ A 7 m
(;.;3a)
wcmrm, A wcmrm, A1 ahe A ahe A7 arm kjbpdmx kjhstahts ahe Wm . fbpdfms tcm rmad part j` tcm tmrb fh tcm grakimts. Datmr jh wm wfdd adsj hmme tcm fbanfhary part j` tcm tmrb fh grakimt wcfkc fs sybgjdfzme gy Fb . . ^m hjtm tcat tcm emrfvatfvm wftc rmspmkt tj x fs x fs a rmad quahtfty ahe kah gm bjvme fhsfem tcm grakimt wftc rmspmkt tj tcm jpmratjrs Wm . ahe Fb . . ^m `urtcmr hjtm tcat ahy rmad varfagdm, sukc as ahe snh x ‛ , kah gm taimh jutsfem tcm jpmratjrs Wm . ahe Fb . fh sugsmqumht emrfvatfjhs. ^m tcus jgtafh
A 1 m 1 + f x + A 7 m ‛ 1 + f x x
v x
2 Wm
2 Wm 1 + f A 1 m
1 + f x
‛ 1 + f x
‛ A 7 m
(;.;3g)
@rjb tcm rmqufrmbmht tcat efspdakmbmhts ahe sdjpms bust gm gjuheme as x YMquatfjh (;.1>k)R (;.1>k)R wm kjhkduem A 1 2 ?
(;.;3k)
Ucm kjheftfjh tcat sdjpm bust gm zmrj at tcm jrfnfh YMquatfjh YMquatfjh (;.13a)R (;.13a)R yfmdes tcm `jddjwfhn Wm 1 + f A 7 2 ?
(;.;3e)
Ucm mqufdfgrfub kjheftfjh YMquatfjh YMquatfjh (;.1>m)R (;.1>m)R yfmdes 7
?
‛ 1 + f x
i Wm A 7 m
e x
‛ A7 ‛ 1 + f x 7 i A 7 1 ‛ f i 2 7 i Wm -------------------m 2 ------ Wm --------------------2 -- Wm A 7 1 ‛ f 2 _ ? 1 ‛ f 7 1 + f
(;.;3m)
^m rmprmsmht A 7 2 G 1 + fG 7 , wcmrm, G wcmrm, G1 ahe G ahe G7 arm rmad kjhstahts. [ugstftutfhn A [ugstftutfhn A7 fhtj Mquatfjhs (;.;3e) ahe (;.;3e) ahe (;.;3m) wm (;.;3m) wm jgtafh tcm `jddjwfhn
Wm 1 + f G 1 + fG 7 i Wm G 1 + fG 7 1 ‛ f
2 Wm G1 ‛ G 7 + f G 1 + G 7 2 ?
jr
2 i Wm G 1 + G 7 ‛ f G 1 ‛ G 7 2 _
G 1 ‛ G 7 2 ? jr
(;.;3`)
G 1 + G 7 2 _ i
(;.;3n)
[jdvfhn tcm agjvm twj mquatfjhs wm jgtafh G1 2 G7 2 _ 7 i
A 7 2 _ 1 + f 7 i
ahe
(;.;3c)
[ugstftutfhn Mquatfjhs (;.;3k) ahe (;.;3k) ahe (;.;3c) fhtj (;.;3c) fhtj Mquatfjh (;.;3a) wm (;.;3a) wm jgtafh tcm efspdakmbmht as v 2
_ ‛ 1 + f x ------ 7 i Wm 1 + f m
x?
(;.;6)
^m hjw baim usm j` tcm `jddjwfhn femhtftfms njvmrhfhn twj kjbpdmx `uhktfjhs ` `uhktfjhs ` ahe n ahe n Wm `n 2 Wm ` Wm n ‛ Fb ` Fb n
Fb `n 2 Wm ` Fb n + Fb ` Wm n
(;.5?a)
^m adsj hjtm tcat ‛ 1 + f x
m
‛ x
2 m
kjs x ‛ f sfh x
‛ 1 + f x
Wm m
‛ x
2 m
kjs x
‛ 1 + f x
Fb m
‛ x
2 ‛ m
sfh x
(;.5?g)
]sfhn tcm agjvm tcrmm mquatfjhs wm jgtafh `rjb Mquatfjh (;.;6) tcm (;.;6) tcm `jddjwfhn v 2
_ ‛ 1 + f x ------ ‛ Fb 1 + f Fb m ‛ 1 + f x 7 i Wm 1 + f Wm m
‛ x
2
_ ------ 7 i m
kjs x + sfh x
x?
(;.51)
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
Batcmbatfkad emtafds j` tcm emrfvatfjh j` `uhea(;.;6) gy taifhn tcm agsjdutm vadum j` x j` x ahe ahe mxtmhe jur sjdutfjh tj hmnatfvm vadums j` x j` x.^m .^m 1 ^m nmhmradfzm tcm sjdutfjh fh Mquatfjh (;.;6) gy trahsdatm tcm jrfnfh tj argftrary tj jgtafh 7 ; _ ‛ 1 + f x ‛ (;.57a) v 2 ------- Wm 1 + f m 5 7 i = 0 ;.3.7 @uheabmhtad sjdutfjhs `jr emrfvatfvms j` efspdakmbmht > 3 Ucm emrfvatfvms yfmde tcm `jddjwfhn mquatfjhs. 6 7 _ _ 7 ‛ 1 + f x ‛ 7 ‛ 1 + f x ‛ v 2 ------ Wm ‛ 1 + f snh x ‛ m 2 ‛ --------- snh x ‛ Wm 1 + f m jr 1? 7 i 7 i x 11 7 7 17 v _ _ ‛ 1 + f x ‛ ‛ 1 + f x ‛ -------- --------Wm 2 ‛ snh x ‛ 7 f m 2 ‛ ‛ Fb 7 f Fb m ‛ 1 + f x ‛ jr 1; x 7 i 7 i snh x ‛ Wm 7 f Wm m 15 v 2 ‛ _ 7 i snh x ‛ m ‛ x ‛ sfh x ‛ 2 N x _ 1= (;.5;) 1 x 10 1> 7 ; ; _ _ 7 ‛ 1 + f x ‛ ‛ 1 + f x ‛ v 2 --------13 -------- Wm Wm 1 ‛ f m (;.5;a) snh x ‛ 1 + f 7 f m 2 ‛ 7 7 i i 16 x 7? ; 5 5 71 _ 7 _ 7 ‛ 1 + f x ‛ ‛ 1 + f x ‛ v 2 -------- -----------(;.5;g) snh x ‛ Wm 1 ‛ f m snh x ‛ Wm m 2 77 ; i i x 7; 5 (;.5;a) ahe (;.5;g), (;.5;g), wm kah wrftm tcm mxprmssfjhs `jr bjbmht ahe scmar `jrkm as 75 Hjtfhn tcat MF 2 i 5 ahe Mquatfjhs (;.5;a) ahe 7 ; 7= v _ v _ ‛ 1 + f x ‛ ‛ 1 + f x ‛ 70 B z 2 MF 7 2 ‛ ------ Wm 1 ‛ f m P z 2 ‛ MF ; 2 ‛ --- snh x ‛ Wm m (;.5;k) 5 7 x x 7> 73 ]sfhn Mquatfjhs (;.5?a), (;.5?a), (;.5?a) ahe (;.5?a) ahe (;.5?g) wm (;.5?g) wm jgtafh mxprmssfjhs `jr bjbmht ahe scmar `jrkm as 76 ‛ x ‛ B z 2 ‛ _ 5 m (;.55) kjs x ‛ ‛ sfh x ‛ 2 N7 x _ ;? ‛ x ‛ ;1 P z 2 ‛ _ 7 snh x ‛ m (;.5=) kjs x ‛ 2 N ; x _ ;7 ;.5. ;; Ucm mxprmssfjhs `jr N 1 x , N 7 x , ahe N ; x agjvm rmpjrtme fh Uagdm ;.5. ;5 ;= ;.3.; Emrfvatfvms ahe fhtmnrads j` `uheabmhtad sjdutfjhs wftc rmspmkt tj sjurkm pjfht ;0 ]sfhn Mquatfjhs (;.7;) ahe (;.7;) ahe Mquatfjhs wm jgtafh< ;> ‛ 1 + f x ‛ ;3 ‛ 1 + f x ‛ m ‛ 1 + f x ‛ 2 ‛ m E x 2 (;.50) 2 1 + f snh x ‛ m ;6 x 5? 51 Mquatfhn tcm rmad ahe fbanfhary parts wm jgtafh tcm `jddjwfhn twj `jrbudas
m ‛ x ‛ kjs x ‛ 2 snh x ‛ m ‛ x ‛ kjs x ‛ + sfh x ‛ 57 E 1 x 2 (;.5>) 5; 55 m ‛ x ‛ sfh x ‛ 2 snh x ‛ m ‛ x ‛ ‛ kjs x ‛ + sfh x ‛ E 7 x 2 (;.53) 5= 50 5> Kjhsfemr tcm fhtmnrad ‛ 1 + f x ‛ 53 snh x ‛ ‛ 1 + f x ‛ snh x ‛ 1 ‛ f ‛ 1 + f x ‛ ‛ 1 + f x ‛ 1 ‛ f snh x ‛ m ---------------------------------------------------------------------e 2 ‛ -------------------------m 2 ‛ ----------------------------------------m 2 ‛ L x 2 m 56 1 + f 7 1 ‛ f7 =? Mquatfhn tcm rmad ahe fbanfhary parts wm jgtafh tcm `jddjwfhn twj `jrbudas =1 1 ‛ x ‛ ‛ x ‛ =7 sfh x ‛ e 2 ------ snh x ‛ m L 1 x 2 m (;.56) kjs x ‛ + sfh x ‛ 7 =; =5 1 ‛ x ‛ ‛ x ‛ L 7 x 2 m (;.=?) kjs x ‛ e 2 ------ snh x ‛ m kjs x ‛ ‛ sfh x ‛ == 7 =0 => =3 =6 0? 01
07
;
Fh`dumhkm @uhktfjhs 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
5
[tagfdfty
DMAWHFHN JGLMKUFPM 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5;
1. ]hemrstahe tcm kjhkmpt j` gukidfhn j` kjdubhs. [tagfdfty, adsj kaddme gukidfhn, fs ah fhstagfdfty j` mqufdfgrfub fh strukturms tcat jkkurs `rjb kjbprmssfvm djaes jr strmssms. A strukturm jr fts kjbpjhmhts bay `afd eum tj gukidfhn at djaes tcat arm `ar sbaddmr tcah tcjsm tcat prjeukm bat mrfad strmhntc `afdurm. Pmry j`tmh gukidfhn fs a katastrjpcfk `afdurm. Fhstagfdfty eum tj kjbprmssfvm fhpdahm strmssms j` tcfh pdatms ahe scmdds fs kaddme djkad gukidfhn. Fhstagfdfty j` a strukturad bmbgmr eum tj fhpdahm djaes fs kaddme strukturad gukidfhn. Jhm efbmhsfjhad strukturad bmbgmr tcat kah gmhe eum tj axfad djaes fs kaddme a kjdubh. Fh tcfs kcaptmr wm rmstrfkt jursmdvms tj stuey j` gukidfhn j` kjdubhs. Dmjhare Mudmr –––(1>?>‛1>37) was tcm `frst tj stuey gukidfhn j` pfh kjhhmktme kjdubhs ahe tcm `jrbuda cm emvmdjpme fs habme a`tmr cfb. Cfs wjri jh kjdubh gukidfhn was uhapprmkfatme eurfhn cfs tfbm gy praktfkfhn mhnfhmmrs as ft efe hjt akkuratmdy prmefkt `afdurm fh tcm strukturad bmbgmrs tcmh fh usm. Mhe m``mkts, fbpmr`mktfjhs, strmssms mxkmmefhn yfmde strmss, scjrtmr kjdubh dmhntcs, arm sjbm j` tcm kjbpdfkatfjhs tcat cavm gmmh aeeme tj Mudmr‖s wjri tj fbprjvm akkuraky j` prmefktfjhs ahe wm scadd smm tcmsm fh tcfs kcaptmr. @jr add tcm rm`fhmbmhts, tcm Mudmr gukidfhn `jrbuda fs stfdd usme tcrmm kmhturfms datmr `jr kjdubh emsfnh ahe fs stfdd vadfe `jr djhn kjdubhs wftc pfh-suppjrtme mhes. [ukc fs tcm pjwmr j` djnfkad tcfhifhn.
5.1
M]DMW G]KIDFHN
@fnurm 5.1a 5.1a scjws a sfbpdy suppjrtme kjdubh tcat fs axfaddy djaeme wftc a `jrkm _ `jrkm _ . ^m scadd fhftfaddy assubm tcat gmhefhn fs agjut tcm z tcm z axfs axfs as jur mquatfjhs jh gmab em `dmktfjh wmrm emvmdjpme wftc lust tcfs assubptfjh. ^m scadd rmdax tcfs assubptfjh at tcm mhe tj nmhmratm tcm `jrbuda `jr a krftfkad gukidfhn djae. y B z H 2 _
x
_
v( x) x)
_
A
A
D
(a)
(g)
@fnurm 5.1 [fbpdy suppjrtme kjdubh.
Dmt tcm gmhefhn em`dmktfjh at ahy djkatfjh x djkatfjh x gm gm nfvmh gy v( x), x), as scjwh fh @fnurm 5.1g 5.1g. Ah fbanfhary kut fs baem at sjbm djkatfjh x djkatfjh x,, ahe tcm fhtmrhad gmhefhn bjbmht fs erawh akkjrefhn tj jur sfnh kjhvmhtfjh. Ucm fhtmrhad axfad `jrkm H wfdd wfdd gm mquad tj _ . Gy gadahkfhn tcm bjbmht at pjfht A A wm jgtafh B z + _ v ?. [ugstftutfhn tcm bjbmht‛kurvaturm rmdatfjhscfp 7
B z 2 MF zz e v ex
7
, wm jgtafh tcm ef``mrmhtfad mquatfjh< 7
e v MF zz -------- + _ v 2 ? 7 ex
(5.1a)
F` gukidfhn kah jkkur agjut ahy axfs a he hjt lust tcm z tcm z axfs, axfs, as wm fhftfaddy assubme, tcmh t cm sugskrfpts zz sugskrfpts zz fh fh tcm arma bjbmht j` fhmrtfa scjude gm erjppme. Ucm gjuheary vadum prjgdmb kah gm wrfttmh as scjwh gy Mquatfjhs (5.1g) ahe (5.1g) ahe (5.1k). (5.1k).
07
0;
5
[tagfdfty 7
Ef``mrmhtfad Mquatfjh
7 e v -------- + v 2 ? 7 ex
Gjuheary Kjheftfjhs
v ? 2 ?
wc m rm
2
_ ----- MF
(5.1g)
v D 2 ?
(5.1k)
Ucm sjdutfjh tj tcm ef``mrmhtfad mquatfjh, fs v x 2 A kjs x + G sfh x
(5.7)
@rjb tcm gjuheary kjheftfjhs wm jgtafh twj mquatfjhs wcfkc fh batrfx `jrb kah gm wrfttmh as
A ? G 2 ?
?
1 kjs D
sfh D
(5.;)
@jr a hjhtrfvfad sjdutfjh―tcat fs, wcmh A wcmh A ahe ahe G G arm arm hjt gjtc zmrj―tcm kjheftfjh fs tcat tcm emtmrbfhaht j` tcm batrfx bust gm zmrj. Ucfs yfmde sfh D 2 ?
(5.5)
Agjvm mquatfjh fs tcm kcaraktmrfstfk mquatfjh, jr tcm gukidfhn mquatfjh. mquatfjh. Ucm rjjts j` tcm kcaraktmrfstfk mquatfjh arm
D
2 h . [ugstftutfhn `jr `jr ahe ahe sjdvfhn `jr _ `jr _ , wm jgtafh 7
7
h MF _ h 2 ----------------, 7 D
h 2 1 7 ;
(5.=)
Mquatfjh (5.=) rmprmsmhts (5.=) rmprmsmhts tcm vadums j` djae _ djae _ (tcm (tcm mfnmhvadums) at wcfkc gukidfhn wjude jkkur. ^cat fs tcm djwmst vadum j` _ j` _ at at wcfkc gukidfhn wfdd jkkur: Kdmardy, `jr tcm djwmst vadum j` _ j` _ , h scjude mquad 1 fh Mquatfjh (5.=). (5.=). @urtcmrbjrm bfhfbub vadum j` F scjude gm usme. usme. Ucm krftfkad gukidfhn djae fs 7
MF _ kr 2 ----------7 D _ kr , tcm krftfkad gukidfhn djae, fs adsj kaddme Mudmr djae.
(5.0)
• Gukidfhn w fdd jkkur agjut tcm axfs tcat cas bfhfbub arma bjbmht j` fhmrtfa. fhmrtfa.
5.1.1
Gukidfhn bjems
@rjb Mquatfjh (5.;) wm (5.;) wm smm tcat A tcat A 2 2 ?. Ucm sjdutfjh `jr v kah gm wrfttmh as x v 2 G sfh h -- D
(5.>)
Mquatfjh (5.>) rmprmsmhts (5.>) rmprmsmhts tcm gukidme bjem (mfnmhvmktjrs). Ucm fbpjrtahkm j` makc gukidme bjem scapm kah gm apprmkfatme gy mxabfhfhn @fnurm 5.7. 5.7. F` gukidme bjem 1 fs prmvmhtme `rjb jkkurrfhn gy fhstaddfhn a rmstrafht (jr suppjrt), t cmh tcm kjdubh wjude gukidm at tcm hmxt cfncmr bjem at krftfkad djae vadums tcat arm cfncmr tcah tcjsm `jr tcm djwmr bje ms. _jfht F _jfht F jh jh tcm em`dmktfjh kurvms emskrfgfhn tcm bjem scapms cas twj attrfgutms< ft fs ah fh`dmktfjh pjfht ahe tcm banhftuem j` em`dmktfjh at tcfs pjfht fs zmrj. Wmkadd tcat tcm kurvaturm e 7 v/ex v/ex7 at ah fh`dmktfjh pjfht fs zmrj. Cmhkm tcm fhtmrhad bjbmht B z at tcfs pjfht fs zmrj. F` rjddmr sup pjrts arm put at ahy jtcmr pjfhts tcah tcm fh`dmktfjh pjfhts F , as prmefktme gy Mquatfjh (5.>), (5.>), tcmh tcm gjuheary-vadum prjgdmb (smm _rjgdmb 5.6) 5.6) wfdd cavm ef``mrmht mfnmhvadums (krftfkad djaes) ahe mfnmhvmktjrs (bjem scapms).
Bjem scapm 1
Bjem scapm 7
_ kr
_ kr
D
Bjem scapm ;
D/7
D/7
D/; D/; 7
MF _ kr 2 ----------7 D
_ kr
7
5 MF _ kr 2 --------------7 D
D/;
D/; 7
6 MF _ kr 2 --------------7 D
@fnurm 5.7 Fbpjrtahkm j` gukidme bjems.
Fh bahy sftuatfjhs ft bay hjt gm pjssfgdm tj put rjddmr suppjrts fh jremr tj kcahnm a bjem tj a cfncmr krftfkad gukidfhn djae. Gut gukidfhn bjems ahe gukidfhn djaes kah adsj gm kcahnme gy usfhn mdastfk suppjrts as embjhstratm fh Mxabpdm 5.1. 5.1. ^atmr tahis tcat suppdy watmr tj a kjbbuhfty cas kjdubhs wftc mdastfk rfhn suppjrts tj fhkrmasm tcm krftfkad gukidfhn djae.
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
Mudmr gukidfhn 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
5.1.7
05
M``mkts j` mhe kjheftfjhs
Mquatfjh (5.0) fs (5.0) fs appdfkagdm jhdy tj sfbpdy suppjrtme kjdubhs. Cjwmvmr, tcm prjkmss usme tj jgtafh tcm `jrbuda kah gm usme `jr jtcmr typms j` suppjrts. Uagdm 5.1 scjws 5.1 scjws tcm krftfkad mdmbmhts fh tcm emrfvatfjh prjkmss ahe tcm rmsudts `jr tcrmm jtcmr suppjrts. Ucm `jrbuda `jr krftfkad djaes `jr add kasms scjwh fh Uagdm 5.1 kah 5.1 kah gm wrfttmh as _ kr 2
7 MF Dm`7 `
(5.3)
wcmrm D wcmrm Dm`` fs tcm m``mktfvm dmhntc j` tcm kjdubh. Ucm m``mktfvm dmhntc `jr makc kasm fs nfvmh fh tcm dast rjw j`Uagdm j` Uagdm 5.1. 5.1. Ucfs em`fhftfjh j` m``mktfvm dmhntc wfdd pmrbft us tj mxtmhe rmsudts tcat wfdd gm emrfvme fh [mktfjh 5.7 `jr 5.7 `jr sfbpdy suppjrtme fbpmr`mkt kjdubhs tj fbpmr`mkt kjdubhs wftc tcm suppjrts scjwh fh kasms 7 tcrjunc 5 fh Uagdm 5.1. 5.1. Uagdm 5.1
Gukidfhn j` kjdubhs wftc ef``mrmht suppjrts Kasm ; Kasm 1
Kasm 7
_
D
7
e v MF -------- + _ v 2 ? 7 ex v? 2 ? vD 2 ?
y
7
e v MF -------- + _ v 2 _ v D 7 ex ev ? 2 ? e x
v ? 2 ?
Jhm mhe `fxme, jtcmr mhe pfhhme 7
e v MF -------- + _ v 2 W G D ‛ x 7 ex v ? 2 ?
ev ? 2 ? e x
vD 2 ?
sfh D 2 ?
x
y
Jhm mhe `fxme, jtcmr mhe `rmm
_fhhme at gjtc mhes
D
x
y
y
Kcaraktmrfstfk mquatfjh
_
D
x
x
Gjuheary kjheftfjhs
_
_
D
Ef``mrmhtfad mquatfjh
Kasm 5a
kjs D 2 ?
@fxme at gjtc mhes 7
e v MF -------- + _ v 2 W G D ‛ x + B G 7 ex v ? 2 ?
ev ? 2 ? e x
vD 2 ?
ev D 2 ? e x
7 1 ‛ kjs D ‛ D sfh D 2 ?
tah D 2 D
_ MF
2
Krftfkad djae _ kr
7
MF -----------7 D
M``mktfvm dmhntc D dmhntc Dm`` D a.
W G ahe B ahe B G arm tcm `jrkm ahe bjbmht rmaktfjhs.
5.1.;
Kdassf`fkatfjh j` kjdubhs
7
7
MF MF ------------ 2 -------------7 5 D 7 D 7
7 D
7
7?.1; MF MF --------------------- 2 -----------------7 D ?.> D 7
?.> D
7
7
5 MF MF --------------- 2 -----------------7 D ?.= D 7
?.= D
Fh Mquatfjh (5.0), (5.0), F kah kah gm rmpdakme gy Ar gy Ar 7, wcmrm A wcmrm A fs fs tcm krjss-smktfjhad arma ahe r fs fs tcm bfhfbub raefus j` nyratfjh. ^m jgtafh 7 _ kr M kr 2 ------2 -------------------- A D m` ` r 7
(5.6)
wcmrm D wcmrm Dm`` /r fs tcm sdmhemrhmss ratfj ahe kr fs tcm kjbprmssfvm axfad strmss lust gm`jrm tcm kjdubh wjude gukidm. Mquatfjh (5.6) fs (5.6) fs vadfe jhdy fh tcm mdastfk rmnfjh―tcat fs, f` kr 4 yfmde . F` kr 8 yfmde , tcmh mdastfk `afdurm wfdd gm eum tj strmss mxkmmefhn tcm batmrfad strmhntc. Ucus kr 5.; scjws tcm `afdurm yfmde em`fhms tcm `afdurm mhvmdjpm `jr a kjdubh. @fnurm 5.; scjws mhvmdjpms `jr stmmd, adubfhub, ahe wjje. As hjhefbmhsfjhad varfagdms arm usme fh tcm pdjts fh @fnurm 5.;, 5.;, tcmsm pdjts kah adsj gm usme `jr bmtrfk uhfts. Hjtm tcat tcm sdmhemrhmss ratfj fs em`fhme usfhn m``mktfvm dmhntcs9 cmhkm tcmsm pdjts arm appdfkagdm tj kjdubhs wftc ef``mrmht suppjrts. Ucm `afdurm mhvmdjpms fh @fnurm 5.; scjw 5.; scjw tcat as tcm sdmhemrhmss ratfj fhkrmasms, tcm `afdurm eum tj gukidfhn wfdd jkkur at strmss vadums sfnhf`fkahtdy djwmr tcah tcm yfmde strmss. strmss. Ucfs uhemrskjrms tcm fbpjrtahkm j` gukidfhn fh tcm emsfnh j` bmbgmrs uhemr kjbprmssfjh. Ucm `afdurm mhvmdjpms, as scjwh fh @fnurm 5.;, 5.;, empmhe jhdy jh tcm batmrfad prjpmrty ahe arm appdfkagdm tj kjdubhs j` ef``mrmht dmhntcs, scapms, ahe typms j` suppjrt. Ucmsm `afdurm mhvmdjpms arm usme `jr kdassf`yfhn kjdubhs as scjrt jr djhn. [cjrt kjdubh emsfnh fs gasme jh usfhn yfmde strmss as tcm `afdurm strmss. Djhn kjdubh emsfnh fs gasme jh usfhn krftfkad gukidfhn strmss as tcm `afdurm strmss. Ucm sdmhemrhmss ratfj at pjfht A pjfht A `jr `jr makc batmrfad fs usme `jr smparatfhn scjrt kjdubhs `rjb djhn kjdubhs `jr tcat
0=
5
[tagfdfty
batmrfad. _jfht A _jfht A fs fs tcm fhtmrsmktfjh pjfht j` tcm strafnct dfhm rmprmsmhtfhn mdastfk batmrfad `afdurm ahe tcm cypmrgjda kurvm rmprmsmhtfhn gukidfhn `afdurm.
1 7 ; 5 = A A A 0 > 3 6 1? 11 17 1; 15 1= @fnurm 5.; @afdurm mhvmdjpms `jr Mudmr kjdubhs. Fhtmrbmefatm kjdubh fs a tcfre kdassf`fkatfjh usme f` tcm krftfkad strmss fs gmtwmmh yfmde strmss ahe udtfbatm strmss. Ucm tahnmht 10 1> bjeudus tcmjry j` gukidfhn akkjuhts `jr ft gy rmpdakfhn tcm bjeudus j` mdastfkfty gy tcm tahnmht bjeudus j` mdastfkfty, tcat fs, 13 7 M t F _ kr 2 ------------(5.1?) 16 7 7? D m`` 71 wcmrm M wcmrm M t fs tcm tahnmht bjeudus, wcfkc empmhes jh tcm strmss dmvmd _ dmvmd _ kr / A. A. ]sfhn ah ftmratfvm prjkmeurm ahe Mquatfjh (5.1?), (5.1?), tcm 77 krftfkad gukidfhn djae kah gm emtmrbfhme. 7; MSAB_DM 5.1 75 7= A dfhmar sprfhn fs attakcme at tcm `rmm mhe j` a kjdubh, as scjwh fh @fnurm 5.5. 5.5. Assubm tcat gmhefhn agjut tcm y tcm y axfs axfs fs prmvmhtme. 70 (a) Emtmrbfhm tcm kcaraktmrfstfk mquatfjh `jr tcfs gukidfhn prjgdmb. [cjw tcat tcm krftfkad djae _ djae _ kr `jr (g) i ? ahe (k) i fs as 7> nfvmh fh Uagdm 5.1 `jr 5.1 `jr kasms 7 ahe ;, rmspmktfvmdy. 73 y 76 D x _ ;? ;1 i ;7 ;; @fnurm 5.5 Kjdubh wftc mdastfk suppjrt fh Mxabpdm 5.1 ;5 i ---iD AH[. -------- ‛ 1 kjs D ‛ ---------- sfh D 2 ? ;= ; 7 MF MF ;0 ;> MSAB_DM 5.7 ;3 ;6 Emtmrbfhm tcm baxfbub em`dmktfjh j` tcm kjdubh scjwh fh @fnurm 5.= fh 5.= fh tmrbs j` tcm bjeudus j` mdastfkfty M mdastfkfty M , tcm dmhntc j` tcm 5? kjdubh D kjdubh D,, tcm arma bjbmht j` fhmrtfa F fhmrtfa F , tcm axfad `jrkm _ `jrkm _ , ahe tcm fhtmhsfty j` tcm efstrfgutme `jrkm w. y 51 w 57 x _ 5; A 55 D 5= 5.7. @fnurm 5.= Gukidfhn j` gmab wftc efstrfgutme djae fh Mxabpdm 5.7. 50 7 5> D _ wM F wD AH[. v bax 2 ‛ ---------- smk -- ------ ‛ 1 + --------7 53 7 MF 3 _ _ 56 =? MSAB_DM 5.; =1 Fh tmrbs j` tcm tcm tcmrbad kjm``fkfmht j` mxpahsfjh, r fs fs tcm raefus j` nyratfjh, ahe dmhntc D, emtmrbfhm tcm krftfkad kcahnm j` =7 tmbpmraturm at wcfkc tcm gmab scjwh fh @fnurms 5.0 wfdd 5.0 wfdd gukidm. Assubm a uhf`jrb fhkrmasm j` tmbpmraturm. =; y =5 == x =0 => D =3 5.;. @fnurm 5.0 Kjdubh gukidfhn `rjb tmbpmraturm kcahnm fh Mxabpdm 5.;. =6 7 7 0? AH[. U krft 2 D r 01
Fbpmr`mkt kjdubhs 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
5.7
00
FB_MW@MKU KJD]BH[
Fh tcm emvmdjpbmht j` tcm tcmjry `jr axfad bmbgmrs ahe tcm sybbmtrfk gmhefhn j` gmabs, wm jgtafhme tcat tcm kjheftfjh `jr emkjupdfhn axfad em`jrbatfjh `rjb gmhefhn em`jrbatfjh `jr dfhmar, mdastfk, ahe cjbjnmhmjus batmrfad< tcm appdfme djaes bust pass tcrjunc tcm kmhtrjfe j` tcm krjss smktfjhs, ahe tcm kmhtrjfes j` add krjss smktfjhs arm jh a strafnct dfhm. Cjwmvmr, tcm rmqufrmbmhts `jr emkjupdfhn tcm axfad `rjb tcm gmhefhn prjgdmb bay hjt gm bmt `jr a hubgmr j` rmasjhs, sjbm j` wcfkc arm nfvmh cmrm< • Ucm kjdubh batmrfad batmrfad bay kjhtafh sbadd cjdms, bfhutm bfhutm krakis, jr jtcmr batmrfad fhkdusfjhs fhkdusfjhs.. Cmhkm tcm cjbjnmhmfty rmqufrmbmht jr tcm rmqufrmbmht tcat tcm kmhtrjfes j` add krjss smktfjhs gm jh a strafnct dfhm bay hjt gm bmt. • Ucm batmrfad batmrfad prjkmssfhn bay kausm kausm djkad strafh caremhfhn. caremhfhn. Cmhkm tcm kjheftfjh kjheftfjh j` dfhmar ahe mdastfk batmrfad batmrfad gmcavfjr akrjss tcm mhtfrm krjss smktfjh bay hjt gm bmt. • Ucm tcmjrmtfkad tcmjrmtfkad emsfnh emsfnh kmhtrjfe ahe tcm aktuad aktuad kmhtrjfe arm j``smt j``smt eum tj bahu`akturfhn bahu`akturfhn tjdmrahkms. tjdmrahkms. • Djkad kjheftfjhs kjheftfjhs at tcm tcm suppjrt kausm kausm tcm rmaktfjh rmaktfjh `jrkm tj gm j``smt j``smt `rjb tcm kmhtrjfe. kmhtrjfe. • Ucm trahs`mr trahs`mr j` djaes `rjb jhm bmbgmr bmbgmr tj ahjtcmr ahjtcmr bay hjt jkkur at tcm kmhtrjfe. kmhtrjfe. Ucm agjvm partfad dfst kah gm kjhsfemrme as fbpmr`mktfjhs fh tcm kjdubh, wcfkc kausm tcm appdfkatfjh j` axfad djaes tj gm j``smt `rjb tcm kmhtrjfe j` tcm krjss smktfjh. Ucfs j``smt djaefhn fs tmrbme mkkmhtrfk djaefhn jh kjdubhs. Fh tcfs smktfjh wm stuey tcm fbpakt j` mkkmhtrfkfty fh djaefhn jh gukidfhn. y
B z H 2 _
x
v( x) x) m
_
m
_ A
A D (g)
(a)
@fnurm 5.> Mkkmhtrfkaddy djaeme kjdubh.
@fnurm 5.>a 5.>a scjws a sfbpdy suppjrtme kjdubh jh wcfkc ah mkkmhtrfk kjbprmssfvm axfad djae fs appdfme at a efstahkm m `rjb tcm kmhtrjfe j` tcm krjss smktfjh. @fnurm 5.>g 5.>g scjws tcm `rmm-gjey efanrab j` tcm kjdubh smnbmht. Gy gadahkfhn tcm bjbmht at pjfht A pjfht A wm wm jgtafh B + _ (v (v + m) ?. [ugstftutfhn tcm bjbmht‛kurvaturm rmdatfjhscfp wm jgtafh tcm ef``mrmhtfad mquatfjh z + _ 7
_m 7 e v -------- + v 2 ‛ ------ 7 MF ex
(5.11)
wcmrm fs fs nfvmh gy Mquatfjh (5.1g). (5.1g). Ucm gjuheary kjheftfjhs arm tcat efspdakmbmhts at x at x ? ahe x ahe x D arm D arm zmrj, as nfvmh gy Mquatfjh (5.1k) ahe (5.1k) ahe (5.1k (5.1k). ). Ucm cjbjnmhmjus sjdutfjh tj Mquatfjh (5.11 (5.11)) fs nfvmh gy Mquatfjh (5.7). (5.7). Ucm partfkudar sjdutfjh tj Mquatfjh (5.11) fs (5.11) fs v _ ( x) x) ‛ m. Ucus tcm tjtad sjdutfjh fs v x 2 v C + v _ 2 A kjs x + G sfh x ‛ m
(5.17)
@rjb gjuheary kjheftfjhs wm jgtafh v ? 2 A kjs ? + G sfh ? ‛ m 2 ? v D 2 A kjs D + G sfh D ‛ m 2 ?
jr
A 2 m
(5.1;)
7
jr
m 1 ‛ kjs D m 7 sfh D 7 G 2 --------------------------------- 2 ----------------------------------------------------------- 2 m tah D 7 sfh D 7 sfh D 7 kjs D 7
(5.15)
[ugstftutfhn `jr A `jr A ahe ahe G G fh fh Mquatfjh (5.17), (5.17), wm jgtafh tcm em`dmktfjh as v x 2 m kjs x + tah D 7 sfh x ‛ 1
(5.1=)
As D/7 D/7 /7, /7, tcm `uhktfjh tah ( ( D/7) D/7) ahe tcm efspdakmbmht `uhktfjh v( x) x) gmkjbms uhgjuheme. Ucus tcm krftfkad djae vadum kah gm `juhe gy sugstftutfhn `jr fh fh tcm mquatfjh D/7 D/7 /7 /7 tj jgtafh tcm sabm krftfkad vadum as nfvmh gy Mquatfjh (5.0). (5.0). Fh jtcmr wjres, tcm gukidfhn djae vadum ejms hjt kcahnm wftc tcm mkkmhtrfkfty j` tcm djaefhn . ^m wfdd baim usm j` tcfs jgsmrvatfjh tj mxtmhe jur `jrbudas tj jtcmr typms j` suppjrt kjheftfjhs. Fh tcm mfnmhvadum apprjakc efskussme fh [mktfjh 5.1, 5.1, wm wmrm uhagdm tj emtmrbfhm tcm efspdakmbmht `uhktfjh gmkausm wm cae ah uhemtmrbfhme kjhstaht G kjhstaht G fh fh Mquatfjh (5.>). (5.>). Gut cmrm tcm efspdakmbmht `uhktfjh fs kjbpdmtmdy emtmrbfhme gy Mquatfjh (5.1=). (5.1=). Ucm baxfbub em`dmktfjh (gy sybbmtry) wfdd gm at tcm bfepjfht. [ugstftutfhn x D/7 D/7 fhtj Mquatfjh (5.1=), (5.1=), wm jgtafh v bax 2 m kjs D 7 + tah D 7 sfh D 7 ‛ 1 2 m smk D 7 ‛ 1 [ugstftutfhn `jr `rjb `rjb Mquatfjh (5.1g), (5.1g), wm jgtafh D _ v ba x 2 m smk -- ------ ‛ 1 7 MF
D __ kr 2 m smk -- ------------ 7 _ kr MF ‛ 1 jr
(5.10a)
0>
5
[tagfdfty _ v ba x 2 m smk -- ------ ‛ 1 7 _ kr
(5.1>)
Ucm baxfbub hjrbad strmss fs tcm sub j` kjbprmssfvm axfad strmss ahe baxfbub kjbprmssfvm gmhefhn strmss< _ B ba x yba x ba x 2 --- + ---------------------- A F
(5.13a)
Ucm baxfbub gmhefhn bjbmht wfdd gm at tcm bfepjfht j` tcm kjdubh, ahe fts vadum fs B bax _ (m + vbax). [ugstftutfhn `jr vbax wm jgtafh D _ _ _y bax bax 2 --- + ------------- m smk -- ------ 7 MF F A
(5.13g)
Mquatfjh (5.13g) was (5.13g) was emrfvme `jr sfbpdy suppjrtme kjdubhs. ^m kah mxtmhe tcm rmsudts tj jtcmr suppjrts gy kcahnfhn tcm dmhntc j` tcm kjdubh tj tcm m``mktfvm dmhntc D dmhntc Dm`` , as nfvmh fh Uagdm 5.1. 5.1. ^m adsj sugstftutm y sugstftutm ybax k, wcmrm k rmprmsmhts tcm baxfbub efstahkm `rjb tcm gukidfhn (gmhefhn) axfs t j a pjfht jh tcm krjss smktfjh. [ugstftutfhn F [ugstftutfhn F Ar 7, wcmrm A wcmrm A fs fs tcm krjss-smktfjhad arma ahe r fs fs tcm raefus j` nyratfjh, wm jgtafh D m` ` _ _ mk ba x 2 --- 1 + -----7 smk -------- ------- A 7 r MA r
(5.16)
Mquatfjh (5.16) fs (5.16) fs kaddme tcm smkaht tcm smkaht `jrbuda. Ucm quahtfty mk/r quahtfty mk/r 7 fs kaddme tcm mkkmhtrfkfty ratfj. ratfj. Gy mquatfhn bax tj `afdurm strmss `afd fh Mquatfjh (5.16), (5.16), wm jgtafh tcm `afdurm mhvmdjpm `jr ah fbpmr`mkt kjdubh. Ucm `afdurm mhvmdjpm mquatfjh kah gm wrfttmh fh hjhefbmhsfjhad `jrb as
D m` ` _ / A A mk ---------- 1 + ----- smk -------7 `a fd r 7 r
`a fd _ A --------- M ---------- `a fd
2 1
[tmmd ?.?
^jje yf md e
?.?
------------- 2 ?.??1 ?.?? 1 M
ud t
-------- 2 ?.??73 M
?.1
?.1
?.7
?.7
?.5
?.5 ?.0 ?.3
(5.7?)
?.0 ?.3
Mudmr Gukidfhn Gukidfhn Kurvm
1.?
1.?
mk ----7 r
mk ----7 r
Adubfhub ?.?
yf md e
------------- 2 ?.??5 ?.?? 5 M
?.1 ?.7 ?.5 ?.0 ?.3 1.?
mk ----7 r
@fnurm 5.3 @afdurm mhvmdjpms `jr fbpmr`mkt kjdubhs.
Mudmr Gukidfhn Gukidfhn Kurvm
Mudmr Gukidfhn Gukidfhn Kurvm
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
Kdjsurm 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
03
Mquatfjh (5.7?) kah (5.7?) kah gm pdjttme `jr ef``mrmht batmrfads, as scjwh fh @fnurm 5.3. 5.3. Ucmsm kurvms kah gm usme `jr bmtrfk as wmdd `jr ].[. kustjbary uhfts, sfhkm tcm varfagdms usme fh krmatfhn tcm pdjts arm hjhefbmhsfjhad. Ucm kurvms kah gm usme `jr ahy batmrfad tcat cas tcm sabm vadum `jr `jr yfmde / M M . Ucm `afdurm strmss fh tcm kasms j` stmmd ahe adubfhub wjude gm tcm yfmde strmss yfmde, wcmrmas `jr wjje ft wjude gm tcm udtfbatm strmss udt . Ucm kurvms kah adsj gm usme `jr ef``mrmht mhe kjheftfjhs gy usfhn tcm apprjprfatm Dm`` as nfvmh fh Uagdm 5.1. 5.1.
MSAB_DM 5.5 A wjjemh gjx kjdubh ( M 13?? isf) fs kjhstruktme gy ljfhfhn `jur pfmkms j` dubgmr tjnmtcmr, as scjwh fh @fnurm 5.6. 5.6. Ucm djae _ djae _ 3? ifps fs appdfme at a efstahkm j` m 2 ?.00> fh. `rjb tcm kmhtrjfe j` tcm krjss smktfjh. (a) F` tcm dmhntc fs D 1? `t, wcat arm tcm baxfbub strmss ahe tcm baxfbub em`dmktfjh: em`dmktfjh: (g) F` tcm addjwagdm strmss fs ; isf, wcat fs tcm baxfbub pmrbfssfgdm dmhntc D dmhntc D tj tj tcm hmarmst fhkc: y 5 fh. fh. x m
_
5 fh. fh.
3 fh. fh.
A D
3 fh. fh.
5.5. @fnurm 5.6 Mkkmhtrfkaddy djaeme gjx kjdubh fh Mxabpdm 5.5.
AH[.
vbax ?.71 fh.9 bax 7.= isf (K)9 D (K)9 D 1>> fh.
MSAB_DM 5.= A wjjemh gjx kjdubh ( M M 13?? isf) fs kjhstruktme gy ljfhfhn `jur pfm kms j` dubgmr tjnmtcmr, as scjwh fh@fnurm fh @fnurm 5.6. 5.6. Ucm udtfbatm strmss fs = isf. Emtmrbfhm tcm baxfbub djae _ djae _ tcat tcat kah gm appdfme. AH[. _ bax 155 ifps
5.;
KDJ[]WM
Gukidfhn j` kjdubhs fs a``mktme gy smvmrad `aktjrs. ^m saw tcat mhe kjheftfjhs kah gm akkjuhtme `jr gy em`fhfhn m``mktfvm kjdubh dmhntc wcfkc bay gm ef``mrmht `rjb tcm aktuad kjdubh dmhntc. Mdastfk suppjrts kah gm akkjuhtme gy fhkjrpjratfhn dfhmar ahe tjrsfjhad sprfhns fh tcm batcmbatfkad bjemds. Ujdmrahkms, bahu`akturfhn `daws, suppjrt ahe djaefhn bfsadfnhbmhts kah gm fhkjr pjratme gy usm j` mkkmhtrfkfty `aktjr. [trmss dmvmds gmyjhe yfmde strmss kah gm akkjuhtme `jr gy usm j` tahnmht bjeudus. Umbpmraturm, batmrfad hjh-cjbjnmhmfty, ahe varfjus jtcmr kjbpdmxftfms tcat wm saw fh Kcaptmr 7 kah 7 kah gm fhkjrpjratme gy kcahnfhn tcm axfad ahe / jr gmhefhn `jrbudas as wm efe fh Kcaptmr 7. 7. Ucfs kcaptmr fs stahe adjhm fh tcm gjji‖s kurrmht `jrb ahe cas hj fbpakt jh sugsmqumht kcaptmrs. Cjwmvmr, tcm `fmde j` sta gfdfty fs vmry darnm ahe j`tmh tcmrm fs ah mhtfrm kjursm emvjtme tj tcm stuey j` stagfdfty.
5.5
[VHJ_[F[ J@ MQ]AUFJH[ Mudmr Gukidfhn 7
Gukidfhn djae
7 7
h MF MF -- , _ kr 2 ------------ 9 _ h 2 -----------------7 7 D D
x v 2 G sfh h -- D 7
Kjdubh kdassf`fkatfjh kr
7 M t F _ kr M 2 ------2 ---------------------- 9 _ kr 2 ------------7 7 A D m` ` r D m``
Fbpmr`mkt kjdubh Efspdakmbmht
_ v bax 2 m smk -- ------- ‛ 1 7 _ kr
[trmss
Dm` ` _ _ mk bax 2 --- 1 + ----- smk -------- ------- 7 A 7 r MA r
06
5
[tagfdfty 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
=
_DAUM[
DMAWHFHN JGLMKUFPM[ 1 1. ]hemrstahe tcm kdassfkad tcmjry j` gmhefhn j` tcfh pdatms ahe fts dfbftatfjhs. 7 7. ]hemrstahe cjw kjbpdmxftfms kah gm aeeme tj tcm kdassfkad pdatm tcmjry. tcmjry. ; ;. ]hemrstahe tcm sjdutfjh tmkchfqums `jr rmktahnudar ahe kfrkudar pdatms. 5 @djjrs, kmfdfhns, wfhejw pahms, efsk graims, scfp emkis, truki gmes, arm abjhn tcm kjuhtdmss appdfkatfjh j` strukturad = bmbgmr kaddme pdatm. A pdatm fs a `dat sjdfe gjey wcjsm tcfkihmss fs sbadd kjbparme tj tcm jtcmr efbmhsfjhs ahe fs suglmktme tj 0 gmhefhn djaes. Ucm tcmjry j` pdatm fs a twj-efbmhsfjhad ahadjn j` gmab tc mjry. ^m ^m apprjxfbatm tcm varfagdms fh tcm tcfkihmss > efrmktfjh as tcm pdatm fs t cfh. ^m fhtmnratm tcm strmssms akrjss tcm tcfkihmss tj jgtafh fhtmrhad `jrkms ahe bjbmhts, wcfkc wm tcmh 3 rmdatm tj mxtmrhad `jrkms ahe bjbmhts gy mqufdfgrfub. 6 ^m start gy emvmdjpfhn tcm sfbpdmst pjssfgdm tcmjry, tcat fs, tcm k dassfkad tcfh pdatm tcmjry (sm m [zfdare Y16>5R, Ufbjs1? cmhij ahe ^jfhjwsiy-Irfmnmr Y16=6R). Ucm dfbftatfjhs ahe assubptfjhs tcat arm baem tj nmt tcm sfbpdmst pdatm tcmjry arm femh11 tf`fme wftc tcm fhtmht j` erjppfhn tcmb tj fhkduem kjbpdmxftfms fh nmjbmtry, djaefhn, ahe batmrfads. Ucm kjbpdmxftfms tcat wfdd gm 17 kjhsfemrme fh mxabpdms ahe pjst-tmxt prjgdmbs arm emskrfgme fh [mktfjh =.7.6. =.7.6. ]hemrstahefhn bmtcjejdjny j` aeefhn kjbpdmxf1; tfms jpmhs tcm ejjr `jr bahy bjrm pjssfgfdftfms tcat arfsm gy fhkduefhn kjbgfhatfjhs j` kjbpdmxftfms. 15 Ucm em`dmktfjh j` pdatms fs njvmrhme gy a `jurtc jremr partfad ef``mrmhtfad mquatfjh kaddme tcm gf-carbjhfk mquatfjh. Fh 1= [mktfjhs =.; ahe =.; ahe =.5 wm =.5 wm stuey twj sjdutfjh tmkchfqums `jr rmktahnudar pdatms kaddme tcm Havfmr‖s Bmtcje ahe Haeaf-Dmvy‖s 10 bmtcje. Fh [mktfjh =.=, =.=, wm emvmdjp tcm mquatfjhs `jr kfrkudar pdatms. Uj jgtafh sjdutfjh `jr kfrkudar pdatms wm sfbpdf`y gy assub1> fhn axf-sybbmtry fh nmjbmtry, batmrfad prjpmrtfms, ahe djaefhn. 13 16 7? =.1 DFBFUAUFJH[ JH _DAUM UCMJWV. 71 77 7; 75 7= p z ( x, x, y) y) 70 x (u) 7> 73 Bfe pdahm 76 c( x, x, y) y) ;? t ;1 ht ;7 hh ;; y (v) ;5 h z (w) ;= ;0 @fnurm =.1 _datm nmjbmtry ;> @fnurm =.1 empfkts =.1 empfkts pdatm nmjbmtry. ^m wfdd emvmdjp tcm tcmjry `jr tcfh pdatms `jr sbadd em`dmktfjhs kaddme tcm Ifrkccj``-Djvm tcm;3 jry. jry. Ucm `jddjwfhn dfbftatfjhs wfdd appdy. ;6 5? 1. Ucm bfe pdahm j` tcm pdatm fs `dat. [cmdd tcmjry akkjuhts `jr kurvme bfe pdahm. 51 7. Ucm tcfkihmss j` tcm pdatm c( x, x, y) y ) bay vary nraeuaddy vary nraeuaddy akrjss akrjss tcm pdah `jrb gut bust gm ah jremr j` banhf57 tuem sbaddmr tcah tcm efbmhsfjhs fh tcm x tcm x jr jr y y efrmktfjh. efrmktfjh. 5; 06
>?
=
_datms ;. Ucm trahsvmrsm djaefhn p djaefhn p z ( x, x, y) y) bay vary nraeuaddy vary nraeuaddy akrjss akrjss tcm pdah `jrb
1 5. [ueemh kcahnms fh c( x, x, y) y) ahe p ahe p z ( x, x, y) y) wfdd rmsudt fh strmss kjhkmhtratfjh hmar tcm rmnfjhs j` rapfe kcahnm gut 7 tcmfr fbpakt wfdd efm jut rapfedy as pmr [afht Pmhaht‖s Pmhaht‖s prfhkfpdm. ^m ^m wfdd fnhjrm tcmsm rmnfjhs j` strmss kjh- ; kmhtratfjh. Jur tcmjry wfdd nfvm tcm hjbfhad vadums wcfkc arm usme fh emsfnh a`tmr budtfpdyfhn wftc apprj- 5 = prfatm strmss kjhkmhtratfjh `aktjrs. 0 =. ^m wfdd emvmdjp tcm fhftfad tcmjry `jr purm gmhefhn, tcat fs, wftc hj fhpdahm `jrkms. Fh pjst tmxt prjgdmbs wm wfdd rmdax tcfs dfbftatfjh gy kjhsfemrfhn `jrkms hh ahe ht , gut tcmrm wfdd gm hj fhpdahm `jrkms jh tjp ahe gjt- > 3 tjb sur`akm j` tcm pdatm. 6 0. Mdastfkfty tmdds us tcat tcm hjrbad strmss zz jh tcm sur`akm scjude mquad tj tcm trahsvmrsm djae p djae p z . ^m wfdd 1? assubm tcat tcm baxfbub zz wfdd gm ah jremr j` banhftuem dmss tcah tcm baxfbub hjrbad strmssms xx 11 ahe yy, cmhkm kah gm hmndmktme. 17 >. Ucm efspdakmbmhts fh tcm x tcm x,, y, y, ahe z ahe z , efrmktfjhs arm nfvmh gy u, v, ahe w, rmspmktfvmdy. Jur tcmjry wfdd gm 1; vadfe `jr trahsvmrsm em`dmktfjh fh tcm rahnm ? w ?.7c ?.7 c . Ucfs dmaes tj a dfhmar pdatm gmhefhn tcmjry. 15 Em`dmktfjh rahnm j` ?.;c ?.; c w c fs kaddme tcm Pjh-Iarbah rahnm `jr pdatm gmhefhn ahe dmaes tj hjh-dfhmar 1= tcmjry. Em`dmktfjh rahnm j` w 7 c fs kaddme tcm Pjh-Iarbah rahnm `jr bmbgrahm tcmjry. 10 ^m jhkm bjrm wfdd usm tcm djnfk scjwh fh @fnurm =.7. =.7. ^m baim assubptfjhs tcat wfdd dmt us emvmdjp tcm sfbpdmst pjssf- 1> gdm tcmjry. Ucmsm assubptfjhs wfdd gm pjfhts at wcfkc kjbpdmxfty kah gm aeeme, as was ejhm fh gmab tcmjry. 13 16 7? Efspdakmbmhts 71 77 7; 1 I f 75 h m b a t f k 7= s 70 7> Mxtmrhad `jrkms [trafhs 73 ahe bjbmhts 76 ;? md s M M ;1 h m q j e u ;7 r n y fd d B f g ;; B r f r f a 7 5 m t u b ;5 a tm c j e s ;= B ;0 ;> [tatfk ;3 Mqufvadmhky Fhtmrhad `jrkms [trmssms ;6 ahe bjbmhts ; 5? 51 @fnurm =.7 Djnfk fh strukturad ahadysfs. 57 5; =.7 UCMJWV 55 5= ^m wfdd baim assubptfjhs tj emvmdjp tcm sfbpdmst p jssfgdm tcmjry, wcfkc fs kaddme tcm kdassfkad pdatm tcmjry. Fh[mktfjh Fh [mktfjh =.7.6, =.7.6, wm 50 wfdd kjhsfemr kjbpdmxftfms tcat kah gm aeeme gy erjppfhn apprjprfatm assubptfjh ahe tcmh fbpdmbmhtfhn tcm rmst j` tcm djnfk. 5> Ucm tcmjry jglmktfvms arm< 53 56 1. Uj jgtafh mquatfjhs rmdatfhn tcm gmhefhn strmssms xx, yy, ahe xy tj fhtmrhad `jrkms ahe bjbmhts. =? 7. Uj jgtafh tcm gjuheary vadum prjgdmb njvmrhfhn tcm trahsvmrsm em`dmktfjhw em`dmktfjh w j` tcm pdatm. =1 =7 =.7.1 Ifhmbatfks =; ^m baim assubptfjhs sfbfdar tj tcm gmab tcmjry ahe jgtafh ah apprjxfbatfjh `jr efspdakmbmhts ahe strafh. =5 == Assubptfjh 1 Em`jrbatfjhs arm hjt a `uhktfjhs j` tfbm. =0 Ucm agjvm assubptfjh mdfbfhatms eyhabfk m``mkts ahe baim tcm tcmjry appdfkagdm tj statfk prjgdmbs. => tcm z efrmktfjh, efrmktfjh, fs sfnhf`fkahtdy sbaddmr tcah gmhefhn. Assubptfjh 7 [quascfhn―efbmhsfjhad kcahnms fh tcm z =3 Ucm agjvm assubptfjh fbpdfms tcat tcm hjrbad strafh fh tcm z tcm z efrmktfjh efrmktfjh fs sfnhf`fkahtdy sbaddmr tcah tcm hjrbad strafhs fh =6 x ahe x ahe y y efrmktfjh tcat wfdd arfsm `rjb gmhefhn, tcat fs, zz 2 w z ? . Ucus, w kahhjt gm a `uhktfjh j` z j` z as scjwh gmdjw. 0? u 2 u x y z v 2 v x y z w 2 w x y (=.1a) 01
Ucmjry
>1
1 Assubptfjh ; _dahm smktfjhs gm`jrm em`jrbatfjh rmbafh pdahms a`tmr em`jrbatfjh. 7 Ucm agjvm assubptfjh fbpdfms tcat efspdakmbmhts u ahe v arm dfhmar `uhktfjh j` z. j` z. ^m wfdd datmr scjw tcat f` tcm pdatm fs ; cjbjnmhmjus fh tcm tcfkihmss efrmktfjh tcmh z tcmh z fs fs bmasurme `rjb tcm bfe pdahm j` tcm pdatm. 5 @fnurm =.; scjws =.; scjws ah mxannmratme em`jrbme scapm j` tcm pdatm as vfmwme `rjb tcm y tcm y axfs axfs ahe tcm x tcm x axfs. axfs. Dfhm AKG Dfhm AKG gmfhn gmfhn part = j` tcm pdahm rmbafhs strafnct eurfhn em`jrbatfjh gut ft rjtatms gy ah ahndm ahe agjut x ahe x ahe y y axfs, axfs, rmspmktfvmdy. K fs fs a x y 0 pjfht wcmrm z wcmrm z 2 2 ? ahe ft efspdakms gy abjuht u j x y ahe v j x y . Hjtfhn tcat tcm dfhm rjtatfjh bjvms pjfht G fh G fh tcm hmnatfvm x hmnatfvm x > ahe y ahe y efrmktfjh efrmktfjh wm kah wrftm tcm efspdakmbmht `fmde as 3 (=.1g) u uj x y ‛ z sfh y v v j x y ‛ z sfh x w w x y 6 1? A A 11 x x 17 K K 1; G G u x , y v x , y j j 15 1= w x , y w x , y 10 z z 1> A1 A1 13 K 1 K 1 16 ‛ 1 w ‛ 1 w tah tah G1 G1 7? x y 71 y x 77 7; A1 A1 75 7= K 1 K 1 z z 70 G1 G1 7> z sfh z sfh y y x x 73 76 ;? (a) (g) ;1 @fnurm =.; Mxannmratme em`jrbme nmjbmtry. Wjtatfjh j` dfhm fh tcfkihmss efrmktfjh agjut (a) y (a) y axfs axfs (g) x (g) x axfs. axfs. ;7 Hjtm jur efspdakmbmht `fmde wfdd gm dfhmar fh z fh z jhdy jhdy f` tcm rjtatfjhs x ahe y arm `uhktfjhs j` x j` x ahe ahe y y gut gut hjt j` z j` z . ^m cavm ;; ;5 assubme tcmrm arm hj fhpdahm `jrkms ahe cmhkm fhpdahm efspdakmbmhts u j ahe v j wfdd gm zmrj as scjwh gmdjw. ;= v j x y 2 ? F` tcmrm arm hj fhpdahm `jrkms. (=.1k) u j x y 2 ? ;0 Assubptfjh 5 _dahm smktfjhs pmrpmhefkudar tj tcm pdatm bfe sur`akm rmbafh hmardy pmrpmhefkudar hmardy pmrpmhefkudar a`tmr em`jrbatfjh. ;> Ucm rfnct ahndm at K rmbafhs rmbafhs a rfnct ahndm at K 1 akkjrefhn tj tcm agjvm assubptfjh. Ucfs fs jhdy pjssfgdm f` tcm ahndms j` rjta;3 tcm x ahe ahe y y axfs axfs as scjwh fh @fnurm =.;, =.;, tcat fs ;6 tfjh arm mquad tj tcm ahndms j` tcm tahnmht sur`akm w x y baims wftc tcm x 5? (=.1e) x 2 tah ‛ 1 w x y 2 tah ‛ 1 w y 51 57 Assubptfjh = Em`dmktfjh ahe fts emrfvatfvms arm sbadd. 5; ^ftc tcm agjvm assubptfjh wm kah apprjxfbatm sfhm ahe tahnmht `uhktfjhs as scjwh gmdjw. 55 (=.1m) sfh x x sfh y y tah ‛ 1 w x w x tah ‛ 1 w y w y 5= (=.1e) ahe (=.1m) fhtj (=.1m) fhtj Mquatfjh (=.1g) wm (=.1g) wm jgtafh tcm efspdakmbmht `fmde 50 [ugstftutfhn Mquatfjhs (=.1e) ahe (=.7) u ‛ z w x v ‛ z w y w w x y 5> 53 @rjb tcm agjvm efspdakmbmht `fmde wm `fhe tcm `jddjwfhn scmar strafhs 56 (=.;) xz 2 w x + u z ? yz 2 w y + v z ? =? Ucm zmrj scmar strafhs arm a kjhsmqumhkm j` Assubptfjh 5. 5. Fh sybbmtrfk gmhefhn j` gmabs, wm rmdaxme tcfs typm j` assubp=1 tfjh ahe emvmdjpme tcm mquatfjhs `jr Ufbjscmhij gmabs. Ufbjscmhij gmabs. Fh a sfbfdar bahhmr, rmdaxfhn Assubptfjh 5 emvmdjps 5 emvmdjps tcm mquatfjhs `jr =7 Bfhedfh-Wmfsshmr pdatm pdatm tcmjry efskussme fh Mxabpdm =.;. =.;. =; Assubptfjh 0 [trafhs arm sbadd. =5 ^m jgtafh tcm strafhs fh pdatm gmhefhn as == 7 7 7 =0 w w w u v u v => (=.5) 2 ‛ z 7 2 ‛ z 7 + 2 ‛ 7 z xx 2 yy 2 xy 2 x y x y y x x y =3 Mquatfjh (=.5) scjws (=.5) scjws tcat tcm strafhs vary dfhmardy wftc z wftc z . Ucm baxfbub strafh vadums wfdd gm mftcmr jh tcm tjp jr gjttjb sur=6 7 7 7 7 0? `akm j` tcm pdatm. Ucm em`jrbme pdatm `jrbs a kurvme sur`akm wcjsm kurvaturms arm nfvmh gy w x , w y , ahe 7 01 w x y .
>7
=
_datms
1 7 Ucm sfbpdmst batmrfad bjemd fs a dfhmar rmdatfjhscfp gmtwmmh strmssms ahe strafhs nfvmh gy tcm Nmhmradfzme Cjjim‖s daw. Uj usm ; tcfs sfbpdm bjemd wm baim tcm `jddjwfhn assubptfjhs. 5 = Assubptfjh > Ucm batmrfad fs fsjtrjpfk. 0 Assubptfjh 3 Ucm batmrfad fs dfhmardy mdastfk. > Assubptfjh 6 Ucmrm arm hj tcmrbad jr jtcmr hjh-bmkcahfkad strafhs. 3 Ucmrm arm hj fhpdahm `jrkms aktfhn jh tjp ahe gjttjb sur`akms j` tcm pdatm as pmr dfbftatfjh =. ^m `urtcmr assubme tcat tcm 6 strmss zz kah gm hmndmktme fh kjbparfsjh tj tcm strmssms xx ahe yy. Fh jtcmr wjres, tcm tjp ahe gjttjb sur`akms arm `rmm sur`akms, 1? tcat fs tcm pjfhts jh tjp ahe gjttjb sur`akm arm fh pdahm strmss. @jr tcfh pdatms wm kah tcus say tcm pdatm tcm pdatm fs fh statm j` pdahm strmss strmss.. 11 @jr pdahm strmss wm kah wrftm tcm `jddjwfhn mquatfjhs 17 M M 1; (=.=) xx 2 ------------------ xx + yy yy 2 ------------------ yy + xx xy 2 N xy 7 7 15 1 ‛ 1 ‛ 1= wcmrm, M wcmrm, M fs fs tcm bjeudus j` mdastfkfty, fs tcm _jfssjh‖s ratfj, ahe N fs tcm scmar bjeudus j` mdastfkfty. @jr fsjtrjpfk batmrfad 10 N 2 M 7 1 + . [ugstftutfhn Mquatfjh (=.5) fhtj (=.5) fhtj Mquatfjh (=.=), (=.=), wm jgtafh 1> 7 7 7 7 7 13 ‛ M z w ‛ M z w w w ------------------- w + (=.0) + 2 2 ‛ 7 N z xx 2 ------------------ yy xy 16 x y 1 ‛ 7 x 7 y 7 1 ‛ 7 y 7 x 7 7? 71 =.7.; [tatfk mqufvadmhky 77 ^m rmpdakm tcm strmssms gy statfkaddy mqufvadm ht fhtmrhad `jrkms ahe bjbmhts (strmss rmsudtahts) gy fhtmnratfhn akrjss tcm tcfkihmss 7; 75 c as scjwh fh tcm mquat fjhs gmdjw. @fnurm =.5 scjws =.5 scjws tcm pjsftfvm efrmktfjhs j` tcm strmss rmsudtahts. 7= h xx 2 h yy 2 h xy 2 h yx 2 (=.>) xx e z yy e z xy e z 70 c c c 7> b xx 2 b yy 2 b xy 2 b yx 2 (=.3) z xx e z z yy e z z xy e z 73 c c c 76 q x 2 q y 2 (=.6) xz e z yz e z ;? c c ;1 Hjtm tcm `jddjwfhn ;7 • Add `jrkm strmss rmsudtahts rmsudtahts wfdd wfdd cavm tcm uhfts tcm uhfts j` `jrkm pmr uhft dmhntc ;; • Add bjbmht strmss rmsudtahts wfdd rmsudtahts wfdd cavm tcm uhfts j` bjbmhts pmr uhft dmhntc. dmhntc. ;5 ;= q y ;0 b yy ;> b ;3 yx h yy q x ;6 h yx 5? b xy 51 x x h xy b xx 57 b xx b xy 5; h xx h xx 55 5= 50 h xy b yx q x 5> h yx b yy 53 56 y y =? q y h yy =1 ( a) (g) =7 @fnurm =.5 Mqufvadmht fhtmrhad `jrkms ahe bjbmhts (strmss rmsudtahts) (a) Fh-pdahm. (g) Gmhefhn. =; Ucm fhtmrhad scmar `jrkms q x ahe q y arm hmkmssary `jr `jrkm mqufdfgrfub fh tcm z tcm z efrmktfjh, efrmktfjh, wcfkc fbpdfms tcat tcm strmssms xz =5 ahe yz kahhjt gm zmrj. Mquatfjh (=.;) adjhn wftc Cjjim‖s daw fbpdfms tcat tcm scmar strmss xz ahe yz bust gm sbadd gut hjt == zmrj. Avmranm vadums j` tcmsm strmssms kah gm `juhe gy efvfefhn tcm fhtmrhad scmar `jrkms q x ahe q y gy tcm tcfkihmss. @jr jur tcm- =0 jry tj gm vadfe tcm baxfbub j` tcmsm avmranm vadums bust gm ah jremr j` banhftuem sbaddmr tcah tcm baxfbub fhpdahm strmssms. => (=.1?) =3 ba x xz ahe ba x yz « ba x xx yy xy =6 F` jur kadkudatfjhs scjw tcat tcm agjvm mquatfjh fs hjt vadfe, tcmh tcm Assubptfjh 5 fs 5 fs hjt vadfe ahe trahsvmrsm scmar 0? bust gm akkjuhtme `jr fh tcm tcmjry. 01
=.7.7
Batmrfad bjemd
Ucmjry
>;
1 =.7.5 Djkatfjh j` hmutrad sur`akm 7 [ugstftutfhnMquatfjh (=.0) ; ^m cavm assubme tcmrm arm hj mxtmrhad fhpdahm ` jrkms, cmhkm tcm fhtmrhad fhpdahm `jrkms bust gm zmrj. [ugstftutfhnMquatfjh (=.>) ahe taifhn tcm kurvaturms jutsfem tcm fhtmnrads as tcmy arm hjt `uhktfjh j` z j` z , wm jgtafh 5 fhtj Mquatfjh (=.>) ahe 7 7 7 7 = w Mz w M z w Mz w M z -------------------e z + -------------------e z 2 ? -------------------e z + -------------------e z 2 ? h xx 2 h yy 2 0 7 7 7 7 7 7 7 x c 1 ‛ y c 1 ‛ y c 1 ‛ x c 1 ‛ 7 > (=.11a) 7 3 w Nz e z 2 ? h xy 2 ‛ 7 6 x y c 1? 11 Assubptfjh 1? Ucm batmrfad fs cjbjnmhmjus akrjss tcm tcfkihmss j` tcm pdatm. 17 Ucm agjvm assubptfjh fbpdfms tcat tcm batmrfad kjhstahts kah gm taimh jutsfem tcm fhtmnrads ahe wm jgtafh 1; 7 7 7 7 w M w M w M w M 15 ------------------- + ------------------- z e z 2 ? ------------------- + ------------------- z e z 2 ? h xx 2 h yy 2 7 7 7 7 7 7 7 1= x 1 ‛ y 1 ‛ c y 1 ‛ x 1 ‛ 7 c (=.11g) 10 7 w 1> h xy 2 ‛ 7 N z e z 2 ? x y c 13 16 Em`jrbatfjh j` pdatm fbpdfms tcat at dmast jhm j` tcm kurvaturm wfdd hjt gm zmrj. Cmhkm, Mquatfjh (=.11g) fbpdfms (=.11g) fbpdfms 7? (=.17) z e z 2 ? 71 c 77 Ucm kjheftfjh fh Mquatfjh (=.17) wfdd (=.17) wfdd gm bmt f` tcm jrfnfh fs at tcm bfe sur`akm. ^m rmkjre tcfs jgsmrvatfjh `jr `uturm usm. 7; • Ucm jrfnfh ahe ahe tcm hmutrad sur`akm sur`akm bust gm at tcm tcm bfe sur`akm `jr `jr purm gmhefhn j` tcfh tcfh pdatms. 75 7= 70 =.7.= [trmss `jrbudas 7> [ugstftutfhn Mquatfjh (=.0) fhtj (=.0) fhtj Mquatfjh (=.3) wm (=.3) wm jgtafh 73 7 7 7 7 7 7 w w M z 7 w w M z 7 76 Mz Mz e z ‛ 7 ------------------e z e z ‛ 7 ------------------e z b xx 2 ‛ 7 ------------------b yy 2 ‛ 7 ------------------;? x c 1 ‛ 7 y c 1 ‛ 7 y c 1 ‛ 7 x c 1 ‛ 7 ;1 (=.1;a) 7 ;7 w 7 b xy 2 ‛ 7 ;; Nz e z x y c ;5 1? j` cjbjnmhmfty akrjss tcm tcfkihmss, wm jgtafh ;= As pmr Assubptfjh 1? j` 7 7 7 7 ;0 w w w w M M 7 7 ;> b xx 2 ‛ ------------------b yy 2 ‛ ------------------ 7 + 7 z e z 7 + 7 z e z 7 7 1 ‛ x y c 1 ‛ y x c ;3 (=.1;g) ;6 7 w 7 5? b xy 2 ‛ 7 N z e z 51 x y c 57 Hjtfhn tcat z tcat z fs fs bmasurme `rjb tcm bfe sur`akm, wm kah wrftm 5; 7 7 ; 55 (=.1;k) z e z 2 z e z 2 c 17 c ‛ c 7 5= 50 ^m jgtafh tcm `fhad `jrb j` bjbmht-kurvaturm mquatfjhs as 7 7 7 7 7 5> w w w w w 53 b xx 2 ‛ E 7 + 7 b yy 2 ‛ E 7 + 7 b xy 2 ‛ E 1 ‛ (=.15) x y y x x y 56 =? wcmrm, tcm kjhstaht E kjhstaht E fs fs kaddme tcm gmhefhn rfnfefty j` tcm pdatm ahe pdatm ahe fs nfvmh gy =1 ; Mc (=.1=) E 2 ------------------------=7 7 17 1 ‛ =; (=.15) fhtj strmss mxprmssfjhs Mquatfjh (=.0) wm (=.0) wm jgtafh tcm `jrbudas `jr strmssms =5 [ugstftutfhn Mquatfjh (=.15) fhtj == b xx z b yy z b xy z (=.10) 2 -------------------2 ------------------- xx 2 ------------------- yy xy =0 c ; 17 c ; 17 c ; 17 => • Ucm gmhefhn strmssms fh cjbjnmhmjus pdatms vary dfhmardy tcrjunc tcm tcfkihmss ahe arm baxfbub at tjp ahe gjttjb =3 sur`akm j` tcm pdatm z 2 c 7 . =6 0? 01
>5
=
=.7.0
_datms
Mqufdfgrfub
^m kjhsfemr a `rmm gjey efanrab j` a ef``mrmhtfad mdmbmht j` tcm pdatm scjwh fh@fnurm fh @fnurm =.=. =.=. q ex y
b ex yy
b ex yx q ey x
ex
p x , y exey z
b ey xy
x
b ey xx
b xx ex ey b xx + x
b xy ex ey b xy + x
ey
b yx ey ex b yx + y b b yy + yy ey ex y
q x ex ey q x + x
J
q y ey ex q y + y
y
@fnurm =.= @rmm gjey efanrab j ` a ef``mrmhtfad pdatm mdmbmht.
Hjtm tcat add strmss rmsudtahts arm em`fhme pmr uhft dmhntc ahe cmhkm bust gm b udtfpdfme gy tcm ef``mrmhtfad dmhntcs j` tcm sfem jh wcfkc tcmy akt. Gy mqufdfgrfub j` `jrkms fh tcm z tcm z efrmktfjh efrmktfjh wm jgtafh
q + q x ex ey ‛ q ey + q + q y ey ex ‛ q ex + p x y exey x y y x x y y q x q y + x y
2 ? jr
2 ‛ p z x y
(=.1>a)
^m kjhsfemr mqufdfgrfub j` bjbmht agjut tcm y tcm y efrmktfjh efrmktfjh at pjfht J. Ucm bjbmht `rjb p z x y ahe q y wfdd cavm prjeukts j` tcrmm ef``mrmhtfad wcfdm tcm rmst j` tcm tmrbs wfdd jhdy cavm prjeukts j` twj ef``mrmhtfad, cmhkm kah gm hmndmktme. Ucm rmbafhfhn tmrbs yfmde
b + b xx ex ey ‛ b ey + b + b yx ey ex ‛ b ex ‛ q ey ex xx yx x xx x yx y q x 2
2 ? jr
b xx b yx + x y
(=.1>g)
[fbfdardy gy kjhsfemrfhn bjbmht agjut tcm x efrmktfjh x efrmktfjh tcrjunc pjfht J wm jgtafh q y 2
=.7.>
b xy b yy + x y
(=.1>k)
Ef``mrmhtfad mquatfjh
[ugstftutfhn Mquatfjh (=.15) fhtj (=.15) fhtj Mquatfjhs (=.1>g) ahe (=.1>g) ahe Mquatfjhs (=.1>k) wm (=.1>k) wm jgtafh 7
7
7
q x
w + w 2 ‛ E x x7 y7
E 1 ‛ w ‛ x y y
q y
E 1 ‛ w 2 ‛ x y x
7
(=.13a)
7
7
w + w ‛ E y y 7 x7
(=.13g)
[ugstftutfhn tcm agjvm mquatfjhs fhtj Mquatfjh (=.1>a) wm (=.1>a) wm jgtafh 7
7
7 E w + w y 7 x 7 x 7
7
7 E 1 ‛ w +7 x y x y
7
7
7 E w + w + x 7 y 7 y 7
2 p z x y
(=.13k)
Ucm agjvm mquatfjh fs tcm ef ``mrmhtfad mquatfjh njvmrhfhn em`dmktfjh j` tapmrme pdatms. Uj nmt `urtcmr sfbpdf`fkatfjh wm baim aeeftfjhad assubptfjhs.
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
Ucmjry
>=
1 Assubptfjh 11 Ucm pdatm fs cjbjnmhmjus fh tcm x tcm x ahe ahe y y efrmktfjh. efrmktfjh. 7 Assubptfjh 17 Ucm pdatm fs j` uhf`jrb tcfkihmss. ; @rjb agjvm assubptfjhs c, M , ahe arm kjhstaht, cmhkm E cmhkm E fs fs a kjhstaht. Mquatfjhs (=.13a), (=.13a), ahe (=.13g) yfmde (=.13g) yfmde tcm `jddjwfhn 5 7 7 7 7 = 7 w w w w 7 q x 2 ‛ E q y 2 ‛ E (=.16a) + 2 ‛ E w + 2 ‛ E w 0 7 7 7 7 x x y x y x y y > 3 wcmrm, 7 w 2 7 w x 7 + 7 w y 7 fs tcm Dapdakm jr carbjhfk jpmratjr. @rjb Mquatfjh (=.13k) wm (=.13k) wm jgtafh 6 5 5 5 w 5 1? w w (=.7?) Ef``mrmhtfad Mquatfjh< E w 2 E +7 7 7 + 2 p z x y 11 x y y 5 x5 17 5 7 7 gf-carbjhfk jpmratjr. 1; wcmrm, w 2 w fs kaddme tcm gf-carbjhfk jpmratjr. 15 1= =.7.3 Gjuheary kjheftfjhs 10 A `jurtc jremr partfad ef``mrmhtfad mquatfjh njvmrhs tcm pdatm em`dmktfjh as smmh `rjb Mquatfjh (=.7?). (=.7?). [j dfim gmab em`dmktfjh 1> wcfkc fs adsj njvmrhme gy `jurtc jremr ef``mrmhtfad mquatfjh, wm hmme tj spmkf`y twj kjheftfjhs at makc gjuheary pjfht. Fh gmabs 13 jhm kjheftfjh was jh em`dmktfjh jr fhtmrhad scmar `jrkm ahe tcm smkjhe kjheftfjh was jh sdjpm jr fhtmrhad bjbmht. Gut fh pdatms 16 tcmrm fs a scmar `jrkm ahe twj bjbmhts at tcm gjuheary. ^cfkc twj ej wm usm: Ucm ahswmr was prjvfeme gy Ifrkccj`` gy Ifrkccj`` , wcj 7? scjwme tcat tcm varfatfjh j` tcm twfstfhn bjbmht b xy (jr b yx) rmsudts fh a scmar `jrkm tcat aees tj tcm mxfstfhn scmar `jrkm. 71 x x 77 b xy 7; b xyey 75 b xy b xy 7= ey ey b xy + ey b + b xy ey y xy y J 70 J 7> ey b xy 73 y ey b + y 76 xy y ;? ( g ) (a) ;1 ;7 @fnurm =.0 [cmar `jrkm nmhmratme `rjb twfstfhn bjbmht. ;; Kjhsfemr a gjuheary x gjuheary x 2 kjhstaht. kjhstaht. ^m kah fhtmrprmt tcm bjbmht b xyey as ey as a kjupdm j` twj `jrkms aktfhn fh jppjsftm efrmk;5 tfjhs tcat arm smparatme gy a efstahkm j` ey as ey as scjwh fh @fnurm =.0. =.0. Fh a sfbfdar bahhmr tcm varfatfjh j` twfstfhn bjbmht nfvmh gy ;= b xy + b xy y ey ey kah adsj gm fhtmrprmtme as a kjupdm j` jppjsftm `jrkms. Ucm rmsudtaht j` tcm twj `jrkms at pjfht J tcus ;0 prjeukms a scmar `jrkm j` b xy y fh tcm efrmktfjh j` tcm scmar `jrkm q x . Ucus, tcm tjtad gjuheary scmar `jrkm fs ;> P x 2 q x + b xy y . [ugstftutfhn `jr q x ahe b xy wm jgtafh ;3 ; ; w w ;6 P x 2 ‛ E (=.71a) + 7 ‛ 5? x y 7 x ; 51 Fh a sfbfdar bahhmr jh a gjuheary y gjuheary y 2 kjhstaht wm jgtafh< 57 ; ; 5; w w + 7 ‛ P y 2 ‛ E (=.71g) 55 y x 7 y ; 5= 50 x 5> 53 b xy 56 =? =1 =7 b yx K y =; =5 == @fnurm =.> Mxpdahatfjh j` kjrhmr `jrkm. =0 Ucm twfstfhn bjbmht prjeukms ahjtcmr `maturm at a kjrhmr. Ucm fhtmrprmtatfjh j` tcm twfstfhn bjbmht as a kjupdm j` => mquad ahe jppjsftm `jrkms efstrfgutme jh twj menms fs scjwh fh @fnurm =.>. =.>. Fh tcm bfeedm j` ah menm tcm `jrkms `rjb twj sukkms=3 sfvm kjupdms kahkmd jut. Gut at tcm kjrhmr a `jrkm `rjb kjupdm b xy ahe `jrkm `rjb b yx arm fh tcm sabm efrmktfjh nfvfhn a rmsudtaht =6 `jrkm. Ucfs `jrkm fs kaddme tcm kjrhm r `jrkm ahe fs nfvmh gy tcm mquatfj h gmdjw. F` yju smm krakis hmar tcm kjrhmr j` `djjr jr kmfdfh n 0? fh jde gufdefhns, tcmh ft fs dfimdy eum tj tcm kjrhmr `jrkm. 01
>0
=
_datms 7
W kjrhmr 2 b xy + b yx 2 7 b xy 2 ‛ 7 E 1 ‛ w x y
(=.77)
^m kah hjw statm add pjssfgdm gjuheary kjheftfjhs. Jh x Jh x 2 kjhstaht
[pmkf`y Y P x jr w ahe b xx jr w x
Jh y 2 kjhstaht Gjuheary Kjheftfjhs< Jh y
[pmkf`y Y P y jr w ahe b yy jr w y
(=.7;)
[pmkf`y Y W kjrhmr jr w
Jh a kjrhmr
Uagdm =.1 Gjuheary kjheftfjhs at menm x menm x 2 2 a
Gjuheary Kjheftfjhs
Uypm j` suppjrt
[pmkf`y Y P x jr w
[pmkf`y b xx jr w y
x 2 a
Kdabpme
x
x 2 a
@rmm
w a y 2 ? -------- x7
P x a y 2 ? jr jr
b xx a y 2 ? jr
;
x 2 a
Mdastfk
w a y 2 ?
7
;
w + 7 ‛ w x y7 x ;
x
;
x
I
x 2 a
;
2 ?
7
w + w y7 x 7
2 ? x 2 a
;
w + 7 ‛ w 7 ; x y x
w a y 2 ? ------ x
x 2 a
P x a y 2 ‛ I D w a y jr ;
I D
7
2 ?
P x a y 2 ? jr
w + 7 ‛ w x y7 x ;
x
2 ?
b xx a y 2 ? jr
x 2 a
[fbpdy [uppjrtme
Wjddmr
a y ----w -- x
w a y 2 ?
x
2
I ---- D -- E w a y
x 2 a
w a y jr b xx a y 2 I ------ x 7
7
w + w 7 7 y x
I w a y 2 ‛ ------ ------ E x x 2 a
^m wfdd kjhsfemr tcm gjuheary at x at x 2 2 a a ahe ahe tcm mxtmhsfjh tj y tj y 2 2 g fs dm`t tj tcm rmaemr. Uagdm =.1 scjws =.1 scjws smvmrad typms j` sup pjrt ahe dfst tcm assjkfatme gjuheary kjheftfjhs jgtafhme usfhn Mquatfjh (=.7;). (=.7;). Ucm sfbpdy suppjrtme gjuheary kjheftfjh hmmes sjbm mxpdahatfjh. ^ftc w a y 2 ? , tcm y tcm y-emrfvatfvms -emrfvatfvms wfdd adsj gm zmrj. [mttfhn tcm smkjhe y smkjhe y emrfvatfvm emrfvatfvm j` w tj zmrj fh tcm zmrj bjbmht kjheftfjh rmsudts fh tcm smkjhe x smkjhe x emrfvatfvm emrfvatfvm j` w tj gm zmrj as scjwh fh Uagdm =.1. =.1. x 2 a
b xx a y
w a y I ------ x
P x a y I D w a y @fnurm =.3 @rmm gjey gjey efanrab efanrab j` ah ah fh`fhftmsfbad fh`fhftmsfbad mdmbmht at x 2 2 a gjuheary.
Ucm gjuheary kjheftfjh `jr mdastfk suppjrt wmrm jgtafhme gy erawfhn a `rmm gjey efanrab j` ah fh`fhftmsfbad mdmbmht at tcm gjuheary mhe as scjwh fh @fnurm =.3. =.3. Ft adsj cfncdfncts tcat sukc `rmm gjey efanrabs bust gm erawh tj nmt tcm rfnct sfnhs `jr fhtmrhad bjbmht ahe fhtmrhad gjuheary scmar `jrkms wcmhmvmr ah mxtmrhad `jrkm jr bjbmht akts jh tcm menm.
=.7.6
Kjbpdmxfty bap
Fh [mktfjh 7.;.;, 7.;.;, fh kjhtmxt j` jhm-efbmhsfjhad strukturad bmbgmrs, twj jgsmr vatfjhs wmrm baem wcfkc arm ahadjnjus tj tcm jhms nfvmh gmdjw. • Ucm `jrbudas `jrbudas `jr strmssm strmssmss nfvmh gy Mquatfjh (=.10) jr (=.10) jr tcmfr mqufvadmht `jrb arm jgtafhme gy sugstftutfhn Mquatfjh (=.15) jr (=.15) jr fts mqufvadmht `jrbs fhtj Mquatfjh (=.0) jr (=.0) jr fts mqufvadmht `jrbs, frrmspmktfvm j` tcm kjbpdmxfty fhkdueme. • Ucm ef``mrmhtfa ef``mrmhtfadd mquatfjh(s) mquatfjh(s) nfvmh gy Mquatfjh (=.7?) jr (=.7?) jr tcmfr mqufvadmht `jrb arm jgtafhme `rjb gy sugstftutfhn Mquatfjh (=.15) jr (=.15) jr fts mqufvadmht `jrb fhtj Mquatfjhs (=.1>a), (=.1>a), (=.1>g), (=.1>g), ahe (=.1>k) jr (=.1>k) jr tcmfr mqufvadmht `jrbs, frrmspmktfvm j` tcm kjbpdmxfty fhkdueme. Gmdjw arm dfstme tcm kjbpdmxftfms tcat arm aeeme fh tcfs kcaptmr tj tcm kdassfkad tcmjry j` pdatms.
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
Ucmjry
>>
1 1. Erjppfhn j` Assubptfjh 1 fbpdfms 1 fbpdfms tcat fhmrtfad `jrkms bust gm fhkdueme fh mqufdfgrfub mquatfjhs. Fh _rjgdmb =.0 wm =.0 wm kjhsfemr tcm vfgratfjh j` pdatms. 7 ; 7. ^m wfdd hjt gm erjppfhn Assubptfjhs 7 jr ; fh tcfs gjji. Cjwmvmr, fh _rjgdmbs =.1 ahe =.1 ahe =.7, =.7, wm wfdd hjt smt tcm fhpdahm efs5 pdakmbmhts tj gm zmrj as wm efe fh Mquatfjh (=.1k). (=.1k). Fh jtcmr wjres, wm wfdd emvmdjp tcm tcmjry wftc fhpdahm ahe gmhefhn efspdakmbmhts sfbudtahmjusdy. = 0 ;. Erjppfhn Assubptfjh 5 5 pmrbfts us tj fhkjrpjratm jut j` pdahm scmar ahe emvmdjp tcm Bfhedfh-Wmfsshmr tcm Bfhedfh-Wmfsshmr pdatm tcmjry as > scjwh fh Mxabpdm =.;. =.;. Ucfs fs ahadjnjus tj tcm Ufbjscmhij gmab tcmjry efskussme fh Mxabpdm 7.0. 7.0. 3 5. Erjppfhn j` Assubptfjhs = jr 0 pmrbfts us fhkjrpjratms nmjbmtrfk hjhdfhmarfty ahe emvmdjp tcm tcmjry `jr Pjh-Iarbah 6 rmnfjh efskussme fh _rjgdmb =.>. =.>. Fh Mxabpdm 3.0 wm 3.0 wm emrfvm tcm gjuheary vadum prjgdmb a`tmr fhkjrpjratfhn darnm strafhs. 1? =. Assubptfjh > fs > fs erjppme fh Mxabpdm =.7 tj =.7 tj emvmdjp tcm tcmjry `jr jrtcjtrjpfk pdatms. 11 0. Assubptfjh 6 fs 6 fs erjppme fh _rjgdmb =.; tj =.; tj fhkjrpjratm m``mkt j` tmbpmraturm fh pdatm tcmjry. 17 >. Assubptfjh 1? j` 1? j` batmrfad jh batmrfad cjbjnmhmfty akrjss tcm tcfkihmss fs erjppme fh Mxabpdm =.1 ahe =.1 ahe _rjgdmbs =.5 ahe =.5 ahe 1; =.= tj =.= tj emvmdjp tcmjrfms `jr dabfhatme pdatms. 15 3. Assubptfjh 17 j` 17 j` uhtapmrme pdatm fs erjppme fh _rjgdmbs =.1? ahe =.1? ahe =.11 tj =.11 tj apprmkfatm tcm fbpakt jh gjuheary kjheftfjhs. 1= Fh _rjgdmb =.3 tcm =.3 tcm ef``mrmhtfad mquatfjh `jr a pdatm jh mdastfk `juheatfjh fs kjhsfemrme. Bjrm tcah jhm kjbpdmxfty bay 10 gm aeeme sfbudtahmjusdy ahe tcm pjssfgdm kjbgfhatfjhs j` kjbpdmxftfms fs vmry darnm, gut cjpm`uddy tcm rmaemr gy mhe j` tcfs 1> kcaptmr wfdd emvmdjp a smhsm j` cjw tcmsm kjbpdmxftfms kah gm aeeme tj kdassfkad tcmjry j` pdatms. 13 Aeeftfjh j` kjbpdmxftfms dmaes tj bjrm nmhmrad tcmjrfms. Ah tcmjrfms. Ah fbpjrtaht kcmki jh tcmsm kjbpdmx tcmjrfms fs tcat wm scjude 16 gm agdm tj rmprjeukm rmsudts j` kdassfkad pdatm tcmjry gy apprjprfatm sugstftutfjh j` varfagdms. varfagdms. Ucfs kcmki wfdd gm pmr`jrbme fh tcm 7? mxabpdms prmsmhtme hmxt. 71 77 MSAB_DM =.1 7; Ucm krjss smktfjh j` dabfhatme pdatm baem `rjb twj batmrfads fs scjwh fh @fnurm =.6. =.6. Gjtc batmrfads cavm tcm sabm _jfssjh‖s 75 ratfj gut ef``mrmht bjeudus j` mdastfkfty. Ucm efspdakmbmht `fmde fs nfvmh 7= u ‛ z ‛ z j w x v ‛ z ‛ z j w y w w x y 70 wcmrm, z fs fs bmasurme `rjb tcm bfe sur`akm ahe z ahe z j fs tcm djkatfjh j` tcm hmutrad sur`akm. (a (a) Emtmrbfhm tcm vadum j` z j` z j assubfhn 7> wcmrm, z =.7 mxkmpt `jr batmrfad cjbjnmhmfty akrjss tcm tcfkihmss arm vadfe. (g (g) Jgtafh strmss `jrbudas ahe 73 add assubptfjhs fh [mktfjh =.7 mxkmpt M tcm ef``mrmhtfad mquatfjh njvmrhfhn tcm em`dmktfjh j` tcm dabfhatme pdatm. ( (k k ) Gy sugstftutfhn 76 1 2 M 7 , scjw tcat tcm rmsudts `jr ;? parts a ahe g nfvm tcm sabm rmsudts as kdassfkad pdatm tcmjry `jr cjbjnmhjus batmrfad. ;1 c1 M 1 ;7 c7 ;; M 7 ;5 ;= z ;0 @fnurm =.6 Menm vfmw j` dabfhatme pdatm fh Mxabpdm =.1. =.1. ;> M 7 ‛ M 1 c1 c 7 5 7 7 AH[. z j 2 ------------------------------------- 9 E mq w 2 p z x y E mq 2 F 1 ‛ F 2 M z ‛ zj e z ;3 c 7 M 1 c 1 + M 7 c 7 ;6 5? MSAB_DM =.7 51 57 @jr ah jrtcjtrjpfk pdatm tcm kjhstftutfvm mquatfjhs arm 5; (=.75) [ 71 2 [ 17 xx 2 [ 11 xx + [ 17 yy yy 2 [ 71 yy + [ 77 xx xy 2 [ ;; xy 55 (a) [tartfhn wftc tcm efspdakmbmht `fmde j` Mquatfjh (=.7) ahe (=.7) ahe assubfhn z assubfhn z fs fs bmasurme `rjb tcm bfe-sur`akm ahe a dd assubptfjhs 5= mxkmpt Assubptfjh > arm > arm vadfe, jgtafh tcm strmss `jrbudas ahe ef``mrmhtfad mquatfjh njvmrhfhn tcm em`dmktfjh j` jrtcjtrjpfk 50 pdatm. (g) [cjw tcat tcm rmsudts rmeukm tj tcjsm `jr fsjtrjpfk batmrfad gy apprjprfatm sugstftutfjh j` batmrfad kjhstahts. 5> 5 5 5 53 b xx z b yy z b xy z w w w AH[. xx 2 -------------------9 E 11 5 + 7 E 17 + 7 E ;; 7 7 + E 77 5 2 p z x y yy 2 ------------------- xy 2 -------------------56 x x y y c ; 17 c ; 17 c ; 17 =? =1 MSAB_DM =.; =7 =; Fh Bfhedfh-Wmfsshmr Fh Bfhedfh-Wmfsshmr pdatm pdatm tcmjry Assubptfjh 5 j` 5 j` pdahms smktfjhs pmrpmhefkudar tj tcm pdatm bfe sur`akm rmbafh hmardy pmrpmh=5 efkudar a`tmr em`jrbatfjh fs e rjppme tj akkjuht `jr scmar. (a (a) [tartfhn wftc tcm efspdakmbmht `fmde gmdjw jgtafh tcm strmss `jrbu== das ahe ef``mrmhtfad mquatfjhs f` add assubptfjhs `rjb 0 tcrjunc 17 arm 17 arm vadfe =0 (=.7=) u ‛ z x x y v ‛ z y x y w w x y => (g) [cjw tcm rmsudts rmeukm tj tcjsm j` kdassfkad pdatm tcmjry gy sugstftutfhn x 2 w x ahe y 2 w y . =3 b xx z b yy z b xy z =6 --AH[. xx 2 ------------------- yy 2 ------------------- xy 2 ------------------- xz 2 q x z yz 2 q y z ; ; ; c c c 17 c 17 c 17 0? 01
>3
=
_datms
1 7 ; 5 =.; HAPFMW‖[ [JD]UFJH @JW WMKUAHN]DAW _DAUM[ = 0 Fh 137?, Havfmr `juhe a sjdutfjh `jr a rmktahnudar pdatm tcat fs sfbpdy suppjrtme jh add fts gjuheary as scjwh fh @fnurm =.1?. =.1?. Fh > tcfs smktfjh wm efskuss Havfmr‖s apprjakc. 3 6 a 1 ? [fbpdy [uppjrtme x 11 e m 17 t p z ( x, x y) ,y) r e j m p t 1; r p j u p [ 15 p g y u d [ p 1= y d f b p [ 10 f b [ 1> 13 [fbpdy [uppjrtme 16 y 7? @fnurm =.1? [fbpdy suppjrtme rmktahnudar pdatm. 71 77 Ucm gjuheary vadum prjgdmb `jr tcm pdatm scjwh fh @fnurm =.1? kah =.1? kah gm wrfttmh as 7; 5 5 5 p x y w + 7 w + w 2 ----------------- z (=.70) 75 7 7 5 5 E 7= x y y x 70 7 7 w x ? w 7> --------Jh y Jh y 2 ? w x ? 2 ? 2 ? --------- ? y 2 ? Jh x Jh x 2 ? w ? y 2 ? 7 7 y 73 x (=.70a) 76 7 7 w x g 2 ? w --------Jh y Jh 2 2 ? y g w x g ;? --------- a y 2 ? Jh x Jh x 2 a w a y 2 ? y7 x7 ;1 Ucm `uhktfjhs sfh b x a ahe sfh b y g cavm twj prjpmrtfms tcat kah gm usme `jr `fhefhn sjdutfjhs. (f) Fts mvmh emrfva- ;7 tfvms rmsudt fh tcm sabm `uhktfjh budtfpdfme gy a kjhstaht ahe wm cavm jhdy mvmh emrfvatfvms fh jur gjuheary vadum prjgdmb. (ff) ;; Ucm prjeukt j` tcmsm twj `uhktfjhs satfs`y add gjuheary kjheftfjhs. [j wm em`fhm ;5 ;= (=.7>) ;0 w x y 2 ^ bh sfh b x a sfh h y g p z x y 2 _ bh sfh b x a sfh h y g b 21 h 21 b 21 h 21 wcmrm, ^ bh ahe _ bh arm kjhstaht kjm``fkfmhts tj gm emtmrbfhme. Ft scjude gm hjtme tcat tcm `uhktfjhs sfh b x a ahe sfh h y g ;> arm fhempmhemht `uhktfjhs ahe tcm fh`fhftm smrfms fs kjbpdmtm, cmhkm tcm efspdakmbmht `fmde wfdd gm kjrrmkt frrmspmktfvm j` tcm djae- ;3 ;6 fhn. Cjwmvmr, empmhefhn upjh tcm djaefhn tcm sjdutfjh bay cavm pjjr kjhvmrnmhkm ratm, tcat fs, ft rmqufrms tjj bahy tmrbs tj 5? prjeukm akkmptagdm akkuraky. 51 [ugstftutfhn Mquatfjhs (=.73) fhtj (=.73) fhtj Mquatfjh (=.70) wm (=.70) wm jgtafh 57 b 5 b 7 h 7 h 5 x y 1 x y (=.73) 5; ^ bh ------- + 7 ------- ------ + ------ sfh b -- sfh h -- 2 ---_ bh sfh b -- sfh h -- a a g g a g E b 21 h 21 a g b 21 h 21 55 As tcm [fhm `uhktfjhs `jrb ah fhempmhemht smt, tcm agjvm mquatfjh fbpdfms tcm kjm``fkfmhts j` tcm smrfms jh tcm dm`t bust mquad tj 5= kjm``fkfmhts j` tcm smrfms jh tcm rfnct, yfmdefhn tcm `jddjwfhn 50 _ 1 5> bh ^ bh 2 ---- --------------------------------------------------------------------------------------------------------- jr 5 7 7 5 E b a + 7 b a h g + h g 53 56 _ bh 1 (=.76) =? ^ bh 2 ----------5 ------------------------------------------------7 E b a 7 + h g 7 =1 =7 Uj emtmrbfhm tcm kjhstahts _ bh wm usm tcm jrtcjnjhadfty kjheftfjhs j` [fhm `uhktfjhs nfvmh gmdjw. =; bh ? (=.;?a) =5 sfh b sfh h e 2 ? 7 b 2 h == =0 Budtfpdyfhn Mquatfjh (=.7>) gy sfh f x a sfh l y g ahe fhtmnratfhn wm jgtafh => a g g a x x y y x y (=.;1) =3 _ bh sfh f -- sfh b -- e x sfh l -- sfh h -- e y 2 p z x y sfh f -- sfh l -- ex e y a a g g a g b 21 h 21 ? ? ? ? =6 [ugstftutfhn 1 2 x a ahe 7 2 y g fhtj Mquatfjh (=.;1) ahe (=.;1) ahe usfhn Mquatfjh (=.;?a) wm (=.;?a) wm jgtafh Mquatfjh (=.;7). (=.;7). 0? 01 AH[.
7 w ‛ x + y x y
p z x y 2 ‛ -----------------9 Nc
7
x y + x y
p z x y 2 -----------------9 E
7 ‛ x + y y x
7 Nc 2 --------------------- ‛ x + y E 1 ‛ y x
Havfmr‖s sjdutfjh `jr rmktahnudar pdatms 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
b 21
h 21
a _ bh -- sfh f 1 sfh b 1 e 1
g -- sfh l 7 sfh b 7 e 7
?
g
2
?
?
a ?
>6
x y p z x y sfh f -- sfh l -- ex e y jr a g
5 g a (=.;7) _ bh 2 -----p x y sfh b x a sfh h y g ex e y ag ? ? z Ucm rfnct cahe sfem j` Mquatfjh (=.;7) kah gm mvaduatme jhkm z x y fs ihjwh, cmhkm wm ihjw add _ bh . Hjw `rjb Mquatfjh (=.76) wm ihjw add ^ bh , cmhkm `rjb Mquatfjh (=.7>) wm ihjw w x y at add pjfhts jh tcm pdatm. F` tcm efspdakmbmht `fmde fs ihjwh, tcmh add bmkcahfks varfagdms kah gm k adkudatme. Uagdm =.7 dfsts =.7 dfsts add tcm emrfvatfvms j` w tcat wm wfdd hmme fh tcm kadkud atfjhs j` tcm bmkcahfks varfagdms.
Uagdm =.7 Emrfvatfvms j` pdatm em`dmktfjh.
w x
2
w y
2
7
w 7 x 7
w y 7
-- x-- y-- a b 21 h 21 ^ bh b kjs b a sfh h g
(=.;;a) w
-- x-- y-- g b 21 h 21 ^ bh h sfh b a kjs h g
(=.;;g)
2 ‛ -- a
2 ‛ -- g
7
w x y
7
7
b 21
h 21
b 21
h 21
;
x y 7 ^ bh b sfh b -- sfh h -- a g
(=.;;k)
x y 7 ^ bh h sfh b -- sfh h -- a g
(=.;;e)
7
2
;
2 ‛ -------- 7 a 7 g x y
------ x y ag b 21 h 21 ^ bh bh kjs b a-- kjs h g--
;
w x;
2 ‛ -- a
;
w 7 x y ;
b 21
h 21
;
2 ‛ -- g
;
h 21
2 ‛ -------- ag 7
w y;
;
b 21
x y ; ^ bh b kjs b -- sfh h -- a g
b 21
h 21
b 21
h 21
x y 7 ^ bh b h sfh b -- kjs h -- a g
(=.;;`)
(=.;;n)
x y 7 ^ bh bh kjs b -- sfh h -- a g
x y ; ^ bh h sfh b -- kjs h -- a g
(=.;;c)
(=.;;f)
(=.;;m)
MSAB_DM =.5 Ucm sfbpdy suppjrtme pdatm fh @fnurm =.1? fs =.1? fs suglmkt tj a uhf`jrb efstrfgutme `jrkm j` p j` pj. (a) Jgtafh a smrfms sjdutfjh `jr em`dmktfjh w ahe gmhefhn bjbmht b xx. (g) Assubm tcm pdatm fs a squarm pdatm g 2 a ahe _jfssjh‖s ratfj fs 2 1 ; , emtmrbfhm tcm baxfbub em`dmktfjh w ahe gmhefhn bjbmht b xx ahe kjbbmht jh tcm kjhvmrnmhkm j` tcm twj smrfms. 10 p j 1 x y --------------------------------------------------------- sfh b -- sfh h -- AH[. w x y 2 ----------0 7 7 7 b 2 1 ; =. h 2 1 ; = a g E b h b a + h g
AH[.
b xx x y 2
10 p j ---------- 5
b 2 1 ; =.
h 2 1 ; =
7 7 b a + h g x y --------------------------------------------------------- sfh b -- sfh h -- 7 7 7 a g b h b a + h g
AH[.
AH[.
b xx ba x
2
10 p j a 7 ---------------- 5
w ba x
b 2 1 ; =.
5
10 p j a 2 ----------------0 E
b 2 1 ; =.
h 2 1 ; =
h 2 1 ; =
b + h ‛ 7 7 ‛ 1 ---------------------------------9 7 7 7 bh b + h
b + h ‛ 7 7 7 7 b + h ‛ 1 ------------------------------------------------------------7 7 7 bh b + h
MSAB_DM =.= Ucm rmktahnudar pdatm @fnurm =.1? fs suglmktme tj a trahsvmrsm djae tcat varfms as z 2 p j x a y g . Emtmrbfhm (f) tcm em`dmktfjh w ahe bjbmht b xx `fmde jh tcm mhtfrm pdatm. (ff (ff)) tcm em`dmktfjh w ahe bjbmht b xx at tcm kmhtmr j` tcm pdatm usfhn at dmast 5 tmrbs fh tcm smrfms assubfhn tcm pdatm fs squarm f.m., g 2 a. ]sm 2 1 ; . 5 pj AH[< w x y 2 ----------0 E AH[< b xx x y 2
5 p j ------- 5
AH[.
b 21
(M1)
b+h ‛ 1 --------------------------------------------------------------------- b a 7 + h g 7 sfh b x-- sfh h y-- a g 7 7 7 h 21 b h b a + h g
b 21
b+h
x y ‛ 1 --------------------------------------------------------- sfh b -- sfh h -- 7 7 7 h 21 a g b h b a + h g
‛ ;
w a 7 a 7 2 1.?1 1.?1;6 ;6 1?
p j a 5 E 9
(M7) ‛ ;
b xx a 7 a 7 2 17.? 17.?;7 ;7 1?
pj a 7
3?
=
=.5
_datms
HAEAF-DMPV [JD]UFJH J@ WMKUAHN]DAW _DAUM[
Ucm Havfmr‖s sjdutfjh fs dfbftme tj pdatms tcat arm sfbpdy suppjrtme jh add sfems. @urtcmrbjrm, tcm ejugdm subbatfjh kah dmae tj sdjw kjhvmrnmhkm j` tcm smrfms `jr kjbpdmx djaefhn, partfkudardy `jr bjbmhts as wm cavm smmh fh tcm prmvfjus smktfjh. HaeafDmvy jvmrkjbms tcmsm ef``fkudty gy fbpjsfhn a ef``mrmht smt j` dfbftatfjhs. Ucm dfbftatfjhs arm< 1. _datm fs sfbpdy suppjrtme jh twj jppjsftm sfems as scjwh scjwh fh @fnurm =.11. =.11. Ucm jtcmr twj sfems kah cavm ahy typm j` suppjrt jr gjuheary kjheftfjhs. Ucm gjuheary kjheftfjhs at x at x 2 2 ? ahe x ahe x 2 2 a kah gm wrfttmh as 7 w -------- ? y x 7
w ? y 2 ?
w a y 2 ?
2 ?
7. Ucm djaefhn p z fs jhdy empmhemht jh x jh x,, tcat fs, p z x y 2
7 w -------- a y x 7 p z x .
2 ?
(=.;5a)
p x z
g/7 x
g/7
a
y
@fnurm =.11 _datm nmjbmtry ahe djaefhn `jr Haeaf-Dmvy‖s bmtcje
^m kjhsfemr tcm em`dmktfjh sjdutfjh fh twj p arts as scjwh gmdjw. w x y 2 w p x + w c x y
(=.;=)
[ugstftutfhn tcm agjvm fhtj tcm gf-carbjhfk mquatfjh (=.7?) wm (=.7?) wm jgtafh< 5
5
w 2
e w p e x
5
5
5
+
5
w c w c w c +7 7 7 + 5 x y y5 x
p z x 2 ----------- E
^rftfhn tcm agjvm mquatfjh as a smt j` twj mquatfjhs scjwh gmdjw. 5
e w p e x
5
p z x 2 ----------- E
5
5
w c w c +7 7 7 x y x 5 =.5.1
(=.;0a)
5
w c + y5
2 ?
(=.;0g)
Cjbjnmhmjus sjdutfjh
Hjtfhn tcm `akt tcat tcm sfh b x a , satfs`y tcm gjuheary kjheftfjhs fh Mquatfjh (=.;5a), (=.;5a), wm kcjjsm tcm `jddjwfhn `jr tcm cjbjnmhmjus sjdutfjh w c x y 2
b 21
C b y sfh b x a
(=.;0k)
[ugstftutfhn tcm agjvm fhtj Mquatfjh (=.;0k) wm (=.;0k) wm jgtafh 7
b21
5
b 5 b 7 e C b e C b ------ C b ‛ 7 ------- a a e y 7 + e y 5 sfh b x a 2 ?
(=.;0e)
@jr tcm agjvm mquatfjh tj gm vadfe at add pjfhts jh tcm pdatm tcm `jddjwfhn mquatfjh bust gm satfs`fme. 7
5
b 5 b 7 e C b e C b ------ C b ‛ 7 ------- a a e y7 + e y 5 2 ?
(=.;0m)
Ucm agjvm mquatfjh fs ah jrefhary ef``mrmhtfad mquatfjh wftc kjhstaht kjm``fkfmhts, wcjsm sjdutfjh kah gm `juhe gy sug y stftutfhn C b 2 m tj jgtafh tcm kcaraktmrfstfk mquatfjh gmdjw.
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
Haeaf-Dmvy sjdutfjh j` rmktahnudar pdatms
31
7 1 jr jr (=.;0`) b a 5 ‛ 7 b a 7 7 + 5 2 ? 7 ‛ b a 7 2 ? 2 b a 9 b a 7 Ucm rmpmatme rjjts fbpdy tcm `jddjwfhn sjdutfjh ; b a y b a y ‛ b a y ‛ b a y C b 2 K 1 b m + K 7 b ym + K ; b m + K 5 b ym (=.;0n) 5 = Ucm agjvm mquatfjh kah adsj gm mxprmssme fh tmrbs j` cypmrgjdfk `uhktfjhs as scjwh gmdjw. 0 C b 2 A b kjsc b y a + Gb b y a sfh c b y a + K b sfh c b y a + Eb b y a kjsc b y a jr > (=.;0c) C b 2 A b + E b b y a kjsc b y a + Kb + G b b y a y sfh c b y a 3 6 Ucm cjbjnmhjus sjdutfjh fs 1? (=.;>) w c x y 2 A b + E b b y a kjsc b y a + Kb + G b b y a sfh c b y a sfh b y a b 21 11 17 =.5.7 _artfkudar sjdutfjh 1; 15 Ucmrm arm twj ways tj `fhe tcm partfkudar sjdutfjh. Fh Bmtcje F wm `fhe a smrfms sjdutfjh ahe fh Bmtcje FF wm usm efrmkt fhtmnra1= tfjh. Bmtcje F fs usm`ud fh rmdatfhn tcm kjhstahts Ab , G b , K b , ahe E b tj tcm djaefhn parabmtmrs fh p z . Bmtcje FF kah nfvm kdjsme wcmh p z fs a sfbpdm `uhktfjh j` x j` x,, sukc as a pjdyhjbfad. ^m kah usm a kjbgfhatfjh j` tcm twj apprjakcms tj jgtafh 10 `jrb sjdutfjhs wcmh p 1> tcm kjbpdmtm sjdutfjh. 13 Bmtcje F< ^m usm smrfms tj rmprmsmht tcm em`dmktfjh a he efstrfgutme djaes as sc jwh gmdjw. 16 w p x 2 ^ b sfh b x a p z x 2 _ b sfh b x a (=.;3) b 21 b 21 7? _ b kah gm `juhe usfhn tcm jrtcjnjhadfty kjheftfjh Mquatfjh (=.;?a) as (=.;?a) as scjwh gmdjw 71 a a _ b sfh b x a sfh h x a e x 2 p z x sfh h x a e x e x jr 77 b 21 ? ? 7; 7 a (=.;6) _ h 2 -- p z x sfh h x a e x 75 a ? 7= [ugstftutfhn Mquatfjhs (=.;3) ahe (=.;3) ahe Mquatfjhs (=.;3) fhtj (=.;3) fhtj Mquatfjh (=.;0a) wm (=.;0a) wm jgtafh 70 5 b x 1 x 7> ^ b ------- sfh b -- 2 ---_ b sfh b -- b 21 a a E b 21 a 73 76 @jr tcm mquadfty tj cjde `jr makc tmrb tcm `jddjwfhn mquatfjhs bust gm satfs`fme ;? a 5 _ b (=.5?a) . ^ b 2 ------- -----;1 b E ;7 ;; Ucm partfkudar sjdutfjh kah gm wrfttmh as ;5 a 5 _ b ------- ------ sfh b x a (=.51a) w p x 2 ;= b 21 b E ;0 ;> [ugstftutfhn Mquatfjhs (=.;0c), (=.;0c), (=.;3), (=.;3), ahe (=.5?a) fhtj (=.5?a) fhtj Mquatfjh (=.;=) wm (=.;=) wm jgtafh tcm kjbpdmtm sjdutfjh as ;3 a 5 _ b b b b b x ;6 ------- ------ + A b + Eb ------- y kjsc ------- y + K b + Gb ------- y sfh c ------- y sfh b -- (=.57) w x y 2 b E a a a a a b 21 5? 51 Ucm kjhstaht _ b fs emtmrbfhme `rjb Mquatfjh (=.;6). (=.;6). Ucm kjhstahts Ab , G b , K b , ahe E b arm emtmrbfhme `rjb tcm 57 gjuheary kjheftfjhs jh y 2 g 7 . 5; Bmtcje FF< w satfs`fms tcm gjuheary kjheftfjh at x at x 2 2 ? ahe x ahe x 2 2 a. Ucus tcm partfkudar sjdutfjh bust satfs`y tcm gjuheary kjhefc 55 tfjhs j` zmrj efspdakmbmht ahe bjbmht at x at x 2 2 ? ahe x ahe x 2 2 a nfvmh gy Mquatfjh (=.;5a). (=.;5a). ^m kjhsfemr a uhf`jrb djae p z 2 p ? . Gy 5= fhtmnratfhn Mquatfjh (=.;0a) `jur (=.;0a) `jur tfbms wm jgtafh 50 p ? x5 ; 7 5> w p 2 ----- ------ + K 1 x + K 7 x + K ; x + K 5 (=.57a) E 75 53 @rjb tcm gjuheary kjheftfjhs fh Mquatfjh (=.;5a) wm (=.;5a) wm jgtafh tcm kjhstahts ahe cmhkm tcm sjdutfjh as fh 56 5 ; =? p ? a x 5 -x- + -x- ---------(=.57g) w 2 ‛ 7 p =1 a a 75 E a =7 @jr jtcmr djaefhns, wm kah `fhe tcm partfkudar sjdutfjh fh a sfbfdar bahhmr. =; =5 MSAB_DM =.0 == Ucm sfbpdy suppjrtme pdatm fh @fnurm =.17 fs =.17 fs suglmkt tj a uhf`jrb efstrfgutme `jrkm j` p j` pj. (a) Jgtafh a smrfms sjdutfjh `jr em`dmk=0 tfjh w usfhn Haeaf-Dmvy Bmtcje. (g) @jr baxfbub em`dmktfjh w ahe gmhefhn bjbmht b xx, kjbparm tcm kjhvmrnmhkm j` => Haeaf-Dmvy Bmtcje wftc Havfmr‖s bmtcje `jr a squarm pdatm ahe _jfssjh‖s ratfj j` 2 1 ; . =3 =6 0? 01
37
=
_datms a [fbpdy [uppjrtme e t m r j g / 7 p p u [ y d p f b [ g / 7
x
e t m r j p p u [ y d p f b [
p z ( x, x y) ,y)
[fbpdy [uppjrtme y
@fnurm =.17 5
AH[.
p? a w x y 2 ---------75 E
-x- a
5
x ; x ‛ 7 -- + -- + a a AH[.
AH[.
=.=
b 2 1 ; =
A b kjsc b y + Gb b y sfh c b y sfh b x
w x ‛ ; -----------ba ------------ 2 1;.? 1; .?7 7 1? + 5 p ? a E B ba x ----------- 2 7 p ? a
?.17= +
b 2 1 ; =
Ab ---------------------- ‛ 1 b ‛ 1 7 5 b 2 1 ; = p? a E
7 A ‛ G 7 b b -- b ----------------------- ‛ 1 b ‛ 1 7 5 ; p a E ?
KFWK]DAW _DAUM[
Ucm mquatfjhs j` gmhefhn j` kfrkudar pdatms arm emrfvme fh tcfs smktfjh. ^m wfdd usm kydfherfkad kjjrefhatms scjwh fh fh@fnurm @fnurm =.1; ahe trahs`jrb tcm mquatfjhs fh [mktfjh =.7 fhtj =.7 fhtj kydfherfkad kjjrefhatms.
u ) x (
u r ) r (
v
y
v ) (
z (w) @fnurm =.1; Kfrkudar pdatm.
^m kah rmdatm tcm kjjrefhatms fh tcm kartmsfah ( x, x, y, z ) ahe kydfherfkad kjjrefhatms (r (r , , z ) as scjwh gmdjw. x 2 r kjs
y 2 r sfh
7
7
r 2 x + y
7
‛ 1
2 tah
y x
(=.5;a)
^m kah wrftm
r x
r y
x
2
y 2 -- 2 sfh r
r + x r x
2 kjs 7
x 7 7
x7
x
7
y ‛ sfh ‛ y x 2 ------------------------- 2 ‛ ---- 2 -------------7 7 r 1 + y x r ^m kah rmdatm tcm `frst ahe emrfvatfvms fh ka rtmsfah tj kydfherfkad as scjwh gmdjw. x 2 -- 2 kjs r
2
7
2
y
2
r + y r y
x kjs 1 x 2 ------------------------- 2 ---- 2 ------------ (=.5;g) 7 7 r 1 + y x r
2 sfh
kjs + ----------- r r
(=.5;k)
r ( kjs ‛ sfh ( kjs ‛ sfh ----------- ) + ----------- ) jr x r r r x r r
2 kjs kjs
y7
‛ sfh ---------- r r
y
7 7 kjs + sfh sfh sfh ‛ ---------------------- ‛ ----------- ‛ sfh ‛ ---------- 7 7 7 r r r r r r
kjs kjs r ( sfh + ----------- ( sfh + -----------)+ ) jr y r r r y r r
(=.5;e)
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
Kfrkudar pdatms 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
7
y 7
2 sfh sfh
kjs 7 ‛ ----------- 7 r r 7
kjs kjs sfh + ------------ kjs + ------------ 7 ‛ ----------- r r r r
3;
7
(=.5;m)
Aeefhn Mquatfjhs (=.5;e) ahe (=.5;e) ahe (=.5;m) wm (=.5;m) wm jgtafh< 7
7 + 7 x7 y 7
7
7 + 1-- + -1--- 7 7 r 7 r r r
1 + -1--- 2 2 -r 2 r r r r 7 7 ^m kah rmdatm tcm efspdakmbmhts fh tcm kartmsfah (u (u, v , w) w) ahe kydfherfkad kjjrefhatms (u (ur , v, w)as w)as scjwh gmdjw. 7
u r 2 u kjs + v sfh
(=.5;`)
v 2 ‛ u sfh + v kjs
(=.5;n)
^m emrfvm tcm kfrkudar pdatm mquatfjhs assubfhn add assubptfjhs j` [mktfjh =.7 arm =.7 arm appdfkagdm. [jbm j` tcm adnmgrafk emtafds arm sifppme as tcmsm emtafds arm sabm as fh [mktfjh =.7. =.7. 1. Efspdakmbmhts [ugstftutfhn Mquatfjh (=.7) fhtj (=.7) fhtj Mquatfjh (=.5;n) wm (=.5;n) wm jgtafh
w w sfh x kjs + ‛ z y
w w x sfh + ‛ z y kjs
ur 2 ‛ z
v 2 ‛ ‛ z
w w r
[ugstftutfhn Mquatfjh (=.5;k) fhtj (=.5;k) fhtj tcm agjvm mquatfjh wm jgtafh
w r
z w v 2 ‛ - r
u r 2 ‛ z
w w r
(=.55)
7. [trafhs
[ugstftutfhn Mquatfjh (=.55) fhtj (=.55) fhtj Mquatfjhs (1.;1a) tcrjunc (1.;1a) tcrjunc (1.;1k) wm (1.;1k) wm jgtafh tcm strafhs as 7
rr
2
w ur 2 ‛ z 7 r r
7
7
1 w 1 w u 1 v 2 ----r + -- 2 ‛ z -- + ----7 7 r r r r r
1 w v 1 u v 1 w r 2 -- r + ‛ ----- 2 ‛ 7 z -- ‛ ---7- (=.5=) r r r r r r
;. [trmssms
@jr pdahm strmss, tcm strmssms kah gm wrfttmh as M rr 2 ------------------ rr + 1 ‛ 7 7
7
rr
‛ zM w w w 2 ------------------+ -+ --- 1 ‛ 7 r 7 r r r 7 7
M 2 ------------------ + rr 1 ‛ 7
7
r
2 N r jr
r
1 w 1 w 2 ‛ 7 Nz - ‛ ----7 r r r
7
7
‛ zM 1 w 1 w w 2 ------------------ -r - r + ----7 7 + 7 7 r 1 ‛ r
(=.50)
5. [trmss Wmsudtahts
^m kah rmpdakm tcm strmssms wftc mqufvadmht `jrkms ahe bjbmhts as scjwh fh Mquatfjh (=.5>). (=.5>). @fnurm =.15 scjws =.15 scjws tcm pjsftfvm efrmktfjhs j` tcm strmss rmsudtahts. b rr 2
c
rr e z
b 2
e z
b r 2
e z
c
e z
b r 2
r
c
q
c
q
r b
b
q b
r
rr
rz e z
q 2
e z z
(=.5>)
c
b b
c
q b
qr 2
r
r
r b
r
r
b
rr
r z
@fnurm =.15 _jsftfvm strmss rmsudtahts.
[ugstftutfhn Mquatfjh (=.50) fhtj (=.50) fhtj Mquatfjh (=.5>) wm (=.5>) wm jgtafh tcm fhtm rhad bjbmhts as scjwh gmdjw. 7
b rr
7 w w + ---- w 2 ‛ E + - r 7 r r r 7 7
7
b
1 w 1 w 2 ‛ E -+ --- r r r 7 7
7
w + 7 r
7
b r
1 w 1 w 2 ‛ E 1 ‛ - ‛ ----7 r r r
(=.53)
[ugstftutfhn Mquatfjh (=.53) fhtj (=.53) fhtj Mquatfjh (=.50) wm (=.50) wm jgtafh tcm strmssms as scjwh gmdjw.
rr
b rr z 2 ------------------- c ; 17
b z 2 ------------------- c ; 17
r
b r z 2 ------------------- c ; 17
(=.56)
35
=
_datms
=. Mqufdfgrfub ahe ef``mrmhtfad mquatfjh
^m kjude kjhsfemr a `rmm gjey efanrab j` a ef``mrmhtfad mdmbmht ahe wrftm tcm mqufdfgrfub mquatfjhs as wm efe fh fh[mk[mktfjh =.7. =.7. Cjwmvmr wm wfdd taim a sfbpdmr adtmrhatfvm rjutm. ^m hjtm tcat q x q y ahe q r q arm vmktjrs fh Kartmsfah ahe kydfherfkad kjjrefhatm systmb rmdatme as wmrm tcm efspdakmbmhts as fh Mquatfjh (=.5;n). (=.5;n). ^m adsj kah sugstftutm `jr q x q y nfvmh gy Mquatfjh (=.16a) ahe jgtafh tcm rmsudts `jr qr q as gmdjw.
7 w kjs ‛ E 7 w sfh x y 7 7 E w sfh ‛ E w kjs x y
q r 2 q x kjs + q y sfh 2 ‛ E q 2 ‛ q x sfh + q y kjs 2
[ugstftutfhn Mquatfjh (=.5;k) fhtj (=.5;k) fhtj tcm agjvm mquatfjhs wm jgtafh qr 2 ‛ E
7 w r
E 7 q 2 ‛ ---- w r
(=.=?)
Fh a sfbfdar bahhmr tcm rmsudtaht scmar `jrkm jh tcm gjuheary fs 1 b r P r 2 q r + -r
P 2 q +
br r
(=.=1)
Ucm ef``mrmhtfad mquatfjh fs tcm gf carbjhfk mquatfjhs wrfttmh as scjwh gmdjw.
7 7 w
2 p z r
7
w c m rm
7
1 + -1--- 2 -r r r r r 7 7
(=.=7)
0. Gjuheary kjheftfjhs
Ucm gjuheary kjheftfjhs jh ah menm `jddjws tcm gasfk rudm tcat mftcmr rmsudtaht scmar `jrkm P r fs spmkf`fme jr tcm em`dmktfjh w , ahe mftcmr ahe mftcmr tcm hjrbad bjbmht brr fs spmkf`fme jr tcm hjrbad sdjpm w r as scjwh gmdjw. At makc gjuheary pjfht spmkf`y< P r
jr
w
ahe
jr
b rr
w r
(=.=;)
Ucm jhm mxkmptfjh tj tcm agjvm fs a sjd fe kfrkudar pdatm, wcfkc cas jhdy jhm gjuheary. ^m ^m hmme twj bjrm kjheftfjhs. Ucmsm twj kjheftfjhs arm tcat em`dmktfjh ahe sdjpm arm gjuheme fh tcm kmhtmr. ^m wfdd smm tcfs typm j` gjuheary kjheftfjh fh tcm hmxt smktfjh.
=.0
ASF[VBBMUWFK _DAUM[
Fh tcfs smktfjh wm kjhsfemr a sfbpdf`fme smt j` mquatfjhs `rjb tcm nmhmrad mquatfjhs j` kfrkudar pdatms. ^m assubm djaefhn, batmrfad prjpmrty, ahe nmjbmtry arm fhempmhemht j` tcm ahnudar kjjrefhatm . Ucfs fs ah axfsybbmtrfk prjgdmb fh wcfkc sjdutfjh fs fhempmhemht j` ahnudar kjjrefhatm , tcat fs, w 2 w r . Cmhkm w 2 ? ahe w r 2 e w e r . [ugstftutfhn tcfs rmsudt fh tcm mquatfjhs j` [mktfjh =.= wm =.= wm jgtafh tcm mquatfjhs nfvmh gmdjw. Efspdakmbmhts
u r 2 ‛ z e w e r
[trafh
rr
7
7
2 ‛ z e w e r
7 e w w 2 ‛ E + -- e r 7 r r
Bjbmhts
b rr
Dapdakm Jpmratjr
7
1e e 2 -- r r e r e r
[trmssms
rr
b rr z 2 ------------------- c; 17
[cmar `jrkms
q r 2 ‛ E
Wmsudtaht scmar `jrkms
P r 2 q r
Ef``mrmhtfad mquatfjh
v 2 ?
w w r
(=.=5a)
z 2 ‛ - e w e r r
(=.=5g) 7
b
1 ew e w 2 ‛ E -+ r e r e r 7
b r 2 ?
(=.=5k)
(=.=5e)
b z 2 ------------------- c ; 17
ew e 1 -- e r e r r e r e r
r
2 ?
(=.=5m)
(=.=5`)
q 2 ?
P 2 ?
(=.=5n)
E e e 1 e ew ---r -- r 2 p z r r e r e r r e r e r
(=.=5c)
Mquatfjh (=.=5c) fs (=.=5c) fs ah jrefhary ef``mrmhtfad mquatfjh tcat kah gm fhtmnratme `jur tfbms tj jgtafh tcm sjdutfjh as scjwh gmdjw. e 1 e ew E r -- r 2 F 1 r + K 1 e r r e r e r
wcmrm
F 1 r 2
rp r e r z
(=.==a)
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
Axfsybbmtrfk pdatms
3=
1 F 1 r 1 e ew (=.==g) E -- r 2 F 7 r + K 1 dh r + K 7 wc m rm F 7 r 2 ----------- e r 7 r e r e r r ; 7 7 7 ew r r r 5 (=.==k) E r wc m rm F ; r 2 rF 7 r e r e r 2 F ; r + K 1 7 dh r ‛ -5--- + K 7 7 + K ; = 7 0 K 1 7 F ; r r 7 (=.==e) Ew r 2 F 5 r + ------ r dh r ‛ r + K 7 ---- + K ; dh r + K 5 wc m rm F 5 r 2 ----------- e r > 5 5 r 3 Ucm fhem`fhftm fhtmnrads arm kaddme tcm djaefhn fhtmnrads ahe kah gm `juhe as p as p z (r ) fs ihjwh9 K 1, K 7, K ;, K 5 arm kjhstahts 6 tj gm emtmrbfhme `rjb tcm gjuheary kjheftfjhs. Ucm partfkudar sjdutfjh fs F 5 r E ahe tcm cjbjnmhjus sjdutfjh fs 1? K 1 5 r 7 dh r ‛ r 7 + K 7 r 7 7 + K ; dh r + K 5 E . Gy sugstftutfhn Mquatfjh (=.==e) fhtj (=.==e) fhtj Mquatfjhs (=.=5k) ahe (=.=5k) ahe (=.=5`) wm (=.=5`) wm jgtafh 11 tcm sjdutfjhs `jr bjbmhts ahe scmar `jrkm as scjwh gmdjw. 17 7 7 b rr 2 ‛ F 7 ‛ 1 ‛ F ; r ‛ K 1 7 1 + dh r + 1 ‛ 7 ‛ K 7 1 + 7 + K ; 1 ‛ r (=.==m) 1; 15 7 7 b 2 ‛ 1 ‛ F ; r + F 7 ‛ K 1 7 1 + dh r ‛ 1 ‛ 7 ‛ K 7 1 + 7 ‛ K ; 1 ‛ r (=.==`) 1= q r 2 ‛ F 1 + K 1 r (=.==n) 10 1> 13 =.0.1 ]hf`jrb djae 16 ]hf`jrb djae, f.m, p z r 2 p j fs ah apprjxfbatfjh usme fh bahy praktfkad appdfkatfjhs. ^m assubm tcm djae akts jh tcm mhtfrm 7? pdatm. ^m jgtafh tcm djaefhn fhtmnrads gy sukkmssfvm fhtmnratfjh as scjwh gmdjw. 71 7 7 5 5 F 1 F ; p j r pj r p j r p j r 77 F 1 r 2 p j r e r 2 ---------F 7 r 2 ---- e r 2 ---------F ; r 2 rF 7 e r 2 ---------F 5 r 2 ---- e r 2 ---------(=.=0) 7; r r 7 5 10 05 75 Fh tcm agjvm mquatfjhs hj fhtmnratfjh kjhstahts arm fhkdueme as tcmsm arm akkjuhtme fh tcm cjbjnmhjus sjdutfjh. F` tcm 7= djaefhn fs jhdy jh part j` tcm pdatm, tcmh tcm djaefhn fhtmnrads wfdd cavm tj gm rm-mvaduatme. Hubmrfkad fhtmnratfjh kjude gm usme 70 `jr bjrm kjbpdmx djaefhns. 7> 73 =.0.7 [jdfe kfrkudar pdatms 76 ;? [jdfe kfrkudar pdatms scjwh fh @fnurm =.1; cas =.1; cas jhdy jhm gjuheary, cmhkm wm cavm jhdy twj gjuheary kjheftfjhs `jr `jur uhihjwh ;1 kjhstahts. ^m jvmrkjbm tcfs prjgdmb gy rmqufrfhn tcat efspdakmbmht w ahe sdjpm e w e r bust gm gjuheme at r 2 ? . Ucfs rmqufrm;7 bmht j` gjuhemehmss `jr em`dmktfjh ahe sdjpm arm kaddme rmnudarfty kjheftfjhs. Ucm `uhktfjh dh r wjude tmhe tj fh`fhfty as r tmhe ;; tj zmrj, tcus f` em`dmktfjh ahe sdjpm arm tj rmbafh gjuheme tcmh tcm kjhstahts budtfpdyfhn dh r bust gm zmrj. Ucus, `rjbMquatfjhs `rjbMquatfjhs ;5 (=.==k) ahe (=.==k) ahe (=.==e) wm (=.==e) wm jgtafh< ;= (=.=>) K 1 2 ? ahe K ; 2 ? ;0 Ucm rmbafhfhn twj kjhstahts kah gm jgtafhme `rjb tcm twj gjuheary kjheftfjhs at tcm jutmr gjuheary as embjhstratme ;> fh Mxabpdm =.>. =.>. ;3 ;6 5? MSAB_DM =.> 51 Ucm menm vfmw j` a sjdfe kfrkudar pdatm tcat fs sfbpdy suppjrt me ahe djaeme gy a bjbmht fs scjwh fh@fnurm fh @fnurm =.1=. =.1=. Emtmrbfhm tcm 57 em`dmktfjh ahe gmhefhn bjbmhts brr ahe b fh tmrbs j` E j` E,, , r , B j, ahe W ahe Wj. 5; Wj 55 B j B j 5= r 50 5> z 53 =.>. 56 @fnurm =.1= [fbpdy suppjrtme sjdfe kfrkudar pdatm fh Mxabpdm =.>. 7 7 B j W j ‛ r =? AH[. w r 2 ---------------------------b rr 2 b 2 ‛ B j =1 7 E 1 + =7 =; =.0.; Ahhudar pdatms =5 == Ucm nmjbmtry j` ah ahhudar pdatm fs scjwh fh @fnurm =.10. =.10. Uwj gjuheary kjheftfjhs wfdd gm fbpjsme jh makc gjuheary, rmsudtfhn =0 fh `jur kjheftfjhs tcat kah gm usme tj mvaduatm tcm `jur kjhstahts fh Mquatfjh (=.==e). (=.==e). ^cmh tcm efstrfgutme djae fs zmrj, tcm djae=> fhn fhtmnrads arm zmrj ahe tcm adnmgra fs strafnct `jrware. ^cmh tcm efstrfgutme `jrkm fs hjh-zmrj tcm rmsudtfhn adnmgra gmkjbms =3 tmefjus. [fbpdf`fkatfjh j` adnmgra kah gm ejhm gy hjh-efbmhsfjhadfzfhn tcm varfagdms ahe mquatfjhs `jr ahhudar pdatm as efskussme =6 fh tcm hmxt smktfjh. 0? 01
30
=
_datms Wf
Wj
r
@fnurm =.10 Ahhudar pdatm.
=.0.5
Hjh-efbmhsfjhad mquatfjhs fh ahhudar pdatms
Ucm adnmgra `jr mvaduatfhn tcm kjhstahts fh ahhudar pdatms wftc efstrfgutme djaes kah gm tmefjus. [fbpdf`fkatfjh fh tcm adnmgra kah gm akcfmvme gy hjh-efbmhsfjhadfzfhn tcm varfagdms. ^m start wftc hjh-efbmhsfjhadfzfhn tcm raefad kjjrefhatmr kjjrefhatm r wftc wftc rmspmkt tj tcm fhhmr raefus W raefus Wf ahe tcm efstrfgutme djae p djae p z wftc rmspmkt tj sjbm kcaraktmrfstfk vadum p vadum pj as scjwh gmdjw. )
r 2 r W f
pj 2 p z p j
)
(=.=3a)
Parfagdms wftc kurvme gars wf dd rm`mr tj tcm hjh-efbmhsfjhad varfagdms. ^cmh wm fbpjsm gjuheary kjheftfjhs jh tcm fhhmr gjuheary ( r 2 1 ) tcm djnarftcb `uhktfjh wfdd gmkjbm zmrj ahe tcm mquatfjhs wfdd gm sfbpdf`fme. As sfbpdf`fme. As was efskussme fh [mktfjh ;.1.=, ;.1.=, wm kah em`fhm jtcmr hjh-efbmhsfjhad varfagdms usfhn tcm em`fhftfjhs j` tcm varfagdms. Ucm varfagdms ahe tcm mquatfjhs wm hmme arm nfvmh gmdjw fh hjh-efbmhsfjhad `jrb. Ucm djaefhn fhtmnrads arm nfvmh `jr uhf`jrb djae. )
W W 2 -----? W f )
ew E e w er er --------- 2 -------------------------; e r pj W f
)
)
)
7
F 7 r 2 r
)
)
7
)
5
)
F ; r 2 r
7
w r 2 F 5 r + A 1 r dh r ‛ r
)
)
)
)
)
b b 2 ----------7 p j W f
)
)
7
F 1 r 2 r
b rr b rr 2 ----------7 p j W f
)
Ew w 2 ----------5 p j W f
)
)
7
)
)
5
)
10
)
7
)
7
(=.=3g)
05
(=.=3k)
)
F 5 r 2 r
5 + A 7 r 5 + A ; dh
)
5
)
)
r + A 5
(=.=3e)
7
7
)
q r q r 2 --------- p j W f
)
)
e w r r r r --------- 2 F ; r + A 1 -------- dh r ‛ -------- + A 7 -------- + A ; 7 7 5 e r
)
)
)
)
(=.=3m)
)
)
)
)
b rr 2 ‛ F 7 ‛ F ; r
)
)
b 2 ‛ F ; r
)
7
)
7
)
A7 1 + 7 r + 1 ‛ 7 7 ‛ ----------------------+ A ; 1 ‛ r 7
1 ‛ ‛ A1 1 + dh )
1 ‛ + F 7 ‛ A 1 1 + dh )
)
)
(=.=3`)
r ‛ 1 ‛ 7 7 ‛ A 7 1 + 7 ‛ A ; 1 ‛ r
)
7
)
)
(=.=3n)
q r 2 ‛ F 1 + A 1 r
)
(=.=3c)
ahe A 5 arm kjhstahts tj gm emtmrbfhm `rjb gjuheary kjheftfjhs. Ucm agjvm mquatfjhs arm vadfe `jr 1 r wcmrm, A 1 A 7 A ; ahe A
)
W
.
MSAB_DM =.3 Ucm menm j` ah ahhudar pdatm tcat fs sfbpdy suppjrtme at tcm jutmr gjuheary ahe djaeme wftc bjbmhts B bjbmhts B f ahe B ahe B j fs scjwh fh @fnurm =.1>. =.1>. Emtmrbfhm tcm em`dmktfjh j` tcm pdatm fh tmrbs j` r , E, E, Wf, Wj, B f, ahe B ahe B j. Wf B j
Wj
B f
r
B f
B j
z
=.3. @fnurm =.1> A sfbpdy suppjrtme ahhudar pdatm fh Mxabpdm =.3. 7
7
7
7
7
7
W j B j ‛ W f B f W j ‛ r B f ‛ B j W f W j r AH[< w r 2 ‛ -------------------------------- --------------------+ -------------------------------------- dh ----7 E 1 + W 7 ‛ W 7 E 1 ‛ W 7 ‛ W7 W j j f j f
(M;)
MSAB_DM =.6 Ah menm vfmw j` ah ahhudar pdatm tcat fs attakcme tj a rfnfe sca`t ahe suglmktme tj a uhf`jrb efstrfgutme djae pj fs scjwh fh @fnurm =.13. =.13. Emtmrbfhm tcm baxfbub em`dmktfjh w ahe baxfbub gmhefhn bjbmhts brr ahe b fh tmrbs j` E j` E,, pj, ahe W ahe Wj. ]sm 2 1 ; ahe W 2 7 Wf
Wj
pj
pj Wfnfe
@fnurm =.13 Ahhudar pdatm attakcme tj a rfnfe sca`t fh Mxabpdm =.6
AH[.
‛ ;
w ba x 2 3.560 560 1?
p W
5
j j -----------
E
brr ba x
7
2 ‛ ?.1>70 p j W j
b ba x
7
2 ‛ ?.?=>== p j W j
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
Kdjsurm
3>
1 =.> KDJ[]WM 7 7.0. ^m usme tcm djnfk tj emvmdjp tcm sfbpdmst pjssf; Fh tcfs kcaptmr wm jhkm bjrm saw tcm bjeudarfty j` djnfk scjwh fh @fnurms 7.0. gdm tcmjry (kdassfkad tcmjry) ahe femhtf`fme tcm assubptfjhs hmmeme `jr sfbpdf`fk atfjh. ^m ^m tcmh fhkjrpjratme kjbpdmxftfms fh 5 = mxabpdms ahe prjgdmbs tj emvmdjp hmw `jrbudas ahe kcmkime tcm vadfefty j` tcmsm `jrbudas gy apprjprfatm sugstftutfjh j` varf0 agdms tj rmprjeukm rmsudts j` tcm kdassfkad pdatm tcmjry. @jr kdassfkad pdatm tcmjrfms wm adsj emvmdjpme sjdutfjh tmkchfqums gy fbpjsfhn dfbftatfjhs jh nmjbmtry, djaefhn, ahe > 3 gjuheary kjheftfjhs. Havfmr‖s sjdutfjh fs rmstrfktme tj rmktahnudar pdatms wftc sfbpdy suppjrt me menms jh add sfems. Haeaf-Dmvy 6 sjdutfjh fs rmstrfktme tj djhn rmktahnudar pdatms wftc twj jppjsftm gjuhearfms jh tcm scjrt sfem tj gm sfbpdy suppjrtme. Gy rmstrfkt1? fhn djaefhn, nmjbmtry, ahe batmrfad prjpmrtfms tj gm axfsybbmtrfk w m wmrm agdm tj `fhe sjdutfjh `j r kfrkudar pdatms. Gy hjhefbmh11 sfjhadfzfhn ahhudar pdatm mquatfjhs wm wmrm agdm tj rmeukm adnmgra fh `fhefhn sjdutfjhs wftc varfmty j` gjuheary kjheftfjhs. ^m wfdd smm emrfvatfjhs j` pdatm mquatfjhs wftc kjbpdmxftfms fh Kcaptmr > `rjb > `rjb mhmrny pmrspmktfvm. ^m wfdd ej tcm emrf17 1; vatfjh usfhn varfatfjhad kadkudus, wcfkc wfdd dmt us emvmdjp mquatfjhs `jr pdatms wftc argftrary scapms. Gy kjbgfhfhn varfatfjhad 3 wm wfdd smm a vmry kjbpakt ahe mdmnaht way j` jgtafhfhn pdatm mquatfjhs. Ft wfdd dmt us 15 kadkudus ahe fhefkfad hjtatfjh fh Kcaptmr 3 wm 1= jgtafh mquatfjhs j` gmhefhn j` pdatm wftc darnm strafhs. 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
33
=.3
=
_datms
[VHJ_[F[ J@ MQ]AUFJH[ Kdassfkad pdatm tcmjry
Efspdakmbmhts
u ‛ z w x
[trafhs
xx
2 ‛ z w x
[trmssms
xx
‛ M z 7 7 7 7 2 ------------------- w x + w y 7 1 ‛
7
h xx 2
[trmss Wmsudtahts
v ‛ z w y
b xx 2
7
e z c xx
z
xx e z
c
xx
2 ? 7
xx
Mqufdfgrfub Mquatfjhs
q x q y + x y
Ef``mrmhtfad Mquatfjh.
E w 2 E
xy e z
c
e z c yy 7
b z 2 --------- yy ----------- c; 17 q x 2
5
5
2 ‛ 7 z w x y
‛ M z 7 7 7 7 2 ------------------- w y + w x 7 1 ‛
yy
7
7
xy
w + 7 w x 7 y7 x 5
h xy 2 h yx 2 7
e z c xy
z
b yy 2
yy e z
c
7
xy
2 ‛ 7 N z w x y
2 ? 7
7
7
2 ‛ E w y + w x
;
Mc E 2 ------------------------7 17 1 ‛
7
yy
q x 2
e z c xz
b xx b yx + x y
2 ‛ E
7 w x
e z c yz
b xy b yy + x y
P x 2
b xy q x + y
2 ‛ E
w + 7 ‛ w 7 ; x y x
b yx x
2 ‛ E
w + 7 ‛ w y x7 y ;
2 ‛ E
7 w y
2 p z x y
[pmkf`y Y P y jr w ahe
Jh a kjrhmr
[pmkf`y Y W kjrhmr jr w
q y 2
5
w y5
+
Jh y Jh y 2 kjhstaht
q y 2
w x
w b yy jr y
P y 2 q y +
;
;
;
;
7
W kjrhmr 2 b xy + b yx 2 7 b xy Jrtcjnjhadfty Kjheftfjhs
7
b z 2 --------- xy ----------- c ; 17
xy
[pmkf`y Y P x jr w ahe b xx jr
Jh x Jh x 2 kjhstaht Gjuheary Kjheftfjhs
2 ‛ E 1 ‛ w x y
2 ‛ p z x y
5
7
2 ‛ E w x + w y
b z 2 --------- xx ----------- c ; 17
[trmss @jrbudas
7
2 ‛ z w y
h yy 2
z
b xy 2 b yx 2
w w x y
bh
?
sfh b sfh h e 2 7 ?
?
b 2 h
kjs b kjs h e 2
w 2 ‛ 7 E 1 ‛ x y
bh
? 7
b 2 h
Wmktahnudar pdatms w x y 2 Havfmr‖s sjdutfjh
b 21
h 21
^ bh sfh b x a sfh h y g
_ bh 1 ^ bh 2 ---------- ------------------------------------------------5 7 E b a 7 + h g 7
5 _ fl 2 -----ag
5
w x y 2 w p x + w c x y 9 e w p ex Haeaf-Dmvf sjdutfjh
w p x 2
5
g a
p x y sfh f x a sfh l y a ex e y
7 2 p z x E 9 _ h 2 -a
b 21
a
p x sfh h x a e x ? z
5
5 _ b ------- sfh b x a b21 a b E
p ? a 5 ; 9 w p 2 ----------- x a ‛ 7 x a + x a 9 75 E 75 E
wc x y 2
z
? ?
Ab + Eb b y a kjsc b y a + K b + Gb b y a sfh c b y a sfh b x a
Kfrkudar pdatms )
r 2 r W f
p j 2 p z pj
)
W 2 W ? W f
)
; ew --------- 2 E e w e r p j W f e r )
Hjh-efbmhsfjhad axfsybbmtrfk pdatm
)
7
)
)
7
)
7
)
F 7 r 2 r )
7
)
7
)
5
)
)
)
)
)
7
)
)
)
7
)
)
)
b rr 2 ‛ F 7 ‛ F ; r
)
)
b 2 ‛ F ; r
)
)
)
7
)
7
)
)
)
7
)
)
q r 2 q r pj W f
F 5 r 2 r )
5
)
05
)
+ A ; dh r + A 5 7
)
+ A ;
r + 1 ‛ 7 ‛ A 7 7 1 + + A ; 1 ‛ r
1 ‛ + F 7 ‛ A 1 7 1 + dh
q r 2 ‛ F 1 + A 1 r
)
10
7 + A7 7 r
1 ‛ ‛ A1 7 1 + dh )
7
)
)
5
)
+ A7 5 r
r e w e r 2 F ; r + A 1 7 r dh r ‛ r
)
)
b 2 b p j W f
F ; r 2 r
w r 2 F 5 r + A 1 5 r dh r ‛ r
)
7
)
b rr 2 brr p j W f
)
F 1 r 2 r
5
w 2 E w p j W f
)
)
7
)
r ‛ 1 ‛ ‛ A 7 7 1 + ‛ A ; 1 ‛ r
)
7
)
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
0
Fhtrjeuktfjh tj Mdastfkfty
DMAWHFHN JGLMKUFPM[ 1 1. Uj gmkjbm `abfdfar wftc tcm gasfk mquatfjhs j` dfhmar mdastfkfty ahe tcm Afry strmss `uhktfjh. 7 7. Uj uhemrstahe tcm appdfkatfjh ahe rmdatfjhscfp j` tcm mquatfjhs j` mdastfkfty tj bmkcahfks j` batmrfads. ; 5 [trmss ahe stf``hmss stf``hmss ahadysfs ahe emsfnh j` tcrmm-efbmhsfjhad tcrmm-efbmhsfjhad (ahe bahy twj-efbmhsfjhad) twj-efbmhsfjhad) bakcfhm kjbpjhmhts kahhjt gm akcfmvme = gy bmahs j` a bmkcahfks j` batmrfads app rjakc. rjakc . Hubm rfkad bmtcjes bmtcj es arm ar m j`tmh j`t mh usm e, gut gu t mvmh tcmsm kadd `jr ah a h uhemrstah uhe mrstahefhn efhn 0 j` mquatfjhs j` mdastfkfty. Fh mdastfkfty, wm stuey cjw tcm varfagdms j` bmkcahfks (efspdakmbmhts, strafhs, strmssms, fhtmrhad > `jrkms, ahe bjbmhts) vary `rjb pjfht tj pjfht jh a mdastfk gjey. Ucfs fs fh kjhtrast tj bmkcahfks j` batmrfads apprjakc fh 3 wcfkc wm avmranm tcm varfagdms j` bmkcahfks gy apprjxfbatfhn tcmfr varfatfjh akrjss tcm krjss smktfjh (rjes, sca`ts, ahe 6 gmabs) gmabs ) jr tcfkih t cfkihmss mss (pdatm ( pdatmss ahe scmdds ) j` ah mdastfk mda stfk gjey. Ucm `fmde j` mda stfkfty stfkft y fs vmry darnm ahe gmyjhe g myjhe tcm skjpm sk jpm j` j ` a sfhs fh1? ndm kjursm, dmt adjhm a kcaptmr. Cmrm wm fhtrjeukm sjbm gasfk kjhkmpts ahe mquatfjhs j` mdastfkfty, ahe usm tcm mquatfjhs j` 11 mdastfkfty tj emrfvm `jrbudas `jr sukc appdfkatfjhs as rjtatfhn efsis ahe tjrsfjh j` hjh-kfrkudar sca`ts 17 Ucm gasfk mquatfjhs j` mdastfkfty arm tcm strafh‛efspdakmbmht mquatfjhs, tcm nmhmradfzme vmrsfjh j` Cjjim‖s daw rmdatfhn 1; strmssms ahe strafhs (stuefme fh Kcaptmr 1), tcm mqufdfgrfub mquatfjhs jh strmssms, ahe tcm kjbpatfgfdfty mquatfjhs `jr strafhs, 15 wcfkc mhsurm sfhndm-vadume sjdutfjhs `jr efspdakmbmhts. 1= Ah fbpjrtaht tmkchfqum `jr sjdvfhn ah mdastfkfty prjgdmb fs tcm usm Afry strmss `uhktfjh `rjb wcfkc strmssms kah gm 10 emrfvme tcat satfs`y tcm mqufdfgrfub mquatfjhs. ^m wfdd smm cjw tcm Afry strmss `uhktfjhs kah gm usme tj jgtafh rmsudts `jr smvmrad 1> bmkcahfks j` batmrfads prjgdmbs. 13 16 0.1 MDA[UFKFUV MQ]AUFJH[ 7? 71 Ucm gasfk mquatfjhs j` dfhmar mdastfkfty arm tcm strafh‛efspdakmbmht mquatfjhs, tcm kjbpatfgfdfty mquatfjhs, wcfkc arm emrfvme 77 `rjb tcm strafh‛efspdakmbmht mquatfjhs, tcm nmhmradfzme vmrsfjh j` Cjjim‖s daw, ahe tcm mqufdfgrfub mquatfjhs (smm Gargmr 7; Y1667R, Gjrmsf ahe Kcjhn Y163>R, ahe @uhn Y160=R). Ucmsm arm efskussme fh tcfs smktfjh. 75 7= 0.1.1 [trafh-efspdakmbmht mquatfjhs 70 Ucm strafh‛efspdakmbmht rmdatfjhscfps fh kartmsfah kjjrefhatms ahe pjdar kjjrefhatms emrfvme fh Kcaptmr 1, arm rmpm atme gmdjw `jr 7> kjhvmhfmhkm. 73 76 u yy v zz 2 w xy yx u v yz 2 zy 2 v + w zx 2 xz 2 w + u (0.1) xx x y y x z z y x z ;? ;1 v u v v u u ;7 (0.7) rr 2 r 2 ----r + 1-- r 2 1-- r + ‛ ----r r r r r r ;; ;5 0.1.7 Kjbpatfgfdfty mquatfjhs ;= ;0 [fx kjbpatfgfdfty mquatfjhs mhsurm tcat tcm strafh `fmdes arm sukc tcat tcm efspdakmbmht `fmdes jgtafhme `rjb fhtmnratfhn Mquatfjh ;> (0.1) arm (0.1) arm sfhndm vadume. ^m emrfvm jhm j` tcm sfx kjbpatfgfdfty mquatfjhs gy taifhn tcm emrfvatfvms j` Mquatfjh (0.1) ahe (0.1) ahe fhtmr;3 kcahnfhn tcmfr jremr as scjwh fh Mquatfjh (0.;a). (0.;a). ;6 7 7 7 7 xy 5? u + v 2 ------------- ; u + -------------; v 2 u + v (0.;a) ------------ 2 51 x y x y y x x y 7 x7 y y 7 x x 7 y 57 [ugstftutfhn Mquatfjh (0.1) fh (0.1) fh tcm rfnct-cahe sfem j` Mquatfjh (0.;a), (0.;a), wm jgtafh Mquatfjh (0.;g). (0.;g). 5; 33
36
0 Fhtrjeuktfjh
tj Mdastfkfty 7
7
xx yy + y 7 x 7
7
2
xy x y
(0.;g)
Ucm jtcmr `fvm kjbpatfgfdfty mquatfjhs kah gm emrfvme fh a sfbfdar bahhmr (smm _rjgdmbs 0.1 ahe 0.1 ahe 0.7). 0.7).
0.1.;
_dahm strmss ahe pdahm strafh
Ucm twj-efbmhsfjhad prjgdmbs j` pdahm strmss ahe pdahm strafh wmrm efskussme fh [mktfjh 1.3.;. 1.3.;. Ucm mquatfjhs j` pdahm strmss ahe pdahm strafh kah gm emrfvme `rjb tcm mquatfjhs j` tcm nmhmradfzme Cjjim‖s daw YsmmMquatfjh Ysmm Mquatfjh (1.57)R (1.57)R ahe arm statme fh sdfnctdy ef``mrmht `jrb `jr kjhvmhfmhkm. @jr pdahm @jr pdahm strmss wm cavm M xx
xx ‛ yy
M yy
yy ‛ xx
N xy
xy
(0.5a)
Adtmrhatfvmdy, Mquatfjh (0.5a) kah (0.5a) kah gm wrfttmh as
xx M xx yy / 1 ‛ 7
yy M yy xx / 1 ‛ 7
xy N xy
(0.5g)
@jr pdahm @jr pdahm strafh wm strafh wm cavm 7 N xx
1 ‛ xx ‛ yy
7 N yy
1 ‛ yy ‛ xx
N xy
xy
(0.=a)
Adtmrhatfvmdy, Mquatfjh (0.=a) kah (0.=a) kah gm wrfttmh as<
xx
7N -------------------- 1 ‛ xx 1 ‛ 7
yy
yy
7N -------------------- 1 ‛ yy 1 ‛ 7
xx
xy N xy
(0.=g)
Ucm agjvm mquatfjhs arm vadfe `jr ahy jrtcjnjhad systmb. Gy rmpdakfhn x rmpdakfhn x wftc wftc r ahe y ahe y wftc wftc wm kah jgtafh tcm rmdatfjhscfp gmtwmmh strmssms ahe strafhs fh pjdar kjjrefhatms. Ucm mquatfjhs `jr pdahm strmss ahe pdahm strafh arm sfbfdar fh `jrb. Ft kah gm vmrf`fme tcat wm kah jgtafh tcm mquatfjhs j` pdahm strmss gy sugstftutfhn /(1+) fh pdakm j` ahe immpfhn N uhkcahnme fh pdahm strafh mquatfjhs. Ucfs emvfkm kah gm usme fh jgtafhfhn a sjdutfjh `jr pdahm strmss `rjb tcm pdahm strafh sjdutfjh, as wm scadd emskrfgm fh[mktfjh fh [mktfjh 0.7. 0.7. ^m rmkjre tcfs jgsmrvatfjh as Mquatfjh (0.0) `jr (0.0) `jr `uturm usm. / 1 _dahm strafh _dahm strmss (0.0) NN Ahjtcmr adtmrhatfvm tj tcm `jrb j` pdahm strmss ahe strafh mquatfjhs fs
xx
+ 1 xx ‛ ; ‛ yy
2 -------------------------------------------------------3N
yy
+ 1 yy ‛ ; ‛ xx
2 -------------------------------------------------------3N
wc m rm
2
; ‛ ----------- 1+ ; ‛ 5
_dahm [trmss
(0.>a)
_dahm [trafh
Mquatfjh (0.>a) kah (0.>a) kah gm sjdvme tj jgtafh strmssms fh tmrbs j` strafhs as scjwh gmdjw.
xx 0.1.5
N 2 ----------------- + 1 xx + ; ‛ yy ‛ 1
yy
N 2 ----------------- ; ‛ xx + + 1 yy ‛ 1
(0.>g)
Mqufdfgrfub mquatfjhs
@fnurm 0.1a 0.1a scjws a twj-efbmhsfjhad ef``mrmhtfad mdmbmht fh pdahm strmss (jr pdahm strafh). .^m assubm at makc pjfht tcmrm arm akts gjey `jrkms @ `jrkms @ ahe @ x ahe @ y, ahe gjey bjbmht K z . Gjey `jrkms arm `jrkms pmr uhft vjdubm, sukc as tcat eum tj nravfty. Fh banhmtfk `fmdes wm kah adsj cavm gjey bjbmhts wcfkc cavm uhfts j` bjbmhts pmr uhft vjdubm. @fnurm 0.1g 0.1g scjws tcm `rmm gjey efanrab jgtafhme gy budtfpdyfhn strmssms gy tcm armas j` tcm pdahms ahe gjey `jrkms ahe bjbmhts budtfpdfme gy tcm ef``mrmhtfad vjdubm Gy `jrkm mqufdfgrfub fh tcm x tcm x efrmktfjh efrmktfjh fh @fnurm 0.1g 0.1g, wm jgtafh Mquatfjh (0.3a). (0.3a).
xx yx --------- --------- xx x ex ey ez ‛ xx ey ez yx y ey ex ez ‛ yx ex ez @ x ex ey ez ?
xx yx ---------- + --------- + @ x x y
?
jr
(0.3a)
[fbfdardy gy `jrkm mqufdfgrfub fh tcm y tcm y efrmktfjh efrmktfjh fh @fnurm 0.1g 0.1g, wm jgtafh Mquatfjh (0.3g). (0.3g).
xy yy --------- + ---------- + @ y 2 ? y x
(0.3g)
Kjhsfemr tcm bjbmht mqufdfgrfub agjut tcm kmhtmr j` tcm ef``mrmhtfad mdmbmht fh @fnurm 0.1g 0.1g. Ucm bjbmht at tcm kmhtmr wfdd gm `rjb scmar s trmssms jhdy. ^m ^m kah hmndmkt tcm tmrbs kjhtafhfhn tcm prjeukt j ` `jur ef``mrmhtfads, sfhkm tcmsm t mrbs tmhe tj zmrj `astmr fh tcm dfbft tcah tcm tmrbs wftc tcm prjeukts j` tcrmm ef``mrmhtfads. As a rmsudt, wm jgtafh Mquatfjh (0.3m). (0.3m).
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
Mdastfkfty mquatfjhs 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
yy ey y
yy +
yx + yx ey y
y
xy
+ yy ey exez yy y y
xy +
@ y
xy ex x
xy eyez
ey
@ x
xx +
+ yx ey exez yx y + xy ex eyez xy x
@ y(exeyez )
K z (exeyez )
K z
xx
6?
xx eyez
xx ex x
@ x(exeyez )
+ xx ex eyez xx x
ex
x
x
yx exez
yx yy
yy exez (a)
(g)
@fnurm 0.1 (a) A twj-efbmhsfjhad ef``mrmhtfad mdmbmht ahe ( g) fts `rmm gjey efanrab.
xy ex + --------- ex ey ez ------ xy 7 x
ex + xy ey ez ------ 7
yx ey ey + --------- ey ex ez ------ ‛ yx ex ez ------ yx 7 7 y
xy ey ez ex ‛ yx e x ez ey + K z exey ez 2
?
+ K z exey ez 2 ? jr
jr
xy + K z 2 yx
(0.3k)
Ucm agjvm mquatfjh scjws tcat tcm sybbmtry j` scmar strmssms wcfkc wm cavm assubme sj `ar fs jhdy pjssfgdm fh agsmhkm j` gjey bjbmht. ^m wfdd kjhtfhum tj assubm gjey bjbmhts arm zmrj ahe hjt kjhsfemr fts fbpakt fh rmbafhfhn emvmdjpbmhts. Mquatfjh (0.3a), (0.3a), (0.3g), (0.3g), ahe (0.3m) arm (0.3m) arm tcm mqufdfgrfub mquatfjhs at a pjfht j` a twj-efbmhsfjhad mdastfk gjey fh Kartmsfah kjjrefhatms. Ucm mqufdfgrfub mquatfjhs fh tcrmm efbmhsfjhs kah gm jgtafhme gy kjhsfemrfhn tcm mqufdfgrfub j` a ef``mrmhtfad mdmbmht (kugm) fh tcrmm efbmhsfjhs ahe arm nfvmh gmdjw.
xx yx zx ---------- + --------- + --------+ @ 2 ? x y z x
xy yy zy --------- + ---------- + ---------+ @ 2 ? x y z y
xy
yx
2
yz
2
zy
zx
xz yz zz --------+ --------+ ---------+ @ z 2 ? x y z
2
xz
(0.3e) (0.3m)
A ef``mrmhtfad mdmbmht wftc strmssms fh pjdar kjjrefhatms fs scjwh fh @fnurm 0.7a 0.7a. ^m assubm tcm mdmbmht mxtmhes gy ez fh tcm z tcm z efrmktfjh. efrmktfjh. ^m budtfpdy tcm strmss gy tcm ef``mrmhtfad arma j` tcm pdahms jh wcfkc tcmy arm aktfhn tj prjeukm a `rmm gje y efanrab j` tcm ef``mrmhtfad mdmbmht scjwh fh @fnurm 0.7g 0.7g. Ft scjude gm hjtme tcat sfems AG sfems AG ahe KE ahe KE arm arm at ah ahndm j` e tj tcm raefad efrmktfjh. Ucus, tcm `jrkms aktfhn jh sfems AG sfems AG ahe KE ahe KE wfdd wfdd cavm tj gm rmsjdvme fhtj kjbpjhmhts `jrkms fh tcm r ahe ahe efrmktfjh.
r +
r +
r e
r er r
r
E
rr + + e
@
r +
rr er r
+ e
@ r
K
rr
re re
e A e r r r
r e
r + erez E
rr +
@ re erez
erez
rr er r + er e ez r
@ re erez r
K
re
G
e r
r er r + er e ez r
rr re ez A r re ez
e r
r erez
G
erez
(g)
(a)
@fnurm 0.7 _dahm strmss fh`fhftmsfbad (a) strmss mdmbmht. (g) `jrkm mdmbmht fh pjdar kjjrefhatms
Gy mqufdfgrfub j` `jrkms fh tcm r efrmktfjh efrmktfjh wm jgtafh<
rr +
rr e er r + er e ez ‛ rr re ez ‛ r erez kjs ------ 7 r ‛
+
e
+
r +
r e
e erez kjs ------ 7
e e erez sfh ------ + erez sfh ------ + @ r re erez 2 ? 7 7
61
0 Fhtrjeuktfjh
tj Mdastfkfty
^m hjtm tcat kjs e 7 2 1 ahe sfh e 7 2 e 7 `jr tcm fh`fhftmsfbad ahndm e . [ugstftutfhn ahe hmndmktfhn tmrbs wftc `jur ef``mrmhtfads as tcmy wfdd tmhe tj zmrj `astmr tcah jtcmr tmrbs, wm jgtafh
rr er e ez +
rr r rere ez + e erez ‛ ereze + @ r re erez 2 r rr 1 r rr ‛ + -+ --------------------- + @ r 2 r r r
? jr
?
(0.6a)
Fh a sfbfdar bahhmr, gy mqufdfgrfub j` `jrkms fh tcm efrmktfjh wm jgtafh 1 r 7 r -+ + ---------- + @ 2 ? r r r
(0.6g)
Gy taifhn bjbmht at pjfht agjut tcm kmhtmr j` tcm ef``mrmhtfad mdmbmht wm jgtafh tcm sybbmtry j` strmssms.
r 0.1.=
2
r
(0.6k)
Gjuheary kjheftfjhs
Uj sjdvm tcm mdastfkfty mquatfjhs, wm hmme gjuheary kjheftfjhs. At a nfvmh pjfht jh tcm sur`akm j` a gjey, wm bust spmkf`y mftcmr efspdakmbmht jr traktfjh fh a nfvmh efrmktfjh. Ucm traktfjh strmss vmktjr was n fvmh gyMquatfjh gyMquatfjh (1.1?). (1.1?). Ucm gjuheary kjheftfjhs at a pjfht jh a sur`akm arm as nfvmh gmdjw. u 2 u?
jr
xx h x + xy h y + xz h z 2 t x
(0.1?a)
v 2 v?
jr
yx h x + yy h y + yz h z 2 t y
(0.1?g)
w 2 w?
jr
zx h x + zy h y + zz h z 2 t z
(0.1?k)
wcmrm u?, v?, ahe w? rmprmsmht tcm spmkf`fme efspdakmbmht fh tcm x tcm x,, y, y, ahe z ahe z efrmktfjhs, efrmktfjhs, rmspmktfvmdy9 t x, t y, ahe t z rmprmsmht tcm spmkf`fme traktfjh fh tcm x tcm x,, y, y, ahe z ahe z efrmktfjh, efrmktfjh, rmspmktfvmdy9 ahe h x, h y, ahe h z rmprmsmht tcm efrmktfjh kjsfhms j` tcm uhft hjrbad tj tcm sur`akm fh tcm x tcm x,, y, y, ahe z ahe z efrmktfjh, efrmktfjh, rmspmktf vmdy.
0.7
ASF[VBBMUWFK _WJGDMB[
Fh axfsybbmtrfk prjgdmbs tcm djaefhn, tcm nmjbmtry, ahe tcm batmrfad prjpmrtfms arm add fhempmhemht j` ahnudar djkatfjh. ]hemr tcm kjheftfjhs j` axfsybbmtry, tcm rmsudts j` efspdakmbmhts, strafhs, ahe strmssms scjude adsj gm fhempmhemht j` . Ucus, add emrfvatfvms wftc rmspmkt tj wfdd gm zmrj, ahe add varfagdms arm `uhktfjhs j` tcm raefad kjjrefhatm r jhdy9 jhdy9 partfad emrfvatfvms j` r kah kah gm wrfttmh as jrefhary emrfvatfvms j` r . Mquatfjh (0.7) ahe (0.7) ahe Mquatfjh (0.6a) kah (0.6a) kah gm wrfttmh as scjwh gmdjw.
rr 2
eu u --------r 2 --------r er r
u 2 ----r r
e rr rr ‛ ---------+ --------------------- + @ r 2 ? er r
0.7.1
(0.11a)
(0.11g)
Axfsybbmtrfk pdahm strafh
@jr pdahm strafh, Mquatfjh (0.=g) kah (0.=g) kah gm wrfttmh as
rr 2
7N -------------------- 1 ‛ rr + 1 ‛ 7
7N 2 -------------------- 1 ‛ + rr 1 ‛ 7
(0.17a)
[ugstftutfhn Mquatfjh (0.11a) fhtj (0.11a) fhtj Mquatfjh (0.17a) wm (0.17a) wm jgtafh
rr 2
eu u 7N -------------------- 1 ‛ --------r + ----r er r 1 ‛ 7
u r eu r 7N 2 -------------------- 1 ‛ ---- + -------r er 1 ‛ 7
(0.17g)
[ugstftutfhn Mquatfjh (0.17g) fhtj (0.17g) fhtj Mquatfjh (0.11g) ahe (0.11g) ahe sfbpdf`yfhn, wm jgtafh tcm ef``mrmhtfad mquatfjh gmdjw. 7
1 eu r u 1 ‛ 7 + -- -------- ‛ ----r + ------------------------- @ r 2 ? r er r 7 7 1 ‛ N e r 7 e u r
jr
1 ‛ 7 @ 2 ? e 1 -- e ru r + ------------------------- @ 7 1 ‛ N r e r r e r
(0.1;)
Ucm sjdutfjh tj tcm agjvm mquatfjh kah gm wrfttmh as scjwh gmdjw. u r 2 u r c + u r p
u r c 2 K 1 r + K 7 r
(0.15)
wcmrm, u r p fs tcm partfkudar sjdutfjh tcat empmhes upjh tcm vadum j` @ r . ahe u r c fs tcm cjbjnmhmjus sjdutfjh kjrrmspjhefhn tj @ r 2 ?. Ucm kjhstahts K 1 ahe K 7 wcfkc bust gm emtmrbfhme `rjb tcm gjuheary kjheftfjhs j` a partfkudar appdfkatfjh.
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
Wjtatfhn efsis 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
0.7.7
67
Axfsybbmtrfk pdahm strmss
@jr pdahm strmss, Mquatfjh (0.5g) kah (0.5g) kah gm wrfttmh as
rr 2
u eu M M ------------------- rr + 2 ------------------- --------r + ----r r 1 ‛ 7 1 ‛ 7 er
eu u M M ------------------- ----r + --------r 2 ------------------+ 2 rr er 1 ‛ 7 1 ‛ 7 r
(0.1=)
[ugstftutfhn tcm Mquatfjhs (0.1=) ahe (0.1=) ahe (0.1=) fhtj (0.1=) fhtj Mquatfjh (0.11g) ahe (0.11g) ahe sfbpdf`yfhn, wm jgtafh a ef``mrmhtfad mquatfjh< 7
e u 1 eu r u 1 ‛ 7 @ 2 ? ----------r + -- -------- ‛ ----r + ------------------- @ r 7 r er r 7 M er
7
1 ‛ @ 2 ? e 1 -- e ru r + ------------------- @ r M e r r e r
jr
(0.10)
Ucm sjdutfjh tj Mquatfjh (0.10) fs (0.10) fs nfvmh gy Mquatfjh (0.15), (0.15), wftc tcm partfkudar part j` tcm sjdutfjh empmhemht upjh tcm vadum j` @ j` @ r.
0.;
WJUAUFHN EF[I[
A nrfhefhn wcmmd jr a efsi graim kah gm bjemdme as a rjtatfhn efsi. ^m assubm tcat tcm efsi fs tcfh (pdahm strmss) ahe fs rjtatfhn at a kjhstaht ahnudar spmme . Ucfs wjude suglmkt a pjfht j` tcm efsi tcat fs at a raefad efstahkm r tj tj ah akkmdmratfjh j` 7r fh fh tcm 7 raefad efrmktfjh. F` tcm bass emhsfty j` tcm efsi batmrfad fs b tcmh tcm raefad gjey `jrkm wfdd gm @ gm @ r 2 b r wcfkc wm sugstftutm fhtj Mquatfjh (0.10) tj (0.10) tj jgtafh 7
1 ‛ 7 e 1 -- e ru r + ------------------- b r 2 ? M e r r e r
(0.1>)
Ucm partfkudar sjdutfjh tj Mquatfjh (0.1>) fs (0.1>) fs 7
7
1 ‛ b r ; ---------------------------------------------u 2 ‛ r p M 3
(0.13)
Ucm tjtad sjdutfjh nfvmh gy Mquatfjh (0.15) kah (0.15) kah hjw gm wrfttmh as 7
7 ;
K 1 ‛ b r u r 2 K 1 r + ------7 ‛ -----------------------------------------r 3 M
(0.16a)
[ugstftutfhn Mquatfjh (0.16a) fhtj (0.16a) fhtj Mquatfjh (0.1=) ahe (0.1=) ahe (0.1=), (0.1=), wm jgtafh tcm raefad ahe tahnmhtfad hjrbad strmssms as scjwh gmdjw.
rr 2
K 7 1 ‛ M ; + 7 7 ------------------- K 1 1 + ‛ ---------------------- ‛ ----------------- b r 7 7 3 1 ‛ r
(0.16g)
K 7 1 ‛ M 1 + ; 7 7 2 ------------------K 1 1 + + ---------------------- ‛ -------------------- b r 7 7 3 1 ‛ r
(0.16k)
Ucm kjhstahts K 1 ahe K 7 fh Mquatfjhs (0.16a), (0.16a), (0.16g) ahe (0.16g) ahe (0.16k) kah (0.16k) kah hjw gm emtmrbfhme `rjb tcm gjuheary kjheftfjhs `jr a partfkudar appdfkatfjh. [jbm typfkad gjuheary kjheftfjhs arm efskussme gmdjw. 1. A sjdfe rjtatfhn efsi (_rjgdmb 0.17)< 0.17)< Ucm jutmr gjuheary fs strmss `rmm, tcus rr r 2 W j 2 ? . @jr a sjdutfjh tj gm `fhftm at tcm kmhtmr (r (r 2 2 ?) j` a sjdfe efsi rmqufrms K 7 2 ? . 7. A rjtatfhn efsi wftc a cjdm (_rjgdmb cjdm (_rjgdmb 0.1;)< 0.1;)< Ucm fhhmr gjuheary (r (r 2 W 2 Wf) ahe jutmr gjuheary (r (r 2 W 2 Wj) arm strmss `rmm. Ucus tcm gjuheary kjheftfjhs arm rr r 2 W f 2 ? ahe rr r 2 W j 2 ? . ;. A rjtatfhn efsi gjheme jh a rfnfe sca`t< Ucm sca`t< Ucm jutmr gjuheary fs strmss `rmm, ahe tcm pjfht jh tcm fhhmr gjuheary kahhjt gm efspdakme. Ucus tcm gjuheary kjheftfjhs arm ur r 2 W f 2 ? ahe rr r 2 Wj 2 ? .
MSAB_DM 0.1 Ucm baxfbub rjtatfjhad spmme at wcfkc a nrfhefhn wcmmd kah jpmratm fs kaddme tcm —gurstfhn spmme,‒ sfhkm f` tcfs spmme fs mxkmmeme, baxfbub tmhsfdm strmss wfdd kausm tcm wcmmd tj gurst. Kjhsfemr a nrfhefhn wcmmd wftc fhhmr raefusa raefus a, jutmr raefus 7a 7a, _jfssjh ratfj 2 1/ ; , bjeudus j` mdastfkfty M, mdastfkfty M, ahe bass emhsfty j` b. Jgtafh a rmdatfjhscfp gmtwmmh tcm g urst spmme bax ahe tcm baxfbub addjwagdm tmhsfdm strmss addjw fh tmrbs a, M, ahe b. Assubm tcat tcm nrfhefhn wcmmd fs bjuhtme jh a rfnfe sca`t. AH[.
0.5
bax
2 ?.>=>=
addjw b a 7
AFWV [UWM[[ @]HKUFJH
Bahy mdastfkfty prjgdmbs kah gm sjdvme wftc sfnhf`fkaht masm gy usfhn tcm Afry strmss `uhktfjh. Ucm `uhktfjh fs kcjsmh sukc tcat tcm mqufdfgrfub mquatfjhs fh tcm agsmhkm j` gjey `jrkms arm fbpdfkftdy satfs`fme gy tcm strmssms fh twj efbmhsfjhs. Ft bay gm vmrf`fme tcat tcm strmssms em`fhme fh Mquatfjh (0.7?) satfs`y (0.7?) satfs`y Mquatfjhs (0.3a) ahe (0.3a) ahe (0.3g) f` (0.3g) f` @ @ ahe @ x 2 ? ahe @ y 2 ?.
6;
0 Fhtrjeuktfjh
tj Mdastfkfty
xx
7
yy
2 -------- y 7
7
xy
2 -------- x 7
7
2 ‛ ------------ x y
(0.7?)
Ucm `uhktfjh fs kaddme tcm Afry strmss `uhktfjh. Uj emtmrbfhm , wm hmme tj satfs`y tcm kjbpatfgfdfty kjheftfjh j` Mquatfjh (0.;g) ahe (0.;g) ahe add tcm gjuheary kjheftfjhs. [ugstftutfhn Mquatfjh (0.7?) fhtj (0.7?) fhtj Mquatfjh (0.>a) wm (0.>a) wm jgtafh tcm strafhs nfvmh fh Mquatfjh (0.71). (0.71).
xx
1 2 ------3N
7
7
+ 1 ---------7 ‛ ; ‛ ---------7 y x
yy
1 2 ------3N
7
7
7
+ 1 ---------7 ‛ ; ‛ ---------7 x y
xy
1 2 ‛ ---- ------------ N x y
(0.71)
[ugstftutfhn tcm agjvm mquatfjh fhtj Mquatfjh (0.;g) wm (0.;g) wm jgtafh 7 1 1 7 ‛ ; ‛ -------- 7 + 7 ------ 7 ‛ ; ‛ --------7 ----------------------+ 1 + 1 y 7 x7 x7 3 N x7 y 7 y 7 3 N
2
7 7 jr ‛ -1--- ----------- x y N x y
1 1 5 2 ? jr -- + 1 --------- ‛ ; ‛ ------------------ + -- + 1 --------- ‛ ; ‛ ------------------ + -----------------3 y 5 x 7 y 7 3 x 5 x 7 y 7 x 7 y7 5
5
5
5 + + 1 --------5 + -----------------5 3 ‛ 7 ; ‛ + 1 -------- y 5 x 5 x 7 y 7 5
5
5
2 ?
jr
5
5
5 5 + --------5 --- + 7 ----------------- + 1 ------ x 7 y 7 y 5 x 5
2 --------- + 7 -----------------+ --------- 2 ? x 7 y 7 y 5 x 5
2 ? jr
(0.77) 5
Ucm agjvm mquatfjh fs vadfe `jr pdahm strmss ahe pdahm strafh. ^m saw tcm gfcarbjhfk jpmratjr fhKcaptmr fhKcaptmr =. =. F` wm kah `fhe ah Afry strmss `uhktfjh tcat satfs`fms tcm gfcarbjhfk jpmratjr j` Mquatfjh (0.77), (0.77), tcmh wm cavm satfs`fme tcm mqufdfgrfub ahe kjbpatfgfdfty mquatfjhs. F` fh tcm Afry strmss `uhktfjh tcmrm arm sjbm parabmtmrs tcat kah gm kcjsmh sukc tcat tcm gjuheary kjheftfjhs kah gm bmt, tcmh wm cavm tcm kjbpdmtm sjdutfjh tj tcm gjuheary vadum prjgdmb. @jr ah argftrary gjuheary nmjbmtry wftc argftrary kjheftfjhs, tcm prjgdmb fs stfdd ef``fkudt1. ^m wfdd kjhsfemr sjbm sfbpdm kasms `jr wcfkc wm kah `fhe sjbm sjdutfjhs.
0.=
[JD]UFJH GV _JDVHJBFAD[
^m kah `fhe sjdutfjhs tj Mquatfjh (0.77) gy (0.77) gy sugstftutfhn pjdyhjbfads j` ef``mrmht jremr ahe emtmrbfhfhn tcm kjm``fkfmhts j` tcm pjdyhjbfads. Ucmsm sjdutfjhs arm fhempmhemht j` tcm nmjbmtry j` tcm gjey ahe arm usme `jr tmstfhn adnjrftcbs fh hubmrfkad bmtcjes ahe `jr gmhkcbarifhn fh mxpmrfbmhtad bmtcjes. [jbm pjdyhjbfads arm kjhsfemrme hmxt.
0.=.1
Quaeratfk pjdyhjbfads
@jr strmssms tj gm hjhzmrj, tcm djwmst jremr j` pjdyhjbfad fs quaeratfk. Ucm quaeratfk pjdyhjbfad scjwh gmdjw wfdd fbpdfkftdy satfs`y tcm gfcarbjhfk mquatfjh.
2 a7 x
7
7 + g 7 xy + k7 y 7 7
(0.7;a)
[ugstftutfhn Mquatfjh (0.7;a) fhtj (0.7;a) fhtj Mquatfjh (0.7?), (0.7?), wm jgtafh tcm strmssms as
xx
2 k7
yy
2 a7
xy
2 ‛ g 7
(0.7;g)
Add tcrmm strmss kjbpjhmhts arm kjhstaht tcrjuncjut tcm gjey, frrmspmktfvm j` tcm scapm j` tcm gjey. gjey. @jr tcfs statm j` strmss tj gm tcm sjdutfjh, tcm kjheftfjhs jh traktfjhs jh tcm gjuheary, fh akkjreahkm wftc Mquatfjhs (1.1;a) ahe (1.1;g), bust gm as scjwh gmdjw. [ x 2 k 7 h x ‛ g7 h y
[ y 2 ‛ g 7 h x + a 7 h y
(0.7;k)
wcmrm h x ahe h y arm tcm efrmktfjh kjsfhms j` tcm uhft hjrbad tj tcm gjuheary at a pjfht. @jr a rmktahnudar gjey, tcm statm j` strmss fs scjwh fh @fnurm 0.;. 0.;. [mvmrad kasms j` fhtmrmst fh tcfs kjhstaht strmss statm arm grfm`dy emskrfgme as `jddjws< • F` g7 2 a7 2 ?, ahe k7 2 , wm cavm uhfaxfad tmhsfjh. • F` g7 2 ahe a7 2 k7 2 , wm cavm a statm j` purm scmar. • F` g7 2 ? ahe a7 2 k7 2 , wm cavm a cyerjstatfk statm j` strmss (tcm hjrbad strmss fs tcm sabm fh add efrmktfjhs). • F` g7 2 ? ahe a7 2 ‛ k7 2 , wm cavm a statm j` purm scmar fh a kjjrefhatm systmb tcat fs 5=¿ tj tcm xy tcm xy kjjrefhatm kjjrefhatm systmb.
0.=.7
Kugfk pjdyhjbfads
1. Ucm kjbpdmx varfagdm bmtcje (Busicmdfscvfdf Y160;R) ahe tcm @jurfmr trahs`jrb trahs`jrb bmtcje ([hmeejh Y163?R) arm twj bmtcjes tcat arm usme fh `fhefhn bjrm nmhmrad mdastfkfty sjdutfjhs.
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
[jdutfjh gy pjdyhjbfads 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
a7
g7
y
g7 x
k7
k7
g7 g7
65
a7
@fnurm 0.; Kjhstaht strmss statm.
A kugfk pjdyhjbfad wfdd fbpdfkftdy satfs`y tcm gfcarbjhfk mquatfjh. ^m start wftc tcm Afry strmss `uhktfjh em`fhme as
2 a; x
;
0 + g ; x 7 7 y + k; x y7 7 + e ; y; 0
(0.75a)
Ucm strmssms kah gm jgtafhme `rjb Mquatfjh (0.7?) as (0.7?) as
xx
2 k ; x + e ; y
yy
2 a ; x + g ; y
xy
2 ‛ g ; x + k ; y
(0.75g)
Add tcrmm strmss kjbpjhmhts arm dfhmar fh x fh x ahe ahe y y frrmspmktfvm j` tcm scapm j` tcm gjey. gjey. Jhkm bjrm, wm kjhsfemr smvmrad sugkasms `jr tcfs strmss statm. • F` a; 2 g; 2 k; 2 ?, ahe e ; 2 , wm cavm purm gmhefhn j` a rmktahnudar krjss smktfjh as scjwh fh @fnurm 0.5a 0.5a. • F` g; 2 k; 2 a; 2 ? ahe a; 2 , wm cavm purm gmhefhn j` a rmktahnudar krjss smktfjh as scjwh fh @fnurm 0.5g 0.5g.
y
y x
x
(a)
(g)
@fnurm 0.5 Uwj statms j` purm gmhefhn wftc kugfk pjdyhjbfads.
0.=.;
@jurtc-jremr pjdyhjbfads
A `jurtc-jremr pjdyhjbfad wfdd hjt fbpdfkftdy satfs`y tcm gfcarbjhfk mquatfjh, ahe tcm rmdatfjhscfp gmtwmmh tcm kjhstahts bust gm emtmrbfhme. ^m start wftc tcm `jddjwfhn Afry strmss `uhktfjh<
2 a5 x
5
75 + g 5 x; 0 y + k5 x 7 7 y 7 7 + e 5 x y; 0 + m5 y 5 75
(0.7=a)
[ugstftutfhn Mquatfjh (0.7=a) fhtj (0.7=a) fhtj Mquatfjh (0.77), (0.77), wm jgtafh tcm `jddjwfhn kjheftfjh jh tcm kjhstahts< a 5 + 7 k5 + m 5 2 ?
(0.7=g)
[ugstftutfhn Mquatfjh (0.7=a) ahe (0.7=a) ahe (0.7=g) fhtj (0.7=g) fhtj Mquatfjh (0.7?) tcrjunc (0.7?) tcrjunc (0.7?), (0.7?), wm jgtafh
xx
2 k5 x
7
7 + e 5 xy + m5 y 7 7
yy
2 a5 x
7
7 + g5 xy + k5 y 7 7
xy
2 ‛ g5 x
7
7 + k 5 xy + e 5 y 7 7 (0.7=k)
Add tcrmm strmss kjbpjhmhts arm quaeratfk fh x fh x ahe ahe y y frrmspmktfvm j` tcm scapm j` tcm gjey. gjey.
MSAB_DM 0.7 @fnurm 0.= scjws 0.= scjws a kahtfdmvmr gmab wftc a rmktahnudar krjss smktfjh. Mquatfjh (M1) nfvms (M1) nfvms ah Afry strmss `uhktfjh tcat kjude gm a usme. Emtmrbfhm tcm strmss kjbpjhmhts xx, yy, ahe xy fh tmrbs j` _ j` _ 1, _ 7, g, c, x, x, y, y, ahe D ahe D.. y
_ 1
y
_ 7
x D
z
7c
y y ; y 7 2 a 1 ; -- ‛ -- x ‛ D + a 7 -- c c c
7g
0.7. @fnurm 0.= Kahtfdmvmr gmab fh Mxabpdm 0.7.
AH[.
xx
2
; _ _ 7 -----------1; y x ‛ D + --------5 gc 5 gc
a. Ucm `uhktfjh fs ah jutkjbm j` a bmtcje prjpjsme gy K.V. Hmju (Gjrmsf ahe Kcjhn Y163>R). Y163>R).
yy
2 ?
xy
; _ 1 y 7 2 ‛ --------- 1 ‛ -- 5 gc c
(M1)
0 Fhtrjeuktfjh
6=
0.0
tj Mdastfkfty
EF[_DAKMBMHU[ @WJB [UWAFH[ FH 7-E
^m cavm tj fhtmnratm strafh kjbpjhmhts tj jgtafh efspdakmbmhts. Fh mdastfkfty tcmrm fs a nmhmrad apprjakc j` jgtafhfhn efspdakmbmhts `rjb strafhs. ^m cjwmvmr, wfdd usm a sfbpdm apprjakc tcat f s sfbfdar tj tcm nmhmrad apprjakc gut bjrm rmvmadfhn j` tcm fhtmnratfjh prjkmss ahe fs m``mktfvm `jr twj-efbmhsfjhad prjgdmbs. Ucm strafh efspdakmbmht rmdatfjhscfp fh twj-efbmhsfjh arm nfvmh gy
xx
2
u x
yy
2
v y
xy
yx
2
2
u + v y x
(0.70)
Ucm prjkmeurm fs as `jddjws< 1. Fhtmnra Fhtmnratm tm tcm strafh strafh xx wftc wftc rmspmkt rmspmkt tj x tj x ahe aee `uhktfjh ` y tj jgtafh efspdakmbmht u x y . 7. Fhtmnra Fhtmnratm tm tcm strafh strafh yy wftc wftc rmspmkt rmspmkt tj y tj y ahe aee `uhktfjh n x tj jgtafh efspdakmbmht v x y . ;. [ugstftutm [ugstftutm u x y ahe v x y fhtj tcm strafh mxprmssfjh mxprmssfjh xy . ^rftm ^rftm add tmrbs tcat arm `uhktfjhs `uhktfjhs j` x jh x jh jhm sfem j` tcm mquad sfnh ahe add tmrbs tcat arm `uhktfjhs j` y j` y jh tcm jtcmr sfem j` tcm mquad sfnh. Ucfs fbpdfms tcat makc sfem bust mquad tj tcm sabm kjhstaht. 5. Ucm jrefhary jrefhary emrfvatfvms emrfvatfvms j` ` y ahe n x fh tcm agjvm stmp kah gm fhtmnratme. fhtmnratme. Ucm kjhstahts kjhstahts j` fhtmnratfjh fhtmnratfjh kjrrmspjhe tj rfnfe gjey bjem efskussme fh [mktfjh 0.0.1 ahe 0.0.1 ahe arm emtmrbfhme `rjb tcm gjuheary kjheftfjhs.
MSAB_DM 0.; Jgtafh tcm efspdakmbmht `fmde `jr tcm kjhstaht strmss statm nfvmh fh Mquatfjh (0.7;g). (0.7;g). u x y 2
AH[.
0.0.1
+ 1 ; ‛ -----------------k 7 ‛ ----------------- a 7 x + x + y 3N 3N
v x y 2
g + 1 ; ‛ -----------------a 7 ‛ ----------------- k 7 y ‛ ----7- x + y ‛ x 3N 3N N
Wfnfe gjey bjtfjh
Fhtmnratfjh j` strafhs wfdd rmsudt fh a rfnfe gjey bjem fh tcm efspdakmbmht mxprmssfjhs. Ucm rfnfe gjey bjem fs em`fhme gy tcm kjhstahts x y , ahe . Kjhsfemr u x y 2 x ―add pjfhts jh tcm mdastfk gjey bjvm gy a kjhstaht abjuht gut ft prjeukms hj strafh jr strmss. Fh jtcmr wjres, tcm gjey trahsdatms gy ah abjuht x fh tcm x efrmktfjh. [fbfdardy, y rmprmsmhts trahsdatfjh fh tcm y efrmktfjh. Ucm rjtatfjh at ahy pjfht fs em`fhme as
x y
1 u v 2 -- ‛ 7 y x
(0.7>)
F` wm hjw kjhsfemr u x y 2 y ahe v x y 2 ‛ x , ahe sugstftutm fhtj tcm agjvm mquatfjh, tcmh wm jgtafh x y 2 . Jhkm bjrm hj strafh jr strmss fs prjeukme ahe add pjfhts jh tcm mdastfk gjey rjtatm gy tcm sabm abjuht j` . Ucus, tcm kjhstaht rmprmsmhts rfnfe gjey rjtatfjh. Uj mdfbfhatm rfnfe gjey bjtfjh wm hmme tj `fx tcm gjey at dmast at ; pjfhts, wftc jhm pjfht tcat fs hjt kj-dfhmar wftc tcm jtcmr twj. Ucmsm twj. Ucmsm suppjrt pjfhts nfvm us tcm gjuheary kjheftfjhs hmmeme tj emtmrbfhm tcm rfnfe gjey kjhstahts.
MSAB_DM 0.5 Kjhsfemr tcm uhfaxfad tmhsfjh
xx
2
fh pdahm strmss ahe jgtafh tcm efspdakmbmhts `jr tcm `jur kasms scjwh fh@fnurm fh@fnurm 0.0. 0.0.
7a
7a x
Kasm F
y
y
y
7a x
7g
y
x
7g
Kasm FF
7a
Kasm FFF
7 g
x
Kasm FP
@fnurm 0.0
AH[.
0.>
Kasm F< u x y 2
v x y 2 ‛ M y + y ‛ x M x + x + y AH[. Kasm FF< u x y 2 M x + a v x y 2 ‛ M y + y AH[. Kasm FFF< u x y 2 M x + a v x y 2 ‛ M y + g AH[. Kasm FP< u x y 2 M x v x y 2 ‛ M y + g
UJW[FJH J@ HJH-KFWK]DAW [CA@U[
Fh Kcaptmr 1 wm 1 wm saw tcm tcmjry `jr tcm tjrsfjh j` kfrkudar sca`ts. Fh tcfs smktfjh wm emvmdjp a bjrm nmhmrad tcmjry fh wcfkc tcmrm fs hj dfbftatfjh jh krjss-smktfjhad scapm. [afht-Pmhaht [afht-Pmhaht was tcm `frst tj emvmdjp a tcmjry `jr tcm tjrsfjh tjrsfjh `jr hjh-kfrkudar hjh-kfrkudar sca`ts. sca`ts.
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
Ujrsfjh j` hjh-kfrkudar sca`ts 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
60
_rahetd _rahetd datmr emvmdjpme emvmdjpme ah adtmrhat adtmrhatfvm fvm gasme jh Afry‖s strmss `uhktfjh apprjakc. ^m wfdd us m gjtc tcmjrfms. ^m wfdd usm _rahetd‖s apprjakc `jr jgtafhfhn tcm strmssms ahe [afht-Pmhaht‖s [afht-Pmhaht‖s tj jgtafh tcm em`jrbatfjh.
0.>.1
[afht-Pmhaht‖s [afht-Pmhaht‖s bmtcje
[afht-Pmhaht jgsmrvme tcat a efspdakmbmht `fmde `jr a krjss smktfjh j` a hjh-kfrkudar sca`t uhemr tjrsfjh scjwh fh@fnurm fh @fnurm 0.> scjude akkjuht `jr tcm `jddjwfhn< (f) Ucm krjss smktfjh wjude warp uhemr tjrsfjh. (ff) Ucm jhdy hjhzmrj strmss kjbpjhmhts wjude gm xy ahe xz , wcfkc `jr fsjtrjpfk batmrfads fbpdfms tcat tcm jhdy hjhzmrj strafh kjbpjhmhts wjude gm xy ahe xz .. U mxt y
U mxt
z
x
@fnurm 0.> Ujrsfjh j` a hjh-kfrkudar sca`t.
[afht-Pmhaht [afht-Pmhaht prjpjsme a efspdakmbmht `fmde tcat fhkdueme a warpfhn `uhktfjh ahe a `jrb tcat wjude rmsudt fh scmar strafhs xy ahe xz jhdy, as nfvmh gmdjw u 2
e y z
e v 2 ‛ xz e x
e x
w 2 xy
e e x
(0.73)
wcmrm u, v, ahe w arm tcm efspdakmbmhts fh tcm x tcm x,, y, y, ahe z ahe z efrmktfjh, rmspmktfvmdy9 e / ex fs a kjhstaht rmprmsmhtfhn rmprmsmhtfhn tcm ratm j` twfst pmr uhft dmhntc9 ahe tcm warpfhn `uhktfjh y z emskrfgms tcm bjvmbmht j` pjfhts jut j` tcm krjss-smktfjhad pdahm ^ftc tcm efspdakmbmht `fmde ihjwh, tcm djnfk skcmbatfzme fh @fnurm 7.0 kah 7.0 kah gm usme tj jgtafh tcm rmqufrme `jrbudas, as scjwh fh tcm efskussfjh tcat `jddjws. [ugstftutfhn Mquatfjh (0.73) fhtj (0.73) fhtj Mquatfjh (0.1) wm (0.1) wm jgtafh tcm hjhzmrj strafh kjbpjhmhts nfvmh gmdjw.
xy
2
y z e e x y
+
‛ xz e e x x
2
e ‛ z e x y
xz
2
xy e x e x
+
y z e e x z
2
e y + e x z
(0.76)
@rjb Cjjim‖s daw `jr fsjtrjpfk batmrfads, tcm scmar strmssms kah gm wrfttmh as scjwh gmdjw.
xy
‛ z e y e x
2 N
xz
+ y e z e x
2 N
(0.;?)
Ucm scmar strmssms kah gm rmpdakme gy ah mqufvadmht fhtmrhad tjrqum as scjwh fh @fnurm 0.3. 0.3. Gy mquatfhn tcm bjbmht agjut tcm jrfnfh, wm jgtafh Mquatfjh (0.;1). (0.;1). U 2
A
y xz ‛ z xy e A
(0.;1)
eP y 2 xy eA y
y
J
J
z y
eP z 2 xz eA z
x (a)
z
x (g)
@fnurm 0.3 [tatfk mqufvadmhky `jr hjh-kfrkudar sca`ts.
^m kjude sugstft utm Mquatfjh (0.;?) fhtj (0.;?) fhtj Mquatfjh (0.;1) ahe (0.;1) ahe jgtafh jtcmr `jrbudas, gut at tcfs stanm wm wfdd swftkc tj _rahetd‖s bmtcje, wcfkc rmsudts fh a sfbfdar `jrb j` mquatfjhs.
6>
0 Fhtrjeuktfjh
0.>.7
tj Mdastfkfty
_rahetd‖s bmtcje
Ucm hjhzmrj strmss kjbpjhmhts xy ahe xz tcat wfdd gm nmhmratme tj rmsfst tcm mxtmrhad tjrqum bust satfs`y tcm mqufdfgrfub mquatfjhs. Fh tcm agsmhkm j` gjey `jrkms, Mquatfjh (0.3e) kah (0.3e) kah gm wrfttmh as
yx zx (0.;7) --------- + --------2 ? ------ xy --- 2 ? ------ xz --- 2 ? y z x x Mquatfjh (0.;7) fbpdfms tcat scmar strmssms xy ahe xz kahhjt gm `uhktfjhs j` x j` x,, wcfkc fs pjssfgdm as djhn as tcmrm as tcmrm fs hj efstrfgutme tjrqum jh tcm sca`t . Uj satfs`y Mquatfjh (0.;7), (0.;7), wm em`fhm a strmss `uhktfjh as
yx
2
xy
z
2
zx
y
2 ‛
xz
2
(0.;;)
Ucm gjuheary kjheftfjh jh tcm sur`akm j` tcm hjh-kfrkudar sca`t fs zmrj traktfjh. [ugstftutfhn t x 2 ?, xx 2 ?, ahe Mquatfjh (0.;;) fhtj (0.;;) fhtj Mquatfjh (0.1?a), (0.1?a), wm jgtafh
h h 2 z y y z
?
(0.;5)
Ucm jutware hjrbad tj tcm sur`akm j` tcm hjh-kfrkudar sca`t wfdd dfm fh tcm y tcm y-- z z pdahm. pdahm. @fnurm 0.6 scjws 0.6 scjws tcm nmjbmtry gy wcfkc wm kah rmdatm tcm efrmktfjh kjsfhms j` tcm uhft hjrbad tj tcm sur`akm nmjbmtry. Hjtfhn tcat tcm tahnmht tj tcm sur`akm fs 6? tj tcm uhft hjrbad (f.m., z 2 6? + z ), wm jgtafh Mquatfjh (0.;=a) ahe (0.;=a) ahe (0.;=g). (0.;=g). y
h es
ey
z
ez
z
z
x
@fnurm 0.6 Efrmktfjh kjsfhms j` a uhft hjrbad.
h y 2 sfh
z 2 sfh z ‛ 6? 2 ‛ kjs z 2 ‛ e z e s
h z 2 kjs
z 2
kjs
z ‛ 6? 2 sfh z 2
jr
ey es
jr
h y 2 ‛ e z e s
(0.;=a)
h z 2 e y e s
(0.;=g)
[ugstftutfhn Mquatfjh (0.;=a) ahe (0.;=a) ahe (0.;=g) fhtj (0.;=g) fhtj Mquatfjh (0.;5), (0.;5), wm jgtafh Mquatfjh (0.;0). (0.;0). e e z e y ‛ ‛ jr ‛ 2 ? z e s y e s 2 ? e s
(0.;0)
Mquatfjh (0.;0) fbpdfms tcat fs a kjhstaht jh tcm sca`t sur`akm, tcat fs, jh tcm gjuheary j` tcm krjss smktfjh. Ucfs kjhkdusfjh pdays a krftfkad rjdm fh smdmktfjh smdmktfjh j` tcm strmss `uhktfjh, ahe wm rmkjre ft `jr `uturm usm. (0.;>) Ucm strmss `uhktfjh bust gm kjhstaht jh tcm gjuheary j` tcm krjss smktfjh j` tcm hjh-kfrkudar sca`t. [ugstftutfhn Mquatfjh (0.;;) ahe (0.;;) ahe (0.;;) fhtj (0.;;) fhtj Mquatfjh (0.;1) ahe (0.;1) ahe fhtmnratfhn, wm jgtafh Mquatfjh (0.;3), (0.;3), U 2
y ‛ y ‛ z z e y e z 2 ‛ y ez ‛ z ey + + e y e z A
A
wcmrm rmprmsmhts fhtmnratfjh jh tcm kdjsme gjuheary em`fhfhn tcm krjss smktfjh. @rjbMquatfjh @rjb Mquatfjh (0.;>), (0.;>), gjuheary j` tcm krjss smktfjh. Ucm kjhtjur fhtmnrads kah gm wrfttm h as ‛ y ez ‛
z ey
2 ‛
(0.;3)
fs kjhstaht jh tcm
y ez + z ey 2 ‛e yz 2 ?
(0.;6)
e y e z
(0.5?)
Ucm fhtmnrad fh Mquatfjh (0.;6) fs (0.;6) fs zmrj gmkausm f` wm start at ahy pjfht jh tcm gjuheary, nj arjuhe ft, ahe rmturh tj tcm startfhn pjfht, tcm y tcm y ahe ahe z z kjjrefhatms kjjrefhatms ej hjt kcahnm. [ugstftutfhn Mquatfjh (0.;6) fhtj (0.;6) fhtj Mquatfjh (0.;3), (0.;3), wm jgtafh tcm `fhad rmsudt `jr tcm fhtmrhad tjrqum as scjwh fh Mquatfjh (0.5?). (0.5?). U 2 7
A
Uj jgtafh tcm efspdakmbmhts `rjb tcm strmssms fh Mquatfjh (0.;;), (0.;;), wm kjude `frst jgtafh strafhs ahe tcmh fhtmnratm tcm partfad emrfvatfvms. ^m kcjjsm ah masfmr adtmrhatfvm, as emskrfgme fh tcm efskussfjh tcat `jddjws. ^m sugstftutm Mquatfjh (0.;?) fhtj (0.;?) fhtj Mquatfjh (0.;;) tj (0.;;) tj jgtafh
z
e y ‛ z e x
y
2 N
e z + y e x
2 ‛ N
(0.51a)
Uaifhn tcm partfad emrfvatfvm j` `frst mquatfjh wftc rmspmkt tj z tj z ahe ahe tcm partfad emrfvatfvm j` smkjhe mquatfjh wftc rmspmkt tj y tj y ahe ahe aeefhn, wm jgtafh e (0.57) --------- + --------- 2 ‛ 7 N 7 7 e x y z Fh jgtafhfhn Mquatfjh (0.57), (0.57), wm baem usm j` tcm `akt tcat e / ex ahe N arm kjhstahts. ^m hjw cavm add tcm mquatf jhs wm hmme tj sjdvm prjgdmbs j` tjrsfjh j` hjh-kfrkudar sca`ts. Ucm hmxt smktfjh emskrfgms tcm sjdutfjh prjkmeurm. 7
7
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
Bmbgrahm ahadjny `jr tjrsfjh j` hjh-kfrkudar 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
0.>.;
_rjkmeurm `jr sjdvfhn prjgdmbs j` tjrsfjh j` hjh-kfrkudar sca`ts
[tmp 1 [tmp 7 [tmp ; [tmp 5
@jr a nfvmh krjss-smktfjhad scapm, jgtafh a strmss `uhktfjh tcat fs kjhstaht jh tcm gjuheary j` tcm krjss smktfjh fh akkjreahkm wftc Mquatfjh (0.;>). (0.;>). Emtmrbfhm tcm kjhstaht fh tcm strmss `uhktfjh fh tmrbs j` tcm fhtmrhad tjrqum U gy gy fhtmnratfhn jvmr tcm krjss smktfjh `rjb Mquatfjh (0.5?). (0.5?). Emtmrbfhm tcm scmar strmssms `rjb Mquatfjh (0.;;). (0.;;). Emtmrbfhm tcm ratm j` twfst e / ex `rjb Mquatfjh (0.57). (0.57).
MSAB_DM 0.= A sca`t cas ah mddfptfkad krjss smktfjh as scjwh fh @fnurm 0.1?. 0.1?. Emtmrbfhm tcm mquatfjhs `jr baxfbub scmar strmss bax at a krjss smktfjh ahe tcm rmdatfvm rjtatfjh (7 ‛ 1) j` tcm krjss smktfjhs at pjfhts x pjfhts x1 ahe x ahe x7 adjhn tcm dmhntc j` tcm sca`t fh tmrbs j` fhtmrhad tjrqum U , scmar bjeudus N, a, g, x1, ahe x ahe x7. y
z
7a
7g 0.=. @fnurm 0.1? Mddfptfkad krjss smktfjh `jr Mxabpdm 0.=.
AH[.
bax
7
2 7 U ag 9
7 ‛ 1
7
2 U a + g
7
x 7 ‛ x1 N a ; g ;
(M7)
MSAB_DM 0.0 Ah adubfhub (N (N 2 5??? isf) mddfptfkad sca`t fs djaeme as scjwh fh @fnurm 0.11. 0.11. Emtmrbfhm tcm baxfbub scmar strmss fh tcm sca`t ahe tcm rjtatfjh j` smktfjh E smktfjh E wftc wftc rmspmkt tj tcm rjtatfjh j` smktfjh A smktfjh A.. 1? fh-ifps fh-ifps
y
=? fh-ifps fh-ifps
A G
0? fh-ifps fh-ifps
z
; fh. fh.
7? fh. fh. x
;0 fh. fh.
7? fh-ifps fh-ifps
K
E ;? fh. fh.
7 fh.
0.0. @fnurm 0.11 Mddfptfkad sca`t `jr Mxabpdm 0.0.
AH[.
0.3
bax
2 1> isf 9
E ‛ A
2 ?.?56 rae K^
BMBGWAHM AHADJNV @JW UJW[FJH J@ HJH-KFWK]DAW [CA@U[
_rahetd rmadfzme tcat tcm em`dmktfjh j` bmbgrahm uhemr prm ssurm cas tcm sabm ef``mrmhtfad mquatfjh as tjrsfjh j` hjh-kfrkudar ga rs. Ucus, ah ahadjny mxfsts tcat kah gm usme `jr emtmrbfhfhn tcm varfagdms fh tjrsfjh j` hjh-kfrkudar gars gy djjifhn at bmbgrahm em`dmktfjh. Ucfs ahadjny cas prjvme tj gm fbpjrtaht fh mxpmrfbmhtaddy emtmrbfhfhn tcm tjrsfjhad rfnfefty j` kjbpdmx krjss smktfjhad scapms, sukc as krjss smktfjh j` a wfhn j` ah afrkra`t. Kjhsfemr a pdatm wftc a cjdm j` tcm sabm scapm as tcm krjss smktfjh j` a hjh-kfrkudar gar. Cjdm ejms hjt hmme tj gm j` tcm sabm efbmhsfjh as tcm krjss smktfjh j` tcm sca`t, prjvfeme ah apprjprfatm kadfgratfjh fs kjheuktme. [trmtkc a bmbgrahm akrjss ft mhsurfhn gy apprjprfatm bmahs tcat tcm bmbgrahm ejms hjt em`dmkt at tcm cjdm gjuheary.^m rmprmsmht tcfs bmbgrahm rmnfjh asW wftc a gjuheary as scjwh fh @fnurm 0.17a 0.17a. Appdy a prmssurm p prmssurm p uhemrhmatc uhemrhmatc tcm bmbgrahm, kausfhn ft tj em`dmkts gy ub as scjwh fh @fnurm 0.17g 0.17g. Ucm tmhsfjh pmr uhft dmhntc fh tcm bmbgrahm ([ ([ ) fs uhf`jrb fh add efrmktfjhs. @urtcmrbjrm tcm tmhsfjh akts fh tcm tahnmht efrmktfjh j` em`dmktme bmbgrahm at add pjfhts as scjwh fh @fnurm 0.17k 0.17k. Ucm tahnmht efrmktfjhs kah gm rmdatme tj bmb-
66
0 Fhtrjeuktfjh
tj Mdastfkfty
1 7 ub ; x Bmbgrahm _datm 5 p = z 0 z > y W 3 6 y 1? (a) 11 (g) 17 y pe ye z 1; [ez 15 u u 1= b [ey b z 10 z ez z + 1> z u b u b 13 [ey tah 2 tah 2 y y y z z z 16 y z 7? [ez y 71 ey y + (k) (e ) (m) 77 y 7; @fnurm 0.17 Bmbgrahm em`dmktfjh 75 grahm em`dmktfjhs as scjwh fh @fnurms 0.17e 0.17e ahe ahe m. Fh @fnurms 0.17e 0.17e ahe ahe m, ft fs assubme tcat tcm tahnmht ahndms arm sbadd ahe 7= cmhkm tcm Uahnmht `uhktfjh kah gm rmpdakme gy fts arnubmhts. Gy mqufdfgrfub j` `jrkms fh tcm x tcm x efrmktfjh efrmktfjh fh @fnurm 0.17k 0.17k, wm 70 jgtafh< 7> 73 y + pezey 2 ? [e y sfh z + z ez ‛ [e y sfh z + [e z sfh y + y ey ‛ [e z sfh 76 z y ;? Hjtfhn tcat `jr sbadd ahndms tcm wm kah rmpdakm tcm [fhm `uhktfjh gy fts arnubmht wm jgtaf h< ;1 z z y ;7 p [e y z + ez ‛ [e y z + [e z y + y ‛ [e z y + pezey 2 ? jr + 2 ‛ -- jr ;; z [ y z y ;5 7 7 u u p ;= (0.5;) fh W 7 u b 2 7b + 7 b 2 ‛ -- [ ;0 y z ;> (0.55) ub 2 ? jh ;3 ^m smm tcat _jfssjh‖s mquatfjh rmprmsmhts tcm _rahetd‖s `jrbudatfjh j` tcm tjrsfjh prjgd mb ahe tcm em`dmktfjh j` a bmb;6 grahm. Ucm ef``mrmhtfad mquatfjh kah gm sjdvme usfhn zmrj em`dmktfjh kjheftfjh jh tcm gjuheary wcfkc fs ahadjnjus tj zmrj 5? vadum j` jh tcm gjuheary. gjuheary. Ucus, tcm gjuheary vadum prjgdmbs `jr tjrsfjh j` hjh-kfrkudar sca`ts ahe em`dmktfjh j` bmbgrahm 51 arm sfbfdar ahe tcm ahadjny gmtwmmh tcm twj prjgdmbs kah hjw gm erawh as scjwh fh Uagdm 0.1. 0.1. Ft scjude gm bmhtfjhme tcat 57 _jfssjh‖s mquatfjh mquatfjh fs jgtafhme fh bahy `fmd es< tcmrbad prjgdmbs, ef``usfjh prjgdmbs, mdmktr jstatfk, mtk. Bmbgrahm ahadjny kah 5; gm usme fh add j` tcmsm appdfkatfjhs j` _jfssjh‖s mquatfjh. 55 Bmbgrahm em`dmktfjh kah gm usme tj emtmrbfhm tcm tjrqum karryfhn kapakfty j` a krjss smktfjh gy emtmrbfhfhn tcm vjd- 5= 50 ubm uhemr tcm bmbgrahm ahe ejugdfhn ft. Ucm mxpmrfbmht kah adsj tmdd us wcmrm tcm tjrsfjhad scmar strmssms wfdd gm baxfbub 5> gy djjifhn at tcm baxfbub sdjpms fh tcm bmbgrahm em`dmktme scapm. Ucm bmbgrahm ahadjny kah adsj gm usme tj jgtafh 53 apprjxfbatm ahadytfkad sjdutfjhs, partfkudardy `jr tcfh waddme jpmh smktfjhs as wfdd gm scjwh fh [mktfjh 0.6. 0.6. 56 Uagdm 0.1 _rahetd bmbgrahm ahadjny =? Bmbgrahm em`dmktfjh prjgdmb. _rahetd tjrsfjh prjgdmb. =1 7 7 =7 7 7 u b u b p + 2 ‛ 7 N e + 2 ‛ -- =; [ 7 7 e x y7 z 7 y z =5 ub == p 7 e ex =0 => 1 [ N =3 ub z yx 2 xy 2 z =6 ub y zx 2 xz 2 ‛ y 0? u b ey e z ey e z Pjdubm gmhmatc tcm bmbgrahm 2 U 2 7 01 A A x
Ujrsfjh j` tcfh-waddme jpmh smktfjh 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
0.3.1
1??
Bmbgrahm ahadjny `jr krjss smktfjh wftc cjdms
Ucm njvmrhfhn ef``mrmhtfad mquatfjh ahe tcm ahadjny ejms hjt kcahnm wcmh tcm krjss smktfjh cas a cjdm. Cjwmvmr, tcmrm fs hj em`jrbatfjh fh tcm cjdm ahe t cm strmssms jh tcm gjuheary j` tcm cjdm arm zmrj. Akkjrefhn tj tcm ahadjny fhUagdm fh Uagdm 0.1, 0.1, zmrj strmssms fbpdfms zmrj sdjpm j` tcm bmbgrahm at tcm cjdm gjuheary. Ucus, tj bjemd tcm cjdm wm kjvmr tcm bmbgrahm wftc a `dat wmfnctdmss rfnfe pdatm j` tcm sabm scapm as tcm cjdm. Ucm rfnfe pdatm rfsms (trahsdatms) fh tcm x tcm x efrmktfjh efrmktfjh as scjwh fh @fnurm 0.1;. 0.1;. At pjfhts G pjfhts G ahe K tcm tcm sdjpm j` tcm bmbgrahm em`dmktfjh arm zmrj, bjemdfhn tcm zmrj strmss kjheftfjh j` tcm cjdm gjuheary. Jhkm bjrm usfhn tcm ahadjny j` Uagdm 0.1, 0.1, tcm tjrsfjhad rfnfefty ahe baxfbub tjrsfjhad scmar strmssms kah gm `juhe. Ucus, mxpmrfbmhtaddy wm kah emtmrbfhm tjrsfjhad prjpmrtfms j` vmry kjbpdmx krjss smktfjhs.
z A G _datm K
E
y
_datm
x K
G
ub
A
y
E
@fnurm 0.1; Bmbgrahm wftc a cjdm.
MSAB_DM 0.> 5 ]sfhn bmbgrahm ahadjny `jr a kfrkudar sca`t fh tjrsfjh prjvm U 2 N W 7 e ex . z
W
r
y
0.>. @fnurm 0.15 Kfrkudar nmjbmtry fh Mxabpdm 0.>.
0.6
UJW[FJH J@ UCFH-^ADDME J_MH [MKUFJH
Jgtafhfhn ahadytfkad `jrbudas `jr tj rsfjh j` tcfh-waddme jpmh smktfjhs fs ef``fkudt. Cjwmvmr, gy usm j` bmbgrahm ahadjny w m kah nmt apprjxfbatm rmsudts ahe kah vfsuadfzm tcm tjrsfjhad rmspjhsm j` tcm krjss smktfjh.
z
y
g t
(g)
(a)
@fnurm 0.1= Bmbgrahm ahadjny `jr tcfh-waddme jpmh smktfjh.
Uj emvmdjp tcm apprjxfbatm `jrbudas wm wfdd `frst kjhsfemr a tcfh rmktahnudar tcfh rmktahnudar krjss smktfjh. Gy tcfh wm tcfh wm fbpdy tcat tcm djhn sfem j` rmktahndm fs sfnhf`fkahtdy nrmatmr tcah tcm tcfkihmss, f.m., g ¹ t fh @fnurm 0.1=a 0.1=a. ]suaddy ah jremr j` banhftuem quahtf`fms tcm wjre –sfnhf`fkahtdy nrmatmr‖ fh mhnfhmmrfhn, tcus g 1?t 1? t at dmast. ^ftc tcfs rmqufrmbmht wm mxpmkt tcat tcm bmbgrahm em`dmktfjh wfdd gm ejbfhatme gy z gy z ahe ahe ejms hjt kcahnm sfnhf`fkahtdy wftc y wftc y,, tcat fs, u b z . ^ftc tcfs assubptfjh wm kah wrftm Mquatfjh (0.5;) as (0.5;) as 7
e u b 7
e z
p 2 ‛ -- [
(0.5=)
1?1
0 Fhtrjeuktfjh
tj Mdastfkfty
Fhtmnratfhn Mquatfjh (0.5=) wm (0.5=) wm jgtafh< 7
u b 2 ‛ pz 7 [ + K 1 z + K 7
(0.50a)
Ucm fhtmnratfjh kjhstahts kah gm jgtafhme gy hjtfhn tcat tcm bmbgrahm em`dmktfjh fs zmrj jh tcm gjuheary fh tcm tcfki7 hmss efrmktfjh, tcat fs, u b z 2 t 7 2 ? . ]sfhn tcmsm kjheftfjhs wm `fhe tcat K 1 2 ? ahe K 7 2 pt 3 [ . Ucm bmbgrahm em`dmktfjh mquatfjh tcus fs< 7
p t 7 (0.50g) u b 2 ------ --- ‛ z 7 [ 5 Mquatfjh (0.50g) fs (0.50g) fs ah mquatfjh j` a paragjda. Ucm apprjxfbatm em`dmktme scapm j` tcm bmbgrahm fs scjwh fh @fnurm 0.1=g 0.1=g. Kdmardy at y 2 g 7 tcm apprjxfbatm bmbgrahm em`dmktme scapm fs fhkjrrmkt. ^m wfdd efskuss tcfs mhe m``mkt datmr gut wm kjhtfhum uhemr tcm assubptfjh tcat tcm mrrjr fhtrjeukme at tcm mhe fs sbadd. Ucm vjdubm uhemr tcm bmbgrahm fs tcm arma uhemr tcm paragjda tfbms tcm dmhntc ahe kah gm emtmrbfhme as< t 7
t 7
7 gp -t -- ‛ z 7 e z 2 ub e z 2 ----- 7 [ ‛ t 7 5 ‛ t 7 e Hjtfhn tcm ahadjny fh Uagdm 0.1 wm rmpdakm P gy U 7 , p gy 7 , ahe
P 2 g
;
e x
gt e U 2 N ------- jr ; e x
;
gp t ---------17[ 17 [ 1 -- gy N fh Mquatfjh (0.5>) tj (0.5>) tj jgtafh<
(0.5>)
[
;
gt e e ; 2 NI wc m rm I 2 gt ; U 2 N ------(0.53) ; e x e x @jr jtcmr scapms j` krjss smktfjhs tcmrm wfdd gm ef``mrmht vadums j` I, gut wfdd gm jgtafhme `rjb usm j` tcm agjvm `jrbuda. NI fs tcm tjrsfjhad rfnfefty. rfnfefty. Ft bust cjwmvmr gm mbpcasfzme tcat I fs hjt tcm pjdar bjbmht j` fhmrtfa as was tcm kasm `jr kfrkudar krjss smktfjhs. Ucm tjrsfjhad scmar strmss fs rmdatme tj tcm sdjpm j` tcm em`dmktme bmbgrahm kurvm. Fh jur apprjxfbatfjh wm arm jhdy pmrbfttfhn sdjpm fh tcm z efrmktfjh. Uaifhn tcm partfad emrfvatfvm wftc rmspmkt tj z j` Mquatfjh (0.50g), (0.50g), wm jgtafh<
u b z
pz 2 ‛ ----- [
(0.56a)
]sfhn tcm ahadjny fh Uagdm 0.1, 0.1, wm rmpdakm P gy U 7 , p gy 7 e ex , ahe 1 [ gy N fh Mquatfjh (0.56a) tj (0.56a) tj jgtafh<
xy
2 ‛ 7 Nz e ex
(0.56g)
Fh tcfh-waddme kdjsme smktfjhs wm assubme tcat tcm tjrsfjhad scmar strmss fh tcm tcfkihmss efrmktfjh was uhf`jrb. Gut `jr tcfh-waddme jpmh smktfjh, Mquatfjh (0.56g) scjws (0.56g) scjws tcat tcm tjrsfjhad scmar strmss varfms dfhmardy fh tcm tcfki hmss efrmktfjh. ^m adsj hjtm tcat y tcat y fs fs fh tcm efrmktfjh j` tcm kmhtmr dfhm. [j wm wfdd erjp tcm sugskrfpts ahe cmhkm `jrtc rmprmsmht tcm baxfbub tjrsfjhad scmar strmss as fh tcm efrmktfjh j` tcm tahnmht tj tcm kmhtmr dfhm. Ucfs wfdd pmrbft us tj nmhmradfzm jur rmsudts tj argftrary scapme tcfh jpmh smktfjhs.Ucm baxfbub tjrsfjhad scmar strmss wfdd gm at menm j` tcm tcfkihmss, f.m., at z 2 t 7 . ^m sugstftutm e ex `rjb Mquatfjh (0.53) tj (0.53) tj jgtafh tcm banhftuem j` baxfbub tjrsfjhad scmar strmss as<
ba x
2 U t I
(0.=?)
Mquatfjh (0.50g) was emrfvme uhemr tcm assubptfjh g ¹ t . Ucus add jur `jrbudas fh tcfs smktfjh arm suglmkt tj tcfs dfbftatfjh. ^cmh a rmktahnudar krjss-smktfjh ejms hjt bmmt tcfs dfbftatfjh, tcmh ef``mrmht `jrbudas k ah stfdd gm jgtafhme usfhn bmbgrahm ahadjny. @jr tcfh krjss smktfjhs j` ef``mrmht scapms wm wfdd usm tcm agjvm apprjakc assubfhn tcat tcm bmbgrahm ejms `jrb a paragjda fh tcm tcfkihmss efrmktfjhs as embjhstratme fh Mxabpdm 0.3. 0.3.
0.6.1
Mhe m``mkts jh rmktahnudar krjss smktfjhs
Kjhsfemr a kjrhmr j` a rmktahnudar krjss smktfjh as scjwh fh @fnurm 0.10. 0.10. As tcm jutsfem sur`akms j` tcm gar arm `rmm sur`akms, tcm strmssms yx ahe zx arm zmrj, cmhkm gy sybbmtry j` tjrsfjhad scmar strmssms xy ahe xz arm zmrj. Gjtc tcmsm kjbpjhmhts rmakc k m ` a r u [ m m r @
yx zx
2 ?
2 ?
@rmm [ur`akm
@fnurm 0.10 Mhe m``mkt jh tjrsfjhad scmar strmss.
g x
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
Kdjsurm 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
1?7
a baxfbub vadum at tcm bfe pjfht j` makc sfem wcfkc wm kjhkduem gy vfsuadfzfhn tcm sdjpm j` em`dmktme scapm j` a bmbgrahm. Ucus, tcmrm fs a trahsftfjh rmnfjh hmar tjwares tcm menm j` y 2 g 7 fh @fnurm 0.1= wcmrm tcm tjrsfjhad scmar strmss xz , wcfkc wm hmndmktme fs hjh-zmrj. ^m assubm tcat tcfs rmnfjh fs sbadd ahe fts fbpakt kah gm hmndmktme.
MSAB_DM 0.3 Jgtafh tcm tjrsfjhad rfnfefty ahe baxfbub tjrsfjhad scmar strmss `jr tcm tcfh jpmh krjss smktfjh j` uhf`jrb tcfkihmss tcfkihmsst t scjwh scjwh fh U @fnurm 0.1>. 0.1>. Assubm t « a ahe nap at E at E fs fs j` hmndfnfgdm tcfkihmss. Wmpjrt tcm ahswmr fh tmrbs j` fhtmrhad tjrqum , scmar bjeudus N, tcfkihmss t , ahe parabmtmr a. y
z
7a
K E 1.=a 1.=a
0.3. @fnurm 0.1> Nmjbmtry fh Mxabpdm 0.3.
AH[.
0.1?
;
NI 2 > ; Na t 9
7
2 ; U > at
KDJ[]WM
Ucm ifhmbatfk mquatfjhs rmdatfhn strafhs ahe efspdakmbmhts, tcm kjbpatfgfdfty mquatfjhs jh strafhs, tcm nmhmradfzme vmrsfjh j` Cjjim‖s daw, tcm mqufdfgrfub mquatfjhs, ahe tcm gjuheary kjheftfjhs jh traktfjhs jr efspdakmbmhts arm tcm mquatfjhs j` mdastfkfty tcat wmrm fhtrjeukme fh tcfs kcaptmr. @rjb arnubmhts j` axfsybbmtry, tcm mquatfjhs j` mdastfkfty kah gm usme tj jgtafh rmsudts `jr pdahm st rafh prjgdmbs prj gdmbs j` tcfki djhn dj hn kydfhemrs kydfh emrs ahe `jr pdahm strmss prjgdmbs p rjgdmbs j` j ` tcfh ef sis jr t cfki scjrt scj rt kydfhemrs kydfh emrs wft c hj axf ad djae. Mqufdfgrfub mquatfjhs jh strmssms arm fhtrfhsfkaddy satfs`fme gy tcm Afry strmss `uhktfjh, a pjwmr`ud kjhkmpt wcjsm jhm appdfkatfjh fs `jr tcm tjrsfjh j` hjh-kfrkudar sca`ts.
1?;
0 Fhtrjeuktfjh
0.11
tj Mdastfkfty
[VHJ_[F[ J@ MQ]AUFJH[ Mquatfjhs j` Mdastfkfty
[trafhs
u xx x u rr 2 r r 7
Kjbpatfgfdfty
_dahm [trmss
7
+
Mqufdfgrfub
zz
2
u 1 v 2 ----r + -r r
xx yy
w 9 u v yx y x z xy v v 1 u r 2 -- r + ‛ ----r r r
2
M yy
yy ‛ xx
xx M xx yy / 1 ‛ 7
N xy
zy
2
v + w z y
zx
7 N yy
1 ‛ yy ‛ xx
7N
jr
yx h x + yy h y + yz h z 2 t y
w 2 w?
jr
zx h x + zy h y + zz h z 2 t z
N xy
w + u x z
xy
7N -------------------- 1 ‛ 7
u r cjbj eu
u
er
r
7
2
?
r
u r 2 u r c + u r p
u r c 2 K 1 r + K 7 r
2 K 1 r + K 7 r
1 ‛ --------r + ----r
u eu M ------------------- --------r + ----r 7 r er 1 ‛
r
2 ?
e rr rr ‛ ----------- + ----------------------- + @ r 2 ? er r
u 2 ----r r
u r 2 u r cjbj + ur part
rr 2
2
7N
v 2 v?
_dahm [trmss
xz
xy N xy
xx -------------------- 1 ‛ xx yy yy -------------------- 1 ‛ yy xx xy N xy 1 ‛ 7 1 ‛ 7 xx yx zx yy zy zz ----------+ ---------- + ---------- + @ 2 ? ------ xy ---- + ----------+ ----------- + @ 2 ? ------ xz ---- + ------ yz ---- + ---------- + @ z 2 x y z x x y z y x y z xy 2 yx yz 2 zy zx 2 xz
rr 2
2
xy
yy M yy xx / 1 ‛ 7
1 ‛ xx ‛ yy
Axfsybbmtrfk eu rr 2 --------r er
_dahm [trafh
2
7
rr 1 r rr ‛ 1 r 7 r -+ -+ ---------------------- + @ r 2 ? + + ---------- + @ r r r r r r u 2 u? jr xx h x + xy h y + xz h z 2 t x Gjuheary Kjheftfjhs
yz
xy x y
y7 x 7 M xx xx ‛ yy 7 N xx
_dahm [trafh
v yy y
ur eu r 7N 2 -------------------- 1 ‛ ---- + -------r er 1 ‛ 7
1 ‛ 7 @ 2 ? e 1 -- e ru + ------------------------- @ 7 1 ‛ N r e r r e r r 7
eu r u M 2 ------------------- ----r + -------7 er r 1 ‛
1 ‛ e 1 -- e ru + ------------------- @ r 2 ? M e r r e r r
7 ;
K 1 ‛ b r u r 2 K 1 r + -----7- ‛ ------------------------------------------r 3 M Wjtatfhn Efsi
K 7 1 ‛ M 7 7 ; + ------------------- K 1 1 + ‛ ---------------------- ‛ ----------------- b r 7 7 3 1 ‛ r
rr 2
K 7 1 ‛ M 7 7 1 + ; 2 ------------------- K 1 1 + + ---------------------- ‛ -------------------- b r 7 7 3 1 ‛ r
Afry [trmss @uhktfjh [trmssms
xx
Ef``mrmhtfad mquatfjh
5
2
7 y 7
yy
2
7 x7
xy
2 ‛
7
x y
2 ?
Ujrsfjh Ujrsfjh j` hjh-kfrkudar sca`ts
y xz ‛ z xy e A
Ujrsfjh
U 2
[afht Pmhaht‖s Bmtcje
xy
2 N y ‛ z e e x
_rahetd‖s Bmtcje
xy
2
Bmbgrahm Ahadjny Ucfh-waddme jpmh smktfjhs
u 2
A
y z e e x z
ub
v 2 ‛ xz e e x
xz 2
7
xz 2 ‛ y 9
p 7 e e x
w 2 xy e e x
N z + y e e x
1 [ N
2 ‛ 7 N e e x 9 U 2 7
ub z yx
Pjdubm gmhmatc tcm bmbgrahm U 7 U 2 NI e e x
;
I 2 gt ;
bax
9
2 U t I
A
e y
e z
u b y ‛ xz
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
>
Parfatfjhad ahe Mhmrny Bmtcjes
DMAWHFHN JGLMKUFPM 1 1. ]hemrstahe tcm kjhkmpts fh varfatfjhad kadkudus. 7 7. ]hemrstahe tcm usm j` varfatfjhad kadkudus tj jgtafh gjuheary vadum prjgdmbs fh bmkcahfks j` batmrfads. ; ;. ]hemrstahe tcm usm j` varfatfjhad kadkudus fh apprjxfbatm bmtcjes j` Waydmfnc-Wftz ahe @fhftm Mdmbmht Bmtcje. 5 = [trafh mhmrny, tcm pjtmhtfad j` wjri, ahe pjtmhtfad mhmrny arm add `uhktfjhs j` tcm efspdakmbmhts. Ucm efspdakmbmhts arm `uhktfjhs 0 j` tcm pjsftfjh kjjrefhatms. Ucus, strafh mhmrny, tcm pjtmhtfad j` wjri, ahe pjtmhtfad mhmrny arm `uhktfjhs j` `uhktfjhs. [ukc `uhk> tfjhs j` `uhktfjhs arm kaddme `uhktfjhads. Parfatfjhad kadkudus (Dahkzjs Y1630R) fs tcm grahkc j` batcmbatfks fh wcfkc tcm bax3 fbub ahe bfhfbub vadums j` `uhktfjhads arm emtmrbfhme. Emsfnhfhn strukturms `jr bfhfbub wmfnct fs a prjgdmb fh jptfbfzatfjh 6 fh wcfkc tcm wmfnct fs tcm `uhktfjhad tcat fs bfhfbfzme. Fheustrfad mhnfhmmrfhn kjhtafhs bahy appdfkatfjhs j` varfatfjhad kadkudus 1? fhvjdvfhn bfhfbfzfhn kjst, tfbm, ahe sj jh. 11 Ucm prfhkfpdms usme fh mhmrny bmtcjes arm a sug-kdass j` tcm prfhkfpdms em`fhme fh varfatfjhad kadkudus. ^m wfdd usm tcm 17 tcmjrmb j` bfhfbub pjtmhtfad mhmrny tj jgtafh tcm gjuheary vadum prjgdmbs `frst `jr dfhmar prjgdmbs ahe tcmh `jr nmjbmtrfk ahe 1; batmrfad hjhdfhmar prjgdmbs. ^m wfdd adsj usm tcmjrmb j` bfhfbub pjtmhtfad mhmrny tj emvmdjp tc m apprjxfbatm bmtcjes j` Way15 dmfnc-Wftz ahe @fhftm Mdmbmht Bmtcje. 1= 10 >.1 GA[FK KJHKM_U[ FH PAWFAUFJHAD KADK]D][ 1> 13 Jur vjkagudary cas djt j` supmrdatfvm tmrbs dfim darnmst, sbaddmst, qufkimst, dmast, bjst, tcat fs, wjres tcat rm`dmkt mxtrmbm vadums 16 j` a nfvmh attrfgutm. Gut cjw ej wm emtmrbfhm ah mxtrmbm vadum: [uppjsm wm wfsc tj emkfem wcj fs tcm taddmst pmrsjh fh a rjjb: ^cat ej wm ej: ^m kjbparm cmfncts j` twj pmjpdm ahe 7? 71 emkfem wcj fs taddmr. ^m hmxt kjbparm tcm taddmr j` tcm twj pmjpdm wftc tcm hmxt pmrsjh ahe rmpmat tcfs tfdd wm kjhkduem Lfdd fs tcm 77 taddmst pmrsjh fh tcm rjjb. [uppjsm wm hjw asi wcj fs tcm taddmst pmrsjh fh tcm gufdefhn: ^m jhkm bjrm kjbparm tcm cmfncts j` 7; pmjpdm fh tcm gufdefhn ahe kjhkduem tcat Laki fs tcm taddmst pmrsjh. ^m hmxt asi wcj fs tcm taddmst pmrsjh fh tcm uhfvmrsfty fs ahe nj 75 tcrjunc tcm mxmrkfsm j` kjbparfhn cmfncts j` add pmjpdm fh tcm uhfvmrsfty. Ucfs mxabpdm cfncdfncts twj fbpjrtaht pjfhts. 7= 1. Uj `fhe ah mxtrmbm vadum wm hmme tj kjbparm. 70 7. Ucm kjbparfsjh fs fh tcm fbbmefatm hmfncgjrcjje ahe wfdd nfvm us jhdy a djkad mxtrmbub vadum ahe hjt a 7> ndjgad mxtrmbub. 73 Ucm smkjhe pjfht cas trmbmhejus fbpakt jh sjdutfjh prj kmeurms j` hjh-dfhmar prjgdmbs. Gmkausm ndjgad mxtrmbub (say 76 ;? a bfhfbub vadum) kahhjt gm nuarahtmme uhdmss sjbm aeeftfjhad kjheftfjhs arm spmkf`fme. Hjw suppjsm wm wfsc tj `fhe tcm cfncmst pjfht fh jur kfty. ^m kah kjheukt tcm cmfnct kjbparfsjh fh twj ways. ;1 ;7 1. ^m kah wadi tj ef``mrmht pjfhts ahe bmasurm tcm mdmvatfjh― wm wfdd kadd ft tcm e-prjkmss9 sj e x, x, e s, s, eu eu rmp;; rmsmht tcm aktuad bjvmbmht adjhn a patc (kurvm). ;5 7. ^m kah kjheukt a tcjunct mxpmrfbmht. @jr mxabpdm, wftcjut bjvfhn, wm asi tcm qumstfjh, f` wm nj tj tcat ;= pjfht wfdd tcm mdmvatfjh fhkrmasm jr emkrmasm: Ucfs fbanfhary bjvmbmht fs kaddme tcm vfrtuad bjvmbmht ;0 jr tcm -prjkmss. ;> @fnurm >.1 scjws >.1 scjws tcm aktuad efspdakmbmht j` gmab uhemr uhf`jrb djae ahe a tcjunct mxpmrfbmht (vfrtuad efspdakmbmht) ;3 ;6 wm bay kjhsfemr fh ah m``jrt tj `fhe tcm aktuad efspdakmbmht. Ucm `uhktfjhs ahe fts emrfvatfvms arm rmdatme fh tcm aktuad efspdakm5? bmhts as wm bjvm adjhn tcm kurvm. Gut fh tcm vfrtuad efspdakmbmht wm kah kcahnm tcm `uhktfjh ahe ft emrfvatfvms (sdjpm) fhempmh51 emhtdy. Fh bahy aspmkts tcm e-prjkmss ahe tcm -prjkmss gmcavm batcmbatfkaddy fh a sfbfdar bahhmr, gut tcm ef``mrmhkm gmtwmmh >.1g satfs`fms tcm zmrj efs57 tcm twj fs krftfkad ahe wfdd gm mdagjratme `urtcmr a dfttdm datmr. Hjtm tcm vfrtuad efspdakmbmht fh @fnurm >.1g pdakmbmht kjheftfjh at tcm suppjrt ahe fs a kjhtfhujus `uhktfjh. ^m wfdd sjjh baim usm j` tcfs jgsmrvatfjh. 5; 10>
103
>
Parfatfjhad ahe Mhmrny Bmtcjes y
w
Pfrtuad Pfrtuad efspdakmbmht `uhktfjh x
Aktuad efspdakmbmht `uhktfjh (g)
(a)
@fnurm >.1 Ef``mrmhkm gmtwmmh aktuad ahe vfrtuad efspdakmbmht.
^cmh a varfagdm fh a smt kahhjt gm rmprmsmhtme as a `uhktfjh j` tcm jtcmr varfagdms tcmh ft fs safe tj gm fhempmhemht. F` add a‖s fh tcm mquatfjh gmdjw arm hjh-zmrj, tcmh wm kjude sjdvm jhm j` tcm u‖s fh tmrbs j` jtcmrs ahe cmhkm gy em`fhftfjh tcm smt fs hjt fhempmhemht. a 1 u 1 + a 7 u 7 + a; u ;
+ ah uh 2 ?
(>.1)
Adtmrhatfvmdy statme, F` u1, u7, u;,...u ,...uh arm fhempmhemht `uhktfjhs tcmh Mquatfjh (>.1) fbpdfms (>.1) fbpdfms a f 2 ?
f 2 1 tj h
(>.7)
Ahy smt j` fhempmhemht varfagdms (parabmtmrs) tcat emskrfgms tcm systmb nmjbmtry arm kaddme tcm nmhmradfzme kjjrefhatms. Efspdakmbmhts j` pjfhts, rjtatfjh j` dfhms, kjm``fkfmhts j` a pjdyhjbfad rmprmsmhtfhn efspdakmbmhts arm sjbm mxabpdms j` nmhmradfzme kjjrefhatms. Ucm spakm spahhme gy tcm nmhmradfzme kjjrefhatms fs kaddme tcm kjh`fnuratfjh spakm. Ahy kjheftfjh tcat dfbfts tcm kcahnm fh nmjbmtry fh tcm kjh`fnuratfjh spakm fs kaddme tcm ifhmbatfk kjheftfjh. A pjfht kahhjt bjvm, a dfhm kahhjt rjtatm, arm sjbm mxabpdms j` ifhmbatfk kjheftfjhs. @uhktfjhs tcat arm kjhtfhujus ahe kjhtfhujus ahe satfs`y add tcm tcm ifhmbatfk gjuheary kjheftfjhs arm kjheftfjhs arm kaddme ifhmbatfkaddy aebfssfgdm `uhktfjhs. Ucm aktuad efspdakmbmht sjdutfjh fs adways a ifhmbatfkaddy aebfssfgdm `uhktfjh. Gut uhdfim aktuad efspdakmbmhts, tcm ifhmbatfkaddy aebfssfgdm `uhktfjhs kah rmsudt fh `jrkms ahe bjbmhts tcat ej hjt satfs`y tcm mqufdfgrfub jr gjuheary kjheftfjhs jh `jrkms ahe bjbmhts. Pfrtuad efspdakmbmht bust adways gm ifhmbatfkaddy aebfssfgdm. aebfssfgdm.
>.1.1
Mxtrmbub ahe statfjhary Padums
[uppjsm wm waht tj `fhe tcm bfhfbub j` a `uhktfjh j` fhempmhemht varfagdms u1, u7, u;...u ...uh as scjwh gmdjw. @ 2 @ u1 u7
+ uh
(>.;a)
^m kjhsfemr a vfrtuad kcahnm fh tcm kjh`fnuratfjh spakm, tcat fs, spakm spahhme gy tcm fhempmhemht varfagdmsu varfagdms u1, u7, u;...u ...uh. Ucm tjtad vfrtuad kcahnm @ fs tcm sub j` sdjpm budtfpdfme gy vfrtuad kcahnm fh makc efrmktfjh. @ @ @ @ 2 u 1 + u 7 + u ; u1 u7 u;
+
+
@ u uh h
(>.;g)
j` @ . F` @ F` @ fs fs tj gm a bfhfbub at a pjfht fh tcm kjh`fnuratfjh spakm, tcmh tcfs kcahnm j` @ bust @ fs kaddme tcm `frst varfatfjh j` @ gm zmrj.
@ 2 @ u 1 + @ u 7 + @ u ; u1 u7 u;
+
+
@ u uh h
2 ?
(>.;k)
F` u1, u7, u;...u ...uh arm fhempmhemht varfagdms tcmh wm arm `rmm tj bjvm fh ahy efrmktfjh. [j f` wm jhdy wadi fh u1 (add jtcmr vfrtuad efspdakmbmht arm zmrj) tcmh wm cavm @ u 1 2 ? . Fh a sfbfdar bahhmr wm kah wadi fh makc j` tcm efrmktfjhs ahe kjhkduem<
@ uf
f 2 1 7 ;
2 ?
h
(>.5)
@ fs adsj zmrj at baxfbub ahe at saeedm pjfht. Uj emkfem wcmtcmr wm cavm bfhfbub, baxfbub, jr saeedm pjfht, tcat 7 fs ah mxtrmbm vadum (mxtrmbub), wm hmme tj kjhsfemr tcm sfnh j` smkjhe varfatfjh j` @ j` @ , tcat fs sfnh j` @ . A `uhktfjh fs safe tj cavm a statfjhary vadum f` ft ejms hjt kcahnm fh tcm fbbmefatm hmfncgjrcjje j` @ 2 ? . Ucus, tj emkfem ah mxtrmbm vadum wm 7 bust kjhsfemr gjtc @ 2 ? ahe sfnh j` @ , gut `jr statfjhary vadum wm hmme tj kjhsfemr jhdy @ 2 ? . frrmvmr frrmvmrsfgd sfgdmm efspdakm efspdakmbmht bmht
@ ´ ? rmvmrsf rmvmrsfgdm gdm efspdakm efspdakmbmh bmhtt
@ @fnurm >.7 Wmvmrsfgdm ahe frrmvmrsfgdm efspdakmbmhts.
2 ?
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
^jri
106
1 @fnurm >.7 grfhns >.7 grfhns jut ahjtcmr ef``mrmhkm gmtwmmh tcm mxtrmbub ahe statfjhary vadum. Ucm bargdm at tcm gjttjb j` tcm 7 trjunc satfs`fms kjheftfjhs `jr mxtrmbub ahe statfjhary vadum. Gut a bargdm prmvmhtme `rjb rjddfhn ejwh tj tcm gjttjb scjws ah ; mxtrmbm vadum (djwmst pjfht) gut @ wfdd hjt gm zmrj. Fh kjbparfhn kjbparfhn cmfncts, cmfncts, bargdm at tcm djwmst pjfht kah gm bjvme fh ahy 5 efrmktfjh, tcat fs, tcm efspdakmbmht fs rmvmrsfgdm. Gut wcmh tcm bargdm fs prmvmhtme `rjb rjddfhn ejwh tj tcm djwmst pjfht, kjb= parfsjh j` cmfnct fs jhdy pjssfgdm fh jhm efrmktfjh, tcat fs, efspdakmbmht fs frrmvm rsfgdm. @jr @jr statfjhary vadums wm hmme tcm vfrtuad 0 efspdakmbmht tj gm rmvmrsfgdm. rmvmrsfgdm. > 3 >.1.7 @uhktfjhads 6 Wmnudar kadkudus kjhsfemrs tcm varfatfjh j` a `uhktfjh. Parfatfjhad kadkudus kjhsfemrs t cm varfatfjh j` `uhktfjhads. ^m wfdd kjh1? sfemr sjbm prjpmrtfms j` `uhktfjhads fh tcfs smktfjh. Ucmsm prjpmrtfms wfdd gm usme tj sfbpdf`y adnmgra fh sugsmqumht smktfjhs. 11 A `uhktfjh u x fs a rudm j` kjrrmspjhemhkm sukc tcat `jr add x add x fh fh E E tcmrm fs assfnhme a uhfqum mdmbmht u x fh W fh W.. A 17 `uhktfjhad @ u x fs a rudm j` kjrrmspjhemhkm sukc tcat `jr add u x fh W fh W tcmrm fs assfnhme a uhfqum mdmbmht @ u x fh . Fh 1; jtcmr wjres, a `uhktfjhad fs a `uhktfjh j` a `uhktfjh. 15 A dfhmar `uhktfjhad d (u) fs jhm tcat satfs`fms tcm rmdatfjhscfp fh Mquatfjh (>.=a). (>.=a). 1= (>.=a) d 1 u 7 v 1 d u 7 d v 10 1> wcmrm ahe arm ahy skadars. ^jri pjtmhtfad efskussme fh [mktfjh >.7 fs >.7 fs a dfhmar `uhktfjhad j` efspdakmbmhts. 1 7 13 A gfdfhmar `uhktfjhad G( G(u, v) fs jhm tcat fs a dfhmar `uhktfjhad fh makc j` fts arnubmhts j`u j`u ahe v, as scjwh fh Mquatfjh (>.=g). (>.=g). 16 (>.=g) G 1 u 1 7 u 7 v 1 G u 1 v 7 G u 7 v G u 1 v 1 7 v 7 1 G u v 1 7 G u v 7 7? 71 A sybbmtrfk gfdfhmar `uhktfjhad fs a gfdfhmar `uhktfjhad tcat fs sybbmtrfk wftc rmspmkt tj fts arnubmhts, as scjwh fh 77 Mquatfjh (>.=k). (>.=k). [trafh mhmrny efskussme fh [mktfjh >.; fs >.; fs a sybbmtrfk gfdfhmar `uhktfjhad. 7; G u v 2 G v u (>.=k) 75 7= Fh tcm agjvm em`fhftfjhs j` dfhmar ahe gfdfhmar `uhktfjhads, u ahe v kah gm vmktjrs, tcat fs, makc j` tcm arnubmht kah cavm bahy 70 kjbpjhmhts ahe tcm prjpmrty j` tcm `uhktfjhads arm appdfkagdm tj makc j` tcm kjbpjhmht. 7> 73 >.7 ^JWI 76 ;? ^jri fs ejhm gy a `jrkm f` tcm pjfht at wcfkc tcm `jrkm fs appdfme bjvms. F` tcm pjfht at wcfkc `jrkm @ fs appdfme bjvms tcrjunc ;1 ah fh`fhftmsfbad efstahkm e u , tcmh tcm wjri fs em`fhme as ;7 (>.0) e^ 2 @ e u ;; ;5 G G ;= ;0 1 1 ;> ;3 ^ 1 AG 2 ^ 7 AG ^ 1 AG + ^ 7 GA 2 ? ;6 7 7 5? 51 57 A A 5; (a) (g) 55 5= @fnurm >.; Kjhsmrvatfvm wjri. 50 5> Fhtmnratfhn tcm wjri mxprmssfjh fh Mquatfjh (>.0) adjhn (>.0) adjhn tcm patc j` bjvmbmht nfvms tcm tjtad wjri ejhm gy tcm `jrkm. 53 @fnurm >.; scjws >.; scjws twj ways fh wcfkc wjri ejhm fs kjhsmrvme. F` tcm start ahe mhe pjfhts arm tcm sabm ahe tcm wjri fs patc fhem56 pmhemht, tcmh tcm tjtad wjri ejhm bust gm zmrj as tcm wjri mxpmheme fh bjvfhn a pjfht `jrware fs rmkjvmrme (kjhsmrvme) wcmh =? tcm pjfht bjvms gaki tj tcm startfhn pjfht. ^jri ejhm gy a `jrkm fs kjhsmrvme f` ft fs patc fhempmhemht. Kdmardy, f` `rfktfjh fs prms=1 mht, tcm wjri ejhm tj jvmrkjbm `rfktfjh bust vary wftc tcm dmhntc j` patc9 cmhkm tcm `rfktfjhad `jrkm fs hjhkjhsmrvatfvm. Fh a sfb=7 fdar bahhmr, f` a gjey em`jrbs pdastfkaddy, tcm wjri ejhm tj krmatm pmrbahmht em`jrbatfjh wfdd hjt gm rmkjvmrme, ahe jhkm bjrm =; wm cavm a hjhkjhsmrvatfvm systmb. Ucus, mdastfk em`jrbatfjh fs a kjhsmrvatfvm em`jrbatfjh, wcfdm pdastfk em`jrbatfjh fs hjhkjh=5 smrvatfvm. Mdastfk em`jrbatfjh kah, cjwmvmr, gm dfhmar jr hjhdfhmar. Wuggmr cas hjhdfhmar strmss ‛strafh kurvm. ^jri ejhm fh == strmtkcfhn ruggmr fs rmkjvmrme wcmh tcm `jrkms arm rm dmasme ahe tcm ruggmr rmturhs tj tcm uhem`jrbme pjsftfjh. Ucus, tcmrm fs a efs=0 tfhktfjh gmtwmmh hjhkjhsmrvatfvm ahe hjhdfhmar systmbs. => • Hjhdfhmar systmbs ahe hjhkjhsmrvatfvm systmbs arm twj fhempmhemht emskrfptfjhs j` a systmb. =3 Mquatfjh (>.0) scjws (>.0) scjws tcat wjri fs a skadar quahtfty. Cmhkm wjri ejhm gy ef``mrmht `jrkms ahe bjbmhts kah gm aeeme tj =6 jgtafh tcm tjtad wjri ejhm gy tcm `jrkms aktfhn jh a strukturm. ^m hjw kjhsfemr wjri ejhm gy `jrkms ahe bjbmhts jh axfad rje, 0? sca`ts, gmabs, pdatms, ahe fh ah mdastfk pdahm wcfkc wm kah datmr aee tj jgtafh tcm tjtad wjri ejhm jh tcm strukturm. 01
1>?
>
Parfatfjhad ahe Mhmrny Bmtcjes
Axfad Bmbgmr< Kjhsfemr ah axfad bmbgmr suglmktme tj a efstrfgutme `jrkm pmr uhft dmhntc j` x x . Ucus, tcm `jrkm aktfhn jh a ef``mrmhtfad mdmbmht wfdd gm p x x ex . Fh aeeftfjh, wm assubm tcmrm arm b kjhkmhtratme `jrkms @ q aktfhn at pjfhts xq . Budtfpdyfhn tcmsm `jrkms gy tcm efspdakmbmht j` tcm pjfhts ahe aeefhn wm jgtafh tcm tjtad wjri ejhm jh tcm axfad bmbgmr ^ A as scjwh gy tcm Mquatfjh (>.>a). (>.>a). Ucm fhtmnratfjh fs jvmr tcm dmhntc D dmhntc D j` j` tcm axfad bmbgmr. Fh Mquatfjh (>.>a), (>.>a), wm assubme tcat p x x , u x , @ q , ahe u x q arm add pjsftfvm fh tcm pjsftfvm x pjsftfvm x efrmktfjh. ^m hjtm tcat tcm ^ A fs a dfhmar `uhktfjhad j` u as scjwh fh tcm mquatfjh.
aheb Ujrsfjh j` sca`t< Fh a bahhmr sfbfdar tj ah axfad bmbgmr, a sca`t suglmktme tj a efstrfgutme tjrqum pmr uhft dmhntc t x , aheb kjhkmhtratme tjrqums U q appdfme at pjfhts x q wfdd yfmde tcm tjtad wjri ejhm jh tcm sca`t ^ U as scjwh gyMquatfjh gyMquatfjh (>.>g). (>.>g). Ucm quahtftfms t x , x , U q , ahe xq , arm add pjsftfvm kjuhtmr-kdjkiwfsm wftc rmspmkt tj tcm x tcm x axfs. axfs. Gmhefhn j` sybbmtrfk gmabs< Ucm wjri ^ G ejhm jh tcm sybbmtrfk gmab fh a sfbfdar bahhmr fs nfvmh gy tcmMquatfjh tcm Mquatfjh (>.>k). (>.>k). ^cmrm, v ahe y x arm tcm efspdakmbmht ahe tcm efstrfgutme `jrkm pmr uhft dmhntc fh tcm y efrmktfjh, @ q arm tcm trahsvmrsm `jrkms aktfhn at pjfhts x q , ahe q arm gmhefhn bjbmhts aktfhn at pjfhts x q . Efspdakmbmhts ahe `jrkms arm pjsftfvm fh tcm pjsftfvm y efrmktfjhs, ahe bjbmhts ahe sdjpms arm pjsftfvm kjuhtmr-kdjkiwfsm agjut tcm z tcm z efrmktfjh efrmktfjh (gmhefhn axfs). Gmhefhn j` tcfh pdatms< @jr tcfh pdatms, wm wfdd kjhsfemr jhdy tcm trahsvmrsm efstrfgutme `jrkm pmr uhft arma z x y aktfhn fh tcm pjsftfvm z pjsftfvm z efrmktfjh. efrmktfjh. Dfhm djaes, ahe kjhkmhtratme `jrkms ahe bjbmhts kah gm fhkjrpjratme gut rmsudts fh djhn mxprmssfjhs wcfkc ej hjt aee tj tcm kjhkmpts efskussme. Em`fhfhn w x y as tcm pjsftfvm efspdakmbmht fh tcm z efrmktfjh, wm jgtafh tcm wjri ejhm jh tcm pdatm ^ _ as ah fhtmnrad jvmr tcm arma A arma A j` j` tcm pdatm as scjwh fh Mquatfjh (>.>e). (>.>e). Mdastfk pdahm< ^m kjhsfemr jhdy gjey `jrkms @ x x y ahe @ y x y `jr pdahm strmss mdastfkfty prjgdmbs. ^m assubm tcm uhf`jrb tcfkihmss j` tcm pdahm fs c, ahe tcm efspdakmbmhts fh x fh x ahe ahe y y efrmktfjhs arm u x y ahe v x y . Ucm wjri ejhm fh pdahm strmss mdas mdastf tfkf kfty ty^ M fs nfvm nfvmh h gy tcm tcm Mquatfjh (>.>m). (>.>m).
^jri Mxprmssfjhs Uagdm >.1 ^jri ^jri
^ A
Ujrsfjh j` kfrkudar sca`ts
^ U
[ybbmtrfk gmhefhn j` gmabs
^ G
Axfad
p x x u x e x
t x x e x
p y x v x e x
D
D
D
Gmhefhn j` tcfh pdatms
^ _ 2
_dahm strmss mdastfkfty
^ M 2 c
>.;
b q1
b q1
U q x q
b1 q1
x y w x y ex e y
p A z
A
@ q u x q
d u
(>.>a)
d
@ q v x q
b7 q1
(>.>g) B q
ev x e x q
d v
2 d w
(>.>k) (>.>e)
@ x x y u x y + @ y x y v x y ex e y
2 d u v
(>.>m)
[UWAFH MHMWNV
Ucm kcahnm fh fhtmrhad mhmrny fh a gjey eurfhn em`jrbatfjh fs kaddme tcm strafh mhmrny. Ucm mhmrny pmr uhft vjdubm fs kaddme tcm strafh mhmrny emhsfty ahe fs tcm arma uhemr tcm strmss‛strafh kurvm up tj tcm pjfht j` em`jrbatfjh. Mquatfjh (>.3) scjws (>.3) scjws tcm rmdatfjhscfp gmtwmmh tcm strafh mhmrny ahe tcm strafh mhmrny emhsfty, ]
] ? e P
(>.3)
P
wcmrm ] fs fs tcm strafh mhmrny, ] ? fs tcm strafh mhmrny emhsfty ahe P fs fs tcm vjdubm j` tcm gjey. Hjtfhn tcat tcm strafh mhmrny emhsfty fs tcm arma uhemr tcm kurvm scjwh fh @fnurm >.5, >.5, wm jgtafh ,
] ?
e
(>.6)
?
Mquatfjh (>.6) scjws (>.6) scjws tcat strafh mhmrny emhsfty cas tcm sabm efbmhsfjhs as tcat j` strmss gmkausm strafh fs efbmhsfjhdmss. Gut tcm uhfts j` strafh mhmrny emhsfty arm uhfts j` mhmrny pmr uhft vjdubm, wcfkc arm ef``mrmht `rjb tcjsm j` strmss. Ucm uhfts `jr strafh mhmrny emhsfty arm Hmwtjh-bmtmrs pmr kugfk bmtmr (H b/b;), ljudms pmr kugfk bmtmr (L/b;), fhkc-pjuhes pmr kugfk fhkc (fh dg/fh;), ahe `jjt-pjuhes pmr kugfk `jjt (`t dg/`t;). Ahjtcmr rmdatme kjhkmpt fs tcm kjbpdmbmhtary strafh mhmrny emhsfty (] ?), scjwh fh @fnurm >.5 ahe >.5 ahe em`fhme as `jddjws< ] ?
e
(>.1?)
?
Bjst mhnfhmmrfhn strukturms arm emsfnhme `jr dfhmar mdastfk batmrfads. Fh tcm dfhmar rmnfjh, tcm arma uhemr tcm strmss‛strafh kurvm fs a trfahndm, ahe wm jgtafh Mquatfjh (>.11) `jr (>.11) `jr tcm strafh mhmrny emhsfty `uhktfjh. ] ?
7
(>.11)
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
[trafh mhmrny 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
1>1
]j 2 Kjbpdmbmhtary strafh mhmrny emhsfty
e]j 2 e
A
e
]j 2 [trafh mhmrny mhmrny emhsfty
e]j 2 e
J e
@fnurm >.5 Mhmrny emhsftfms.
F` fhstmae j` a kurvm rmprmsmhtfhn hjrbad strmss vs. hjrbad strafh, wm cavm a kurvm j` scmar strmss vs. scmar strafh, tcmh wm wfdd cavm a sfbfdar mxprmssfjh `jr strafh mhmrny emhsfty fh tmrbs j` scmar strmss ahe scmar strafh, as scjwh gy Mquatfjh (>.17). (>.17). ] ?
e 7
(>.17)
?
[trafh mhmrny, cmhkm strafh mhmrny emhsfty, fs a skadar quahtfty. ^m ^m kah aee tcm strafh mhmrny emhsfty eum tj fhefvfeuad strmss ahe strafh kjbpjhmhts tj jgtafh Mquatfjh (>.1;) `jr (>.1;) `jr tcm tjtad dfhmar strafh mhmrny emhsfty eurfhn emhsfty eurfhn em`jrbatfjh. 1 (>.1;) ] ? -- xx xx yy yy zz zz xy xy yz yz zx zx 7 Fh tcm agjvm mquatfjhs tcm strafh fs tcm bmkcahfkad strafh. F` tcmrbad strafhs arm prmsmht tcmh tcmy hmme tj gm sugtraktme `rjb tcm tjtad strafh as scjwh gmdjw. 1 ] ? -- xx xx ‛ U yy yy ‛ U zz zz ‛ U xy xy yz yz zx zx (>.15) 7
>.;.1
[trafh mhmrny fh sybbmtrfk gmhefhn j` gmabs
^m hjw kjhsfemr tcm str afh mhmrny mxprmssfjhs `jr sybbmtrfk gmhefhn agjut tcm z tcm z axfs. axfs. Mxprmssfjhs `jr axfad ahe tjrsfjh kah gm jgtafhme fh a sfbfdar bahhmr ahe fs dm`t as ah mxmrkfsm `jr tcm rmaemr (smm prjgdmbs >.1 ahe >.7). ^cmh sybbmtrfk smktfjhs arm gmht agjut tcm z tcm z axfs, axfs, tcmrm arm twj hjhzmrj strmss kjbpjhmhts, xx ahe xy. ^m scadd kjhsfemr strafh mhmrny eum tj makc smparatmdy. [ugstftutfhn xx M xx ahe xx y( y(e 7 v/ex v/ex 7) fh Mquatfjh (>.1;) ahe (>.1;) ahe hjtfhn tcat (e (e 7 v/ 7 ex ) fs lust a `uhktfjh j` x j` x ahe ejms hjt kcahnm akrjss tcm krjss-smktfjh, wm jgtafh tcm gmhefhn strafh mhmrny ] G as ] G
1 7 -- M xx e P P 7
7
D
7
1 e v -- M y -------- e A e x 7 A 7 ex
D
1 e v -- -------- 7 ex 7 7
7
My e A e x 7
(>.1=a)
A
wcmrm, D wcmrm, D fs fs tcm dmhntc j` tcm gmab ahe A ahe A fs fs tcm krjss smktfjhad arma. Assubfhn batmrfad fs cjbjnmhmjus akrjss tcm krjss smktfjh, tcat fs, M fs, M ejms ejms hjt kcahnm fh tcm fhtmnrad, wm jgtafh
My e A M y e A MF 7
7
(>.1=g)
zz
A
A
[ugstftutfhn Mquatfjh (>.1=g) fhtj (>.1=g) fhtj Mquatfjh (>.1=a) wm (>.1=a) wm jgtafh Mquatfjh (>.1>k). (>.1>k). @jr kjbpjsftm sybbmtrfk gmabs, tcm fhtmnrad jvmr tcm arma A arma A wjude wjude gm rmpdakme gy ah fhtmnrad jvmr makc batmrfad arma, ahe tcm vadum j` tcm fhtmnrad fhMquatfjh fh Mquatfjh (>.1=g) wjude gm tcm sub j` tcm gmhefhn rfnfeftfms j` add tcm batmrfads. Ucm strafh mhmrny eum tj scmar fh gmhefhn fs ] [ 1 7 xy xy e P . ^m hjtm tcat tcm baxfbub scmar strmss xy ahe scmar strafh xy arm ah jremr j` banhftuem sbaddmr tcah tcm baxfbub hjrbad strmss xx ahe tcm baxfbub hjrbad strafh xx . Ucus, ] [ wfdd gm twj jremrs j` banhftuem sbaddmr tcah ] G ahe kah gm hmndmktme fh jur kadkudatfjhs.
>.;.7
[trafh mhmrny fh gmhefhn j` tcfh pdatms
@rjb Kcaptmr =, =, jh gmhefhn j` tcfh pdatms wftc hj fhpdahm `jrkms, wm cavm tcm `jddjwfhn mquatfjhs `jr strafhs ahe strmssms.
xx
2
u x
7
w 2 ‛ z 7 x
yy
2
v y
7
w 2 ‛ z 7 y
xy
2
u + v y x
7
w 2 ‛ 7 z x y
(>.10a)
1>7
>
Parfatfjhad ahe Mhmrny Bmtcjes
1 (>.10g) 7 ; Ucm trahsvmrsm scmar strmssms xz ahe yz arm ah jremr j` banhftuem dmss tcah tcm fhpdahm strmssms, cmhkm tcm strafh 5 mhmrny eum tj tcmsm scmar strmssms wfdd gm hmndfnfgdm kjbparme tj tcm strafh mhmrny eum tj tcm fhpdahm strmssms ahe wm hmndmkt ft. = @jr sfbpdfkfty wm assubm tcat tcm pdatm fs cjbjnmhmjus, cmhkm batmrfad kjhstahts kah gm taimh jutsfem tcm fhtmnrads. [ugstftutfhn 0 Mquatfjhs (>.10a) ahe (>.10a) ahe (>.10g) fhtj (>.10g) fhtj Mquatfjh (>.1;) ahe (>.1;) ahe (>.3) wm jgtafh tcm `jddjwfhn mquatfjhs `jr tcm pdatm strafh mhmrny ] _ . > 7 3 M xy 1 M M ------------------] _ 2 -- ------------------+ + + + -------------------e P jr yy xx yy xx yy 6 7 7 1 ‛ 7 xx 7 1 + 1 ‛ 1 ? P 11 1 M 1 ‛ 7 e P jr 7 7 ---------------] _ 2 -- ------------------+ + 7 + 17 xx yy xx yy xy 7 1 ‛ 7 7 1; P 7 7 7 15 7 7 7 7 7 w 1 M 7 w 7 w 7 w w 7 1= ] _ 2 -- ------------------z 7 + z 7 + 7 z 7 7 + 7 z 1 ‛ e P 7 1 ‛ 7 x y 10 x y x y P 1> ^m kah wrftm tcm vjdubm fhtmnrad as fhtmnrad jvmr tcm arma ( A) A) ahe fhtmnrad jvmr tcm tcfkihmss `rjb (-c (-c/7) tj (c (c/7), wcmrm 13 c fs tcm pdatm tcfkihmss. ^m `urtcmr hjtm tcat tcm kurvaturms arm hjt `uhktfjhs j` z j` z , cmhkm kah gm taimh jutsfem tcm fhtmnrad jvmr 16 tcfkihmss. ^m jgtafh 7? 7 7 7 7 7 7 7 7 71 w c 7 7 w w c 7 7 w c 7 7 w c 7 7 1 M ] _ 2 -- ------------------ z e z + z e z + 7 z e z + 7 1 ‛ z e z e x e y 77 7 1 ‛ 7 A x 7 ‛ c 7 y7 ‛ c 7 x7 y 7 ‛ c 7 x y ‛ c 7 7; ; 7 Hjtfhn tcat tcm pdatm‖s gmhefhn rfnfefty fs nfvmh gy E 2 Mc 17 1 ‛ , wm jgtafh Mquatfjh (>.1>e). (>.1>e). 75 7= Uagdm >.7 [trafh Mhmrny 70 7> [trafh Mhmrny 73 1 eu 7 ] MA e x Axfad (>.1>a) 76 A ex 7 D ;? 1 e 7 ] U -- NL ------ e Ujrsfjh j` kfrkudar sca`ts (>.1>g) ;1 ex x 7 D ;7 7 ;; e 7 v 1 -------] MF e x [ybbmtrfk gmhefhn j` gmabs (>.1>k) ;5 G 7 D zz ex 7 ;= 7 7 7 7 7 7 7 7 ;0 w w w w w E Ucfh _datms (>.1>e) ] _ 2 ---;> + 7 + 7 7 7 + 7 1 ‛ x y e x e y 7 A x 7 y x y ;3 ;6 u 7 Mc u v + v 7 --------------- 1 ‛ u + v 7 ex e y _dahm [trmss Mdastfkfty (>.1>m) ] M 2 ---------------------+ 7 5? 7 x y y 7 y x 7 1 ‛ A x 51 57 >.;.; [trafh mhmrny fh pdahm strmss mdastfkfty 5; 55 Mquatfjh (0.5a) ahe (0.5a) ahe Mquatfjh (0.5g) `jr (0.5g) `jr pdahm strmss arm rm-wrfttmh gmdjw `jr kjhvmhfmhkm. 5= (>.13a) 50 M xx xx ‛ yy M yy yy ‛ xx N xy xy 5> (>.13g) 53 xx M xx yy / 1 ‛ 7 yy M yy xx / 1 ‛ 7 xy N xy 56 @jr pdahm strmss mdastfkfty prjgdmbs add strmssms wftc sugskrfpts z sugskrfpts z arm arm zmrj. @rjb Mquatfjh (>.1;) wm (>.1;) wm jgtafh =? 1 ] ? -- xx xx yy yy xy xy (>.16a) =1 7 =7 [ugstftutfhn Mquatfjh (>.13g) fhtj (>.13g) fhtj Mquatfjh (>.16a), (>.16a), ahe tcmh sugstftutfhn tcm strafhs fh tmrbs j` efspdakmbmhts, wm jgtafh =; 1 M M 7 7 7 =5 ] ? -- ------------------ + 7 xx yy + yy -------------------- xy jr 7 1 ‛ 7 xx 7 1 + == =0 7 7 u 7 M u v v 1 ‛ u v (>.7?) => ] ? ---------------------+ x + 7 x y + y ---------------7 7 y x =3 7 1 ‛ =6 ^m assubm tcm uhf`jrb tcfkihmss j` tcm pdahm fs c, tcus tcm ef``mrmhtfad vjdubm fs e P 2 c e x e y . [ugstftutfhn Mquatfjh 0? (>.7?) fhtj (>.7?) fhtj Mquatfjh (>.3), (>.3), wm jgtafh tcm pdahm strmss mdastfkfty strafh mhmrny ] M as scjwh fh Mquatfjh (>.1>m). (>.1>m). 01
xx
M 2 ------------------ xx + yy 1 ‛ 7
yy
M 2 ------------------ yy + xx 1 ‛ 7
xy
2 N xy
Pfrtuad wjri
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1 >.;.5 [trafh mhmrny fh `jrb j` gfdfhmar `uhktfjhad 7 >.; scjws tcm strafh mhmrny wrfttmh as a gfdfhmar `uhktfjhad. Gy sugstftutfhn u1 2 u7 2 u, 1 2 7 2 , v1 2 v7 2 v, ahe ; Uagdm >.; scjws >.; wm jgtafh tcm mquatfjhs j` Uagdm >.7. >.7. Hjtm tcm gfdfhmar `uhktfjhad fs sybbmtrfk fh 5 w1 2 w7 2 w fhtj tcm mquatfjhs j` Uagdm >.; wm add kasms scjwh fh Uagdm >.;. >.; . = 0 Uagdm >.; Gfdfhmar `uhktfjhad `jrb j` strafh mhmrny. > [trafh Mhmrny 3 eu1 eu 7 6 1 1 ] A -- MA e x -- G u1 u7 Axfad (>.71a) 7 e x e x 7 1? D 11 e 1 e 7 Ujrsfjh j` kfr1 1 17 ] U -- NL e x -- G 1 7 (>.71g) 7 e x e x 7 kudar sca`ts 1; D 15 [ybbmtrfk 7 7 e v 1 e v 7 1 1 1= gmhefhn j` ] G -- MF zz ----------- ----------- e x -- G v1 v 7 (>.71k) 7 7 7 7 e x e x 10 gmabs D 1> 7 7 7 7 7 7 7 7 7 7 w 1 w 7 w 1 w 7 w 1 w 7 w 7 w 1 w 1 w 7 E 1 13 Ucfh pdatms (>.71e) ] _ 2 ---+ + + + 7 1 ‛ x y x y ex e y 2 7-- G w1 w7 7 7 7 7 7 7 7 7 7 x x y y x y x y 16 A 7? u1 u 7 u 7 v 1 u 1 v 7 v 1 v 7 1 ‛ u 1 v 1 u 7 v 7 Mc 71 ---------------] M 2 ---------------------+ 7 x y + x y + y y 7 y + x y + x ex e y _dahm strmss 7 1 ‛ A x x 77 (>.71m) mdastfkfty 7; 1 2 -- G u 1 v 1 u 7 v7 7 75 7= 70 >.5 PFWU]AD ^JWI 7> 73 Pfrtuad wjri bmtcjes arm appdfkagdm tj dfhmar ahe hjhdfhmar systmbs, tj kjhsmrvatfvm as wmdd as hjhkjhsmrvatfvm systmbs. Ucm 76 prfhkfpdm j` vfrtuad wjri fs emkmptfvmdy sfbpdm. Ft statms< ;? (>.77) Ucm tjtad vfrtuad wjri ejhm jh a gjey at mqufdfgrfub fs zmrj. ;1 ;7 Pfrtuad wjri fbpdfms ft fs hjt aktuad wjri gut wjri ejhm gy aktuad `jrkms fh bjvfhn pjfhts tcrjunc vfrtuad efspdakm;; bmhts, jr, vfrtuad `jrkms bjvfhn tcrjunc aktuad efspdakmbmht. ^m wfdd `jkus jhdy jh vfrtuad efspdakmbmhts ahe tcm wjri assjkf;5 atme wftc ft. [ybgjdfkaddy vfrtuad wjri kah gm wrfttmh as ;= (>.7;) ^ ? ;0 Ucm tjtad vfrtuad wjri kah gm efvfeme fhtj wjri ejhm gy mxtmrhad ahe fhtmrhad `jrkms. [fhkm tcm fhtmrhad `jrkms arm adways ;> ;3 jppjsme tj tcm mxtmrhad `jrkms, tcm fhtmrhad vfrtuad wjri wfdd adways gm jppjsftm fh sfnh tj tcm mxtmrhad vfrtuad wjri. ^m rmwrftm (>.7;) as ;6 Mquatfjh (>.7;) as 5? (>.75) ^ mxt ^ fht 51 57 >.= BFHFB]B _JUMHUFAD MHMWNV 5; 55 ^m em`fhm tcm pjtmhtfad mhmrny `uhktfjh as 5= (>.7=) ] ‛ ^ 50 fs tcm strafh mhmrny ahe ^ fs fs tcm wjri pjtmhtfad j` a `jrkm. ]hdfim wjri, wcfkc fs a kjhkmpt assjkfatme wftc ahy `jrkm 5> wcmrm ] fs 53 (kjhsmrvatfvm jr hjhkjhsmrvatfvm) tcat bjvms, tcm —wjri pjtmhtfad j` a `jrkm‒ fs assjkfatme wftc kjhsmrvatfvm `jrkms jhdy ahe 56 fbpdfms tcat tcmrm fs a pjtmhtfad `uhktfjh `rjb wcfkc sukc a `jrkm kah gm jgtafhme. Nravftatfjhad `jrkms, mdmktrjbanhmtfk `jrkms, =? sprfhn `jrkms arm sjbm mxabpdms j` kjhsmrvatfvm `jrkms tcat kah gm jgtafhme `rjb a pjtmhtfad `uhktfjh. F` wm assubm tcat tcm mdastfk (dfhmar jr hjhdfhmar) strukturm ahe arm kjhsmrvatfvm, tcmh tcm —wjri pjtmhtfad j` a `jrkm‒ fs =1 mxtmrhad `jrkms arm aktfhn jh ah mdastfk (dfhmar kadkudatme as tcm wjri tmrb fh Uagdm >.1. >.1. Ucus, `rjb jur pmrspmktfvm, tcm wjri pjtmhtfad ahe tcm wjri ejhm gy tcm `jrkm arm kad=7 =; kudatme sfbfdardy, prjvfeme tcm mxtmrhad `jrkms arm kjhsmrvatfvm ahe arm appdfme tj mdastfk systmbs. Uj jgtafh tcm statmbmht j` tcm tcmjrmb j` bfhfbub pjtmhtfad mhmrny, wm kjhsfemr tcm statmbmht j` vfrtuad wjri as nfvmh =5 (>.75). Wmstrfktfhn jursmdvms tj kjhsmrvatfvm mdastfk systmbs, wm kah statm tcat tcm fhtmrhad vfrtuad wjri fs tcm varfa== gy Mquatfjh (>.75). tfjh fh mda mdast stfk fk str straf afh h mhm mhmrn rny y eurf eurfhn hn em`j em`jrb rbat atfj fjh h ( ^ fht ] ) ahe ahe tcm tcm mxt mxtmr mrha hadd vfr vfrtu tuad ad wjri wjri fs fs tcm tcm var varfa fatf tfjh jh fh tcm tcm wjri wjri pjtm pjtmht htfa fadd =0 tfjh j ` t c m ` j r k m ( ^ ^ ) . U c u s , ` r j b Mquatfjh (>.75) wm jgtafh ^ ‛ ^ ] ‛ ^ jr mxt fh t mx t => =3 (>.70) 2 ? =6 Mquatfjh (>.70) fbpdfms (>.70) fbpdfms tcat at mqufdfgrfub, tcm vfrtuad varfatfjh fh tcm pjtmhtfad mhmrny `uhktfjh fs zmrj―wcfkc jkkurs 0? wcmrm tcm sdjpms j` tcm pjtmhtfad mhmrny `uhktfjh wftc rmspmkt tj tcm parabmtmrs em`fhfhn tcm pjtmhtfad `uhktfjh arm zmrj. 01
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Parfatfjhad ahe Mhmrny Bmtcjes Ucm tcmjrmb j` bfhfbub pjtmhtfad mhmrny kah gm statme as `jddjws.
1 7 ; J` add tcm ifhmbatfkaddy aebfssfgdm efspdakmbmht `uhktfjhs, tcm aktuad efspdakmbmht `uhktfjh fs tcm jhm tcat 5 bfhfbfzms tcm pjtmhtfad mhmrny `uhktfjh at stagdm mqufdfgrfub. = Ft hmmes tj gm mbpcasfzme tcat tcmrm arm bahy ifhmbatfkaddy aebfssfgdm efspdakmbmht `uhktfjhs, ahe tcmrm fs hj rmqufrm- 0 > bmht tcat tcmsm `uhktfjhs satfs`y tcm mqufdfgrfub mquatfjhs jr tcm gjuheary kjheftfjhs jh `jrkms ahe bjbmhts. Ucm aktuad efs pdakmbmht fs ifhmbatfkaddy aebfssfgdm ahe satfs`fms add tcm mqufdfgrfub kjheftfjhs ahe tcm statfk gjuheary kjheftfjhs. Ucus, f` wm 3 6 kcjjsm ah argftrary ifhmbatfkaddy aebfssfgdm `uhktfjh ahe kadkudatm tcm pjtmhtfad mhmrny `uhktfjh, tcm vadum sj jgtafhme wfdd 1 adways gm nrmatmr tcah tcm vadum j` tcm p jtmhtfad mhmrny `uhktfjh at mqufdfgrfub. Fh jtcmr w jres, wm apprjakc tcm pjtmhtfad mhmrny ? `uhktfjh vadum at mqufdfgrfub `rjb agjvm. Ucus, f` wm cavm twj apprjxfbatfjhs `jr tcm efspdakmbmht `uhktfjhs, tcm jhm tcat nfvms 11 17 tcm djwmr pjtmhtfad mhmrny fs tcm gmttmr apprjxfbatfjh. A kjrjddary tj tcm prmkmefhn statmbmht fs tcat f` wm aee ahjtcmr tmrb (fhkrmasm tcm emnrmms j` `rmmejb) fh ah apprjxfbatfjh, tcm pjtmhtfad mhmrny vadum kah jhdy em krmasm, tcat fs, fbprjvm tcm akkuraky 1; 15 j` tcm apprjxfbatfjh. ^m rmkjre tcm `jddjwfhn jgsmrvatfjhs. 1= • Ucm gmttmr apprjxfbatfjh j` efspdakmbmht `uhktfjh fs tcm jhm tcat yfmdes tcm djwmr pjtmhtfad mhmrny. mhmrny. 10 • Ucm nrmatmr tcm emnrmms j` `rmmejb, tcm djwmr wfdd gm tcm pjtmhtfad mhmrny `jr a nfvmh smt j` ifhmbatfkaddy aebfs1> sfgdm `uhktfjhs. 13 @rjb Uagdm >.1, >.1, wm saw tcat tcm wjri fs rmprmsmhtme gy a dfhmar `uhktfjhad< ^ 2 d u . @rjb Uagdm >.;, >.;, wm saw tcat tcm 16 strafh mhmrny fs rmprmsmhtme gy a sybbmtrfk gfdfhmar `uhktfjhad< ] 2 G u u 7 . [ugstftutfhn tcmsm mxprmssfjhs fhtjMquatfjh fhtj Mquatfjh 7? (>.7=) wm (>.7=) wm jgtafh tcm pjtmhtfad mhmrny mxprmssfjh as< 71 1 77 (>.7>) -- G u u ‛ d u 7 7; 75 ^m wfdd baim usm j` tcm agjvm `jrb fh Waydmfnc-Wftz ahe `fhftm mdmbmht bmtcje efskussme dattmr. 7= 70 >.0 BAUCMBAUFKAD _WMDFBFHAWFM[ 7> Ucm `frst varfatfjhs j` dfhm, arma, ahe vjdubm fhtmnrads wfdd nmhmratm tmrbs fhsfem tcm fhtmnrad j` `uhktfjhs ahe fts emrfvatfvms. 73 A`tmr tcm prjkmss j` varfatfjh tcm `uhktfjhs ahe tcmfr emrfvatfvms wfdd hj djhnmr gm fhempmhemht, ahe cmhkm wm wfdd hmme tj trahs- 76 `mr emrfvatfvms `rjb jhm tmrb jhtj ahjtcmr fhsfem tcm fhtmnrads. Ucm trahs`mr j` emrfvatfvms fs akkjbpdfscme< fh jhm-efbmhsfjh gy ;? fhtmnratfjh gy parts9 fh twj-efbmhsfjh gy Nrmmh‖s `jrbuda9 ahe fh tcrmm-efbmhsfjh gy Nauss efvmrnmhkm `jrbuda (Irmyszfn ;1 Y16>6R). ^m grfm`dy rmvfmw tcm batcmbatfks tcat wm wfdd hmme `jr `fhefhn tcm statfjhary vadums j` dfhm, arma, ahe vjdubm fhtmnrads. ;7 Fhtmnratfjh gy parts fs parts fs nfvmh gy tcm mquatfjh gmdjw. ;; ;5 g e n g e ` g (>.73) ;= ` e x 2 `n a ‛ n e x a e x a e x ;0 wcmrm ` x ahe n x arm twj kjhtfhujus `uhktfjhs wftc kjhtfhujus `frst emrfvatfvm jh tcm dfhm gjuheme gy x 2 x 2 a ahe x ahe x 2 2 g. ;> Nrmmh‖s `jrbudas arm `jrbudas arm nfvmh gy tcm mquatfjhs gmdjw. ;3 n ` ` e x e y 2 `ney ‛ n e x e y (>.76a) ;6 x x 5? A 51 n ` (>.76g) 57 ` e x e y 2 ‛ `nex ‛ n e x e y y y 5; A A 55 wcmrm ` x y ahe n x y arm twj kjhtfhujus `uhktfjhs wftc kjhtfhujus `frst emrfvatfvm fh tcm arma A tcat A tcat fs gjuheme gy tcm 5= kurvm as scjwh fh @fnurm >.=a >.=a. 50 5> h 53 s y 56 x =? =1 A =7 y y h s =; =5 == x x =0 (g) => =3 @fnurm >.= (a) Nmhmrfk nmjbmtry (g) Kjjrefhatm trahs`jrbatfjh. =6 Mquatfjhs (>.76a) ahe (>.76a) ahe (>.76g) arm (>.76g) arm usm`ud `jr rmktahnudar nmjbmtrfms. @jr sbjjtc argftrary nmjbmtrfms tcmrm fs ah adtmrha- 0? 01 tfvm `jrb tcat wm wfdd usm. Ucfs fs emskrfgme hmxt.
[tatfjhary vadum j` a em`fhftm dfhm fhtmnrad
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1 @fnurm >.=g >.=g scjws tcm nmjbmtry gy wcfkc wm kah rmdatm tcm Kartmsfah kjjrefhatms ( x, x, y) y) ahe tj hjrbad ahe tahnmhtfad 7 kjjrefhatms (h (h, s) s) as scjwh fh tcm mquatfjhs tcat `jddjw. ; (>.76k) x 2 h kjs ‛ s sfh y 2 h sfh + s kjs 5 (>.76k) kah gm sjdvme `jr h ahe s ahe s tj tj jgtafh = Mquatfjh (>.76k) kah 0 (>.76e) h 2 x kjs + y sfh s 2 ‛ x sfh + y kjs > ^m em`fhm tcm efrmktfjh kjsfhms j` tcm uhft hjrbad as 3 6 h 2 kjs 2 s h 2 sfh 2 ‛ s (>.76m) h x 2 h y 2 1? x y y x 11 F` a pjfht fs rmstrfktme tj tcm gjuheary, tcmh eh 2 ? 9 `rjb Mquatfjh (>.76k) wm (>.76k) wm jgtafh 17 ex 2 ‛ sfh es 2 ‛ h y es ey 2 kjs e s 2 h x es (>.76`) 1; 15 [ugstftutfhn Mquatfjh (>.76`) fhtj (>.76`) fhtj Mquatfjhs (>.76a) ahe (>.76a) ahe (>.76g) wm (>.76g) wm jgtafh ahjtcmr `jrb j` Nrmmh‖s `jrbuda as 1= n ` 10 (>.;?a) ` e x e y 2 `n h x es ‛ n e x e y x x 1> A A 13 16 n ` (>.;?g) ` e x e y 2 `n h y ey ‛ n e x e y 7? y y A A 71 ]sfhn kcafh rudm, wm kah wrftm tcm `jddjwfhn femhtftfms `jr datmr usm. 77 7; ` 2 ` h + ` s 2 ` h ‛ ` h ` 2 ` h + ` s 2 ` h + ` h (>.;1a) ahe 75 x y s x h x s x h y h y s y h y s x 7= Fh tcm kjursm j` emvmdjpbmht, wm wfdd gm taifhn emrfvatfvm j` tcm efrmktfjh kjsfhms wftc rmspmkt tj s tj s.. ^m kjhsfemr tcmsm 70 emrfvatfvms hmxt. 7> 73 h y h x 76 (>.;1g) 2 ‛ sfh 2 ‛ h y 2 ‛ --------- s s W ku r s ;? ;1 h x h y 2 kjs 2 h x 2 --------(>.;1k) ;7 s s W ku r s ;; ;5 wcmrm, W ku r fs tcm raefus j` kurvaturm j` tcm gjuheary at tcm pjfht uhemr kjhsfemratfjh ahe kah gm `uhktfjh j` s. s. ;= Nauss efvmrnmhkm `jrbuda fs `jrbuda fs nfvmh gy tcm mquatfjh gmdjw. ;0 u x u y u z (>.;7) + + e P 2 u x h x + u y h y + u z h z e A ;> x y z ;3 U [ ;6 wcmrm, u x, u y, ahe u z arm tcm kjbpjhmhts j` a vmktjr `uhktfjh wcfkc fs kjhtfhujus wftc kjhtfhujus `frst emrfvatfvms fh a rmnfjh U 5? gjuheme gy a sbjjtc sur`akm [ 9 ahe h x, h y, ahe h z arm tcm efrmktfjh kjsfhms j` a uhft hjrbad jh sur`akm [ . Uj emvmdjp `jrbudas fh 51 wcfkc wm trahs`mr emrfvatfvms x emrfvatfvms x,, y, y, jr z jr z `rjb `rjb jhm `uhktfjh tj ahjtcmr wm dmt makc j` tcm kjbpjhmhts j` u gm mquad tj prjeukt j` twj 57 kjhtfhujus `uhktfjhs `n `uhktfjhs `n wcfdm wcfdm tcm jtcmr twj kjbpjhmhts arm zmrj. Ucfs prjeukms tcm `jrbudas gmdjw. 5; ` n e P 2 `n h e A ‛ ` n e P ` n e P 2 `n h e A ‛ ` n e P 55 x y x x y y 5= U [ U U [ U (>.;;) 50 ` n n e P 2 `n h z e A ‛ ` e P 5> z z 53 U [ U 56 [UAUFJHAWV PAD]M J@ A EM@FHFUM DFHM FHUMNWAD =? >.> [UAUFJHAWV =1 =7 Em`fhftm dfhm fhtmnrads arfsm fh pjtmhtfad mhmrny mquatfjhs `jr jhm efbmhsfjhad strukturad bmbgmrs. Gy tcmjrmb j` bfhfbub =; pjtmhtfad mhmrny, tcat fs, `frst varfatfjh j` pjtmhtfad mhmrny, wm kah jgtafh tcm mqufdfgrfub mquatfjhs. As wm scadd smm, wm adsj =5 jgtafh add pjssfgdm gjuheary kjheftfjhs. Fh jtcmr wjres, wm kah jgtafh tcm kjbpdmtm gjuheary vadum prjgdmb `rjb tcm statfjhary == vadum j` em`fhftm dfhm fhtmnrad s rmprmsmhtfhn pjtmhtfad mhmrny. Fh [mktfjh >.>.1 wm >.>.1 wm kjhsfemr a `uhktfjhad wftc u ahe fts `frst emrfvatfvm. Hmxt wm kjhsfemr `frst ahe smkjhe emrfvatfvm fh =0 tcm `uhktfjhad fh [mktfjh >.>.;. >.>.;. @fhaddy, wm nmhmradfzm tj tcm r tc jremr emrfvatfvm fh tcm `uhktfjhad fh [mktfjh >.>.5. >.>.5. => ^m wfdd gm taifhn emrfvatfvms j` tcm `uhktfjhads wftc rmspmkt tj tcm emrfvatfvms j` u. Uj sfbpdf`y adnmgrafk bahfpudatfjh =3 =6 wm fhtrjeukm tcm `jddjwfhn hjtatfjh ? f ff r 7 7 r r 0? (>.;5) u 2 u u 2 e u e x u 2 e u ex u 2 e u ex 01
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Parfatfjhad ahe Mhmrny Bmtcjes
1 7 ^m wfdd emvmdjp tcm femas gy `frst kjhsfemrfhn a `uhktfjhad tcat empmhes jh tcm `uhktfjh u ahe fts `frst emrfvatfvm. ; g f (>.;=) 5 F u 2 C u u x e x a = ^m kjhsfemr tcm `frst varfatfjh j` F u ahe jgsmrvm tcat tcm dfbfts arm `fxme9 cmhkm tcm varfatfjh kah gm taimh fhsfem tcm 0 fhtmnrad tj jgtafh tcm `jddjwfhn. > 3 g g (>.;0a) 6 F u 2 C u f u x e x 2 C u f u x e x a a 1? Hjtm tcm prjkmss j` varfatfjh fs a prjkmss j` kjhsfemrfhn vfrtuad ef spdakmbmht.@fnurm spdakmbmht. @fnurm >.1 mbpcasfzme >.1 mbpcasfzme tcat `uhktfjh ahe 11 emrfvatfvm arm fhempmhemht fh vfrtuad efspdakmbmht. @urtcmrbjrm, wm kah kjhsfemr vfrtuad efspdakmbmht wftcjut kcahnfhn x kcahnfhn x fh fh @fn17 urm >.1, >.1, wcfkc fbpdfms tcat varfatfjh fh x fh x fs fs hjt kjhsfemrme eurfhn vfrtuad efspdakmbmht. Cmhkm wm jgtafh 1; g C 15 (>.;0g) F u 2 u f + C u e x f 1= a u u 10 Jhkm wm cavm kjhsfemrme vfrtuad efspdakmbmht, wm arm hjw jh a spmkf`fk kurvm ahe `uhktfjh ahe fts emrfvatfvm arm 1> rmdatme, tcat fs, hj djhnmr fhempmhemht. F` wm arm tj eraw ahy kjhkdusfjh gy smttfhn F u 2 ? , tcmh wm hmme tj jgtafh ah mxprms13 sfjh fh jhdy u ―wm pmr`jrb fhtmnratfjh gy parts. 16 ^m `urtcmr hjtm tcm `jddjwfhn 7? 71 eu e u (>.;0k) u f 2 2 -------------77 e x ex 7; [ugstftutfhn Mquatfjh (>.;0k) fhtj (>.;0k) fhtj Mquatfjh (>.;0g), (>.;0g), pmr`jrbfhn fhtmnratfjh gy parts, ahe smttfhn F u 2 ? , wm jgtafh 75 g C e C 7= F u 2 u + u e x 2 ? jr a u 70 u f e x 7> g g C C e C (>.;0e) 73 ‛ F u 2 u e x + f u 2 ? f a u e x u 76 u a ;? Ucmrm arm twj pjssfgdm ways gy wcfkc wm kah bmmt tcm kjheftfjh fh Mquatfjh (>.;0e). (>.;0e). ;1 ;7 _jssfgfdfty 1< 1< ^m bmmt tcm kjheftfjh F u 2 ? fh tcm avmranm jr jvmradd smhsm. Ucfs dmae tj apprjx;; fbatm bmtcjes as tcm kjheftfjh fs hjt satfs`fme at makc ahe mvmry pjfht gmtwmmh a ahe g. ^m wfdd kjh;5 sfemr tcfs apprjakc fh [mktfjhs >.17 ahe >.17 ahe >.1;. >.1;. ;= _jssfgfdfty 7< ^m rmqufrm tcat F u 2 ? at makc ahe mvmry pjfht gmtwmmh a ahe g. Ucfs rmsudts fh ;0 gjuheary vadum prjgdmb efskussme fh [mktfjh >.>.7. >.>.7. ;> Gm`jrm bjvfhn tj tcm hmxt smktfjh wm rmkjre tcm `jddjwfhn jgsmrvatfjhs `rjb tcm agjvm emrfvatfjh. ;3 ;6 • Eurfhn tcm prjkmss prjkmss j` varfatfjh, tcm `uhktfjh ahe fts emrfvatfvm emrfvatfvm arm fhempmhemht. fhempmhemht. 5? • A`tmr tcm prjkmss j` varfatfjh, tcm `uhktfjh ahe fts emrfvatfvm arm hj djhnmr fhempmhemht. (>.;>) 51 • Fhtmnratfjh gy parts wfdd nmhmratm ah mxprmssfjh jhdy fh tmrbs j` varfatfjh j` tcm `uhktfjh. 57 5; 55 >.>.7 Gjuheary vadum prjgdmb 5= ^m `frst kjhsfemr tcm fhtmnrad fh Mquatfjh (>.;0e) tj (>.;0e) tj gm zmrj. @jr ft tj gm zmrj, tcm fhtmnrahe bust gm zmrj at add x add x.. u x kahhjt gm 50 zmrj as ft fs tcm vfrtuad efspdakmbmht. Ucus tcm tmrb fh grakimt fs zmrj at makc ahe mvmry x mvmry x as scjwh gmdjw. 5> 53 C ‛ e C 2 ? (>.;3a) 56 Ef``mrmhtfad Mquatfjh< axg f u e x u =? Ucm agjvm mquatfjh fs kaddme tcm Mudmr-Danrahnm tcm Mudmr-Danrahnm mquatfjh. mquatfjh. ^m hjw kjhsfemr tcm gjuheary tmrb fh Mquatfjh (>.;0e). (>.;0e). At =1 f makc gjuheary mhe, tcm tmrb bust gm zmrj. [j mftcmr C u 2 ? jr u bust gm zmrj at makc mhe as scjwh gmdjw. =7 f (>.;3g) =; Gjuheary Kjheftfjhs< C u 2 ? jr at x at x 2 a ahe at x at x 2 g u 2 ? =5 Mquatfjhs (>.;3a) ahe (>.;3a) ahe (>.;3g) rmprmsmht (>.;3g) rmprmsmht tcm kjbpdmtm gjuheary vadum prjgdmb assjkfatme wftc tcm `uhktfjhad fh Mqua== tfjh (>.;=). (>.;=). ^m emrfvme tcm gjuheary vadum prjgdmb hjt `rjb ef``mrmhtfad kadkudus gut `rjb varfatfjhad kadkudus. Ucm mdmnahkm j` =0 jgtafhfhn gjuheary vadum prjgdmb statmbmht `rjb varfatfjhad prfhkfpdm wfdd gm embjhstratme fh smvmrad mxabpdms. => ^m rmkjre tcm `jddjwfhn jgsmrvatfjhs tcat wm wfdd nmhmradfzm fh [mktfjh >.>.5. >.>.5. =3 f • Mquatfjh (>.;3a) wfdd rmsudt fh a smkjhe jremr ef``mrmhtfad mquatfjh gmkausm tcm cfncmst emrfvatfvm u wfdd gm =6 ef``mrmhtfatme fh tcm smkjhe tmrb. ^cmh wm usm tcm pjtmhtfad mhmrny `jr axfad jr tjrsfjh as tcm `uhktfjhad, wm 0? 01 wfdd jgtafh tcm smkjhe jremr ef``mrmhtfad mquatfjhs (7.10-A) ahe (7.10-A) ahe (7.10-U). (7.10-U).
>.>.1
[tatfjhary vadum j` a `uhktfjhad wftc `frst jremr emrfvatfvms
[tatfjhary vadum j` a em`fhftm dfhm fhtmnrad 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
1>>
• F` tcm tcm `uhktf `uhktfjhad jhad C C fs fs a quaeratfk fh u ahe fts emrfvatfvms, tcmh Mquatfjh (>.;3a) wfdd (>.;3a) wfdd rmsudt fh a dfhmar ef``mrmhtfad mquatfjh. Cjwmvmr, f` tcm `uhktfjhad C `uhktfjhad C fs fs kugfk jr cfncmr fh u ahe fts emrfvatfvms, tcmh wm wfdd jgtafh a hjhdfhmar ef``mrmhtfad mquatfjh. f
• Ucm gjuheary kjheftfjh C u 2 ? wfdd rmsudt fh a kjheftfjh jh `frst jremr emrfvatfvm. ^m wfdd jgtafh kjheftfjh jh tcm fhtmrhad axfad `jrkm jr tjrsfjh wcmh wm usm tcm pjtmhtfad mhmrny `jr axfad jr tjrsfjh as tcm `uhktfjhad. • Ucm gjuheary gjuheary kjheftfjh kjheftfjh u 2 ? fbpdfms tcat u fs spmkf`fme, gut ft kjude gm a hjh-zmrj vadum. • F` tcm `uhktfjhad `uhktfjhad kjhtafhs kjhtafhs bjrm tcah jhm varfagdm varfagdm (u (u), say uf, tcmh wm kjude rmpdakm u wftc uf fh tcm agjvm mquatfjhs. Fh jtcmr wjres, tcm agjvm mquatfjhs arm appdfkagdm `jr makc uf
>.>.;
[tatfjhary vadum j` a `uhktfjhad wftc wftc smkjhe jremr emrfvatfvms
Fh tcfs smktfjh wm jgtafh tcm gjuheary vadum prjgdmb `rjb a `uhktfjhad wftc `frst ahe smkjhe jremr emrfvatfvms scjwh gmdjw. g
F u 2
a
Cu
f f
u f u x
e x
(>.;6a)
^m taim tcm `frst varfatfjh F u ahe tcmh gy sukkmssfvm fhtmnratfjh gy parts wm trahs`mr emrfvatfvms as scjwh gmdjw.
F u
2
g a
C u f f + C u f + C u u u f f u f F u F u
F u
F u
2
2
2
2
C u f u f f
C u f u f f
C u f u f f
C u f u f f
g
+
g a
a
g
+
g a
a
e x 2
C e u f + C u f + C u u u f f e x u f
g
a
C e C f C f ‛ f f u + u u e x u u
C e C e C f ‛ e x f f e x u + u u u u g
g
C e C + ‛ u f f f e x u u a
g
C e C + ‛ u f f f e x u u a
+
g a
a
g
+
a
e x jr
e x jr
C ‛ e C ‛ e C u e x jr u e x u f e x u f f
C ‛ e C u e x u f
g
e x jr
a
+
e e x
7 7
C u e x u f f
2 ?
(>.;6g)
Makc tmrb fh tcm agjvm mquatfjh bust mquad zmrj ahe wm jgtafh tcm gjuheary vadum prjgdmb scjwh gmdjw. Ef``mrmhtfad Mquatfjh<
C u f f
2 ?
C ‛ e C u e x u f jr
+
7
C f f 7 e x u
u f
e
(>.5?a)
2 ?
Gjuheary Kjheftfjhs< ahe
C ‛ e C f f u f e x u
axg
2 ?
at x at x 2 a ahe at x at x 2 g 2 ?
jr
u
(>.5?g)
2 ?
^m rmkjre tcm `jddjwfhn jgsmrvatfjhs tcat wm wfdd nmhmradfzm fh [mktfjh >.>.5. >.>.5. f f
• Mquatfjh (>.5?a) wfdd rmsudt fh a `jurtc jremr ef``mrmhtfad mquatfjh gmkausm tcm cfncmst emrfvatfvm u wfdd gm ef``mrmhtfatme twfkm fh tcm tcfre tmrb. ^cmh wm usm tcm pjtmhtfad mhmrny `jr sybbmtrfk gmhefhn j` gmabs as tcm `uhktfjhad tcmh wm wfdd jgtafh tcm `jurtc jremr ef``mrmhtfad mquatfjh (7.10-G). (7.10-G). • F` tcm tcm `uhktf `uhktfjhad jhad C C fs fs a quaeratfk fh u ahe fts emrfvatfvms, tcmh Mquatfjh (>.;3a) wfdd (>.;3a) wfdd rmsudt fh a dfhmar ef``mrmhtfad mquatfjh. Cjwmvmr, f` tcm `uhktfjhad C `uhktfjhad C fs fs kugfk jr cfncmr fh u ahe fts emrfvatfvms, tcmh wm wfdd jgtafh a hjh-dfhmar ef``mrmhtfad mquatfjh. f f
• Ucm gjuheary kjheftfjh C u 2 ? wfdd rmsudt fh a kjheftfjh jh smkjhe jremr emrfvatfvm. ^cmh wm usm tcm pjtmhtfad mhmrny `jr sybbmtrfk gmhefhn j` gmabs as tcm `uhktfjhad wm wfdd jgtafh a kjheftfjh jh tcm fhtmrhad gmhefhn bjbmht.
1>3
>
Parfatfjhad ahe Mhmrny Bmtcjes f
f f
• Ucm gjuheary gjuheary kjheftfjh kjheftfjh jh C u ‛ e C u ex wfdd rmsudt fh a kjheftfjh kjheftfjh jh tcfre jremr emrfvatfvm.^cmh wm usm tcm pjtmhtfad mhmrny `jr sybbmtrfk gmhefhn j` gmabs as tcm `uhktfjhad wm wfdd jgtafh a kjheftfjh jh tcm fhtmrhad scmar `jrkm. • Ucm gjuheary gjuheary kjheftfjhs kjheftfjhs u hjh-zmrj vadums.
f
2 ? ahe u 2 ? fbpdfms tcat u
f
ahe u arm spmkf`fme, gut tcmy kjude gm a
• F` tcm `uhktfjhad `uhktfjhad kjhtafhs kjhtafhs bjrm tcah jhm varfagdm varfagdm (u (u), say uf, tcmh wm kjude rmpdakm u wftc uf fh tcm agjvm mquatfjhs. Fh jtcmr wjres, tcm agjvm mquatfjhs arm appdfkagdm `jr makc uf.
>.>.5
Nmhmradfzatfjh
Fh tcfs smktfjh wm mxtmhe tcm rmsudts tj a `uhktfjhad kjhtafhfhn r tc jremr emrfvatfvm ahe em`fhm tcm `uhktfjhad as F u 2
g
a
Cu
r
u r ‛ 1 u r ‛ 1 u f u ? x
e x
(>.51a)
]sfhn tcm tcrmm jgsmrvatfjhs baem fh Mquatfjh (>.;>) wm taim tcm `frst varfatfjh F u ahe tcmh gy fhtmnratfjh gy parts trahs`mr jhm j` tcm emrfvatfvms as scjwh gmdjw.
F u
2
r + 1
q
g
? a
C u q e x u q
2
C u q ‛ 1 q 1 u q
r + 1
g r + 1
‛
q
g
1 a
a
C u q ‛ 1 e x + x u q
g
a
C u
?
e x
(>.51g)
^m hjtm tcat tcm fhtmnratfjh gy parts kcahnms tcm sfnh as wm trahs`mr emrfvatfvm `rjb jhm tmrb tj tcm jtcmr. ^m trahs`mr r ? add tcm emrfvatfvms `rjb u tfdd wm jgtafh lust u 2 u fh tcm fhtmnrad. [mttfhn F u 2 ? , wm jgtafh tcm ef``mrmhtfad mquatfjh ahe add pjssfgdm gjuheary kjheftfjhs as scjwh fh @fnurm >.0. >.0. Ucm jgsmrvatfjhs wm baem fh mardfmr smktfjhs kah hjw gm nmhmradfzme as nfvmh gmdjw.
C ‛ e C Ef``mrmhtfad mquatfjh< f u ? e x u
; C e C + ‛ f f f ff 7 ; e x u e x u
e
7
+
+ ‛ 1
r
e
e x
r
C u r
r
2 ?
(>.57a)
Gjuheary kjheftfjhs at x at x 2 a ahe at x at x 2 g
C u f
‛
7
e C e C + ------ f f ‛ --------7 f ff + ex u ex u
r ‛ 1
e r e C 2 ? + ‛ 1 ------------r ‛ 1 ex u r
C ‛ e C + u f f e x u f ff
e r e C 2 ? + ‛ 1 ------------r ‛ 7 ex u r
C u f ff
‛
u
jr
u f
2
jr
u ff
2
r ‛ 7
r ‛ ;
+
e r e C 2 ? + ‛ 1 ------------r ‛ ; ex u r
2 ?
jr
2 ?
jr
2
2 ?
jr
2
2 ?
jr
2
C 2 ? u r
jr
u r ‛ 1
2
? ? ? ? ? ?
(>.57g)
@fnurm >.0 Gjuheary Padum _rjgdmb
1. F` tcm cfncmst emrfvatfvm fh tcm `uhktfjhad fs r , tcmh tcm ef``mrmhtfad mquatfjh wfdd gm j` jremr 7r 7 r. Ucus, `rjb tcm strafh mhmrny mxprmssfjhs `jr axfad ahe tjrsfjh fh Uagdm >.7 wm >.7 wm smm tcat r 2 2 1, cmhkm tcm ef``mrmhtfad mquatfjh wjude gm smkjhe jremr as wm saw fh Kcaptmr 7. Ucm strafh mhmrny `jr gmab gmhefhn scjws r 2 2 7 ahe wm wfdd nmt a `jurtc jremr ef``mrmhtfad mquatfjh. 7. Ucm gjuheary kjheftfjhs wftc tcm varfatfjh sybgjd j` cavm emrfvatfvms `rjb ? tj r -1. -1. Ucmsm quahtftfms bust gm kjhtfhujus as wm arm taifhn tcmfr varfatfjh. Ucm emrfvatfvms arm kaddme tcm prfhkfpad emrfvatfvms. Ucm gjuheary kjheftfjhs arm kaddme ifhmbatfk gjuheary kjheftfjhs fh bmkcahfks j` batmrfads. Mssmhtfad gjuheary kjheftfjhs ahe prfbary varfagdms (fh pdakm j` prfhkfpad emrfvatfvms) arm bjrm nmhmrad habms tcat arm akqufrfhn fhkrmasfhn pjpudarfty. Fh gmab gmhefhn, em`dmktfjh ahe sdjpm arm tcm prfbary varfagdms, wcfdm em`dmktfjh fh axfad ahe rjtatfjh fh tjrsfjh arm tcm prfbary varfagdms. Ucm prfbary varfagdms arm jur nmhmradfzme efspdakmbmhts. Ucus, prfhkfpad emrfvatfvms, prfbary varfagdms, ahe nmhmradfzme efspdakmbmht rm`mr tj tcm sabm varfagdms.
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
[tatfjhary vadum j` a em`fhftm arma fhtmnrad
1>6
1 ;. Ucm gjuheary kjheftfjhs tcat cavm emrfvatfvms j` tcm `uhktfjhads tcat vary `rjb r tj tj 7r -1 -1 arm jur fhtmrhad `jrkms ahe bjbmhts ahe arm kaddme statfkad varfagdms fh bmkcahfks j` batmrfad. A bjrm nmhmrad habm j` 7 tcmsm varfagdms fs smkjheary varfagdms. Ucm gjuheary kjheftfjhs jh tcmsm varfagdms arm kaddme statfkad ; gjuheary kjheftfjhs fh bmkcahfks j` batmrfad. A bjrm nmhmrad habm fs haturad gjuheary kjheftfjhs. 5 Fh gmab gmhefhn tcmsm wfdd rmsudt fh gjuheary kjheftfjhs jh fhtmrhad gmhefhn bjbmht (emrfvatfvm jremr 7) = ahe fhtmrhad scmar `jrkm (emrfvatfvm jremr ;). 0 > 5. F` tcm `uhktfjhad kjhtafhs bjrm tcah jhm varfagdm (u (u), say uf, tcmh wm kjude rmpdakm u wftc uf fh tcm agjvm 3 mquatfjhs. Fh jtcmr wjres, tcm agjvm mquatfjhs arm appdfkagdm `jr makc uf. 6 =. F` tcm `uhktfjhad fs quaeratfk fh u ahe fts emrfvatfvm, tcmh tcm gjuheary vadum prjgdmb wfdd gm dfhmar. Cjw1? mvmr, tcm agjvm mquatfjhs arm appdfkagdm tj ahy `uhktfjhad. 11 17 MSAB_DM >.1 1; 15 Jgtafh tcm gjuheary vadum prjgdmb `jr sybbmtrfk gmhefhn j` gmabs usfhn varfatfjhad prfhkfpdms. Assubm tcmrm arm jhdy efs1= trfgutme `jrkm p `jrkm p y ahe hj kjhkmhtratme `jrkm jr bjbmht aktfhn jh tcm gmab. 10 1> >.3 [UAUFJHAWV [UAUFJHAWV PAD]M J@ A EM@FHFUM AWMA FHUMNWAD 13 16 Fh `fhefhn tcm statfjhary vadum j` dfhm fhtmnrad wm `jddjwme tcm tcrmm stmps scjwh fh Mquatfjh (>.;>), (>.;>), tcat fhkdueme tcm trahs`mr j` 7? emrfvatfvms gy fhtmnratfjh gy parts. Fh statfjhary vadum j` ah arma fhtmnrad tcm trahs`mr j` emrfvatfvm fs akkjbpdfscme gy usfhn 71 Nrmmh‖s tcmjrmb as prmsmhtme fh [mktfjh >.0. >.0. 77 7; >.3.1 [tatfjhary vadum j` a `uhktfjhad wftc `frst jremr emrfvatfvms 75 Fh tcfs smktfjh wm j gtafh tcm gjuheary vadum prj gdmb `rjb a `uhktfjhad wftc `frst emrfvatfvms scjwh gmdjw. ^m fhtrjeukm tcm 7= hjtatfjh tcat a kjbba fbpdfms partfad emrfvatfvm wftc rmnare tj tcm kjjrefhatm tcat `jddjws. Ucus, u , x x 2 u x ahe 70 u , y y 2 u y . A sugskrfpt wftcjut a kjbba fbpdfms a kjbpjhmht ahe wftc a kjbba fbpd fms emrfvatfvm. ]sfhn tcfs hjtatfjh, wm 7> em`fhm tcm `uhktfjhad ahe kjhsfemr fts `frst varfatfjh gmdjw. 73 76 F u 2 C u , x (>.5;a) x u , y y u x y ex e y ;? A ;1 ;7 C u + C u + C u ex e y 2 C u + C u + C u ex e y (>.5;g) F u 2 , x x , y y ;; u , x x u , y y u u , x x x u , y y y u A A ;5 (>.;?a), wm trahs`mr tcm emrfvatfvms fh tcm `frst twj tmrbs ahe smt tcm `frst varfatfjh tj zmrj as scjwh gmdjw. ;= ]sfhn Mquatfjh (>.;?a), ;0 C ‛ C ‛ C u ex e y 2 ? ;> F u 2 C h x + C h y ue s + (>.5;k) u , x x u , y y u x u , x x y u, y y ;3 A ;6 Makc tmrb fh tcm `frst varfatfjh bust gm zmrj at add pjfhts.Ucus, wm kah jgtafh tcm gjuheary vadum prjgdmb wrfttmh gmdjw. 5? C ‛ C ‛ C 2 ? 51 (>.55a) Ef``mrmhtfad Mquatfjh< x y fh A u x u , x x y u , y y 57 5; C h + C h 2 ? 55 Gjuheary Kjheftfjhs< jr x y jh (>.55g) u 2 ? u , x x x u , y y y 5= 50 F` tcm `uhktfjhad fh Mquatfjh (>.5;a) kjhtafhs (>.5;a) kjhtafhs bjrm tcah jhm varfagdm tcmh Mquatfjhs (>.55a) ahe (>.55a) ahe (>.55g) bust (>.55g) bust gm wrfttmh 5> `jr makc j` tcm varfagdms as embjhstratme fh Mxabpdm >.7. >.7. 53 Fh Mquatfjh (>.55g) ft (>.55g) ft fs pjssfgdm tcat fh part j` tcm gjuheary tcm prfbary varfagdm fs spmkf`fme ahe fh ahjtcmr part j` tcm 56 gjuheary tcm smkjheary varfagdm fs spmkf`fme. F` tcm `uhktfjhad F `uhktfjhad F fs assjkfatme wftc tcm Dapdakm tcm Dapdakm jpmratjr jpmratjr sukc as fh cmat kjheuktfjh, =? tcmh spmkf`yfhn tcm prfbary varfagdm wfdd gm spmkf`yfhn tmbpmraturm ahe spmkf`yfhn tcm smkjheary varfagdm wfdd gm spmkf`yfhn cmat. =1 =7 MSAB_DM >.7 =; =5 Jgtafh tcm gjuheary vadum prjgdmb `jr ah mdastfk pdahm fh pdahm strmss wftc jhdy gjey `jrkms. == [UAUFJHAWV PAD]M J@ A @]HKUFJHAD ^FUC [MKJHE [MKJHE JWEMW EMWFPAUFPM[ =0 >.6 [UAUFJHAWV => =3 Fh tcfs smktfjh wm jgtafh tcm gjuheary vadum prjgdmb `rjb a `uhktfjhad wftc `frst ahe smkjhe jremr emrfvatfvms. Fh [mktfjh >.6.1 =6 wm `frst kjhsfemr tcm emrfvatfjh fh Kartmsfah kjjrefhatms tcat fs appdfkagdm tj armas j` rmktahnudar nmjbmtrfms ahe tcmh fh [mktfjh >.6.7 wm kjhsfemr tcm bjrm nmhmrad kasm j` nmjbmtrfms j` argftrary scapms fh wcfkc wm usm hjrbad ahe tahnmhtfad kjjrefhatm sys0? >.6.7 wm 01 tmbs `jr emskrfgfhn tcm gjuheary kjheftfjhs.
13?
>
>.6.1
Parfatfjhad ahe Mhmrny Bmtcjes
Gjuheary vadum prjgdmb `jr rmktahnudar nmjbmtrfms
^m wfdd usm tcm rmktahnudar nmjbmtry scjwh fh @fnurm >.> tj >.> tj jgtafh tcm gjuheary vadum prjgdmb `rjb tcm `uhktf jhad nfvmh gmdjw.
Cu
F u 2
, xx xx
u , xy xy u , yy yy u , x x u , y y u x y ex e y
(>.5=)
A
wcmrm, u , xx xx 2
7
u x
7
7
u y
, u, yy yy 2
7
7 u x y .
, ahe u , xy xy 2
y 5
;
g
1
x
7 a
@fnurm >.> Wmktahnudar nmjbmtry.
^m taim tcm `frst varfatfjh j` tcm `uhktfjhad scjwh fh Mquatfjh (>.5=) tj (>.5=) tj jgtafh
F u
C
C u + C u + C u + C u + C u xx yy xy , x x u , y y , y y u, xx , xx u , yy , yy u , xy , xy , x x xx yy xy
u u + u
2
A
ex e y
(>.50)
Ucm fhtmnrads wftc `frst emrfvatfvms arm strafnct `jrware, gut tcjsm wftc smkjhe jremr emrfvatfvms, partfkudardy tcm krjss emrfvatfvms wftc x wftc x ahe ahe y y,, cavm smvmrad stmps tcat hmme tj gm mdagjratme. Cmhkm, wm mvaduatm tcm fhtmnrad fh Mquatfjh (>.50) as (>.50) as sub j` sfx fhtmnrads. Ucm `frst fhtmnrad tcat cas tcm tmrb C u u fs dagmdme F 1 u ahe ft hmmes hj aeeftfjhad stmps. Fh tcm hmxt twj fhtmnrads wm trahs`mr emrfvatfvms usfhn Mquatfjhs (>.76a) ahe (>.76a) ahe (>.76g) tj (>.76g) tj jgtafh tcm `jddjwfhn.
F 7 u
2
C u , x x , x x
u A
F ; u
2
ex e y 2
A
C u ex e y , y y , y y
u A
C u , x x x
u
2
ex e y 2
C u ex e y , y y y
u A
C ue y ‛ , x x
u
A
C ue x ‛ , y y
u
2 ‛
A
C u x u , x x
C u y u , y y
ex e y
ex e y
(>.50a)
(>.50g)
Fh tcm hmxt fhtmnrad, usfhn Mquatfjh (>.76a) wm (>.76a) wm twfkm trahs`mr emrfvatfvms wftc rmspmkt tj x tj x as as scjwh gmdjw.
F 5 u
2
C u xx , xx , xx xx
u A
F 5 u
2
ex e y 2
C u , x x , xx xx x
u A
C u ey ‛ , x x , xx xx
A
C C 7 C u u u , x x ‛ x u u ey + xx , xx xx xx x7 u , xx , xx A
C u , xx xx x
x u
u
ex e y 2
ex e y
ex e y
(>.50k)
Fh a sfbfdar bahhmr wm kah trahs`mr emrfvatfvms wftc rmspmkt tj y tj y usfhn usfhn Mquatfjh (>.76g) tj (>.76g) tj jgtafh
F = u
2
A
C u , yy u , yy yy yy
7 C C u ex + C u u ‛ u , y y y u yy , yy yy yy y 7 u , yy , yy A
ex e y 2 ‛
ex e y
(>.50e)
Fh tcm dast fhtmnrad wm `frst trahs`mr tcm emrfvatfvm wftc rmspmkt tj x tj x usfhn usfhn Mquatfjh (>.76a) tcmh (>.76a) tcmh wm trahs`mr emrfvatfvm wftc rmspmkt tj y tj y usfhn usfhn Mquatfjh (>.76g) as (>.76g) as scjwh gmdjw.
F 0 u
2
C u , y y , xy xy x
u A
F 0 u F 0 u
2
2
ex e y 2
C u ey ‛ , y y , xy xy
C u ey ‛ , xy xy y
A
C u ey + C ue x + u , xy x u , xy xy y xy
A
C u , xy xy y
x u
u
C u , y y , xy xy
x u
u
A
ex e y
ex e y
7 C u x y u , xy xy
ex e y
(>.50m)
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
[tatfjhary vadum j` a `uhktfjhad wftc smkjhe jremr 1 Ucm `frst fhtmnrad fh mquatfjh agjvm kah gm `urtcmr mvaduatme gy fhtmnratfjh gy parts. ^m hjtm tcat tcm gjuheary fhtmnrad 7 fh tcm agjvm mquatfjh wftc lust ey rm`mr ey rm`mr tj tcm gjuhearfms x gjuhearfms x 2 2 ? ahe x ahe x 2 2 a fh @fnurm >.>. >.>. ; y ; y 1 C u ey 2 C u ey + C u ey 5 u , xy xy y xy y y 7 u , xy y 5 u , xy xy y = 0 y ; y 1 y ; y 1 > C C C C u ey 2 u ‛ ue y + u ‛ C ue y u , xy u, xy xy u , xy xy 3 y 7 y u , xy y 5 y u , xy xy y xy xy y 7 y 5 6 1? C u ey 2 C u ‛ C u + C u ‛ C u ‛ C ue y 11 u , xy u, xy ; u , xy 7 u , xy 1 u , xy 5 y u , xy xy y xy ; xy 7 xy 1 xy 5 xy 17 1; C u ey 2 ‛ 5 ‛ 1 i C u ‛ C ue y (>.50`) 15 u , xy u, xy i y u , xy i 2 1 xy y xy i xy 1= wcmrm, tcm sugskrfpts 1 tcrjunc 5 rm`mr tj tcm `jur kjrhmrs fh @fnurm >.>. >.>. [ugstftutfhn Mquatfjh (>.50`) fhtj (>.50`) fhtj Mquatfjh (>.50m) wm (>.50m) wm 10 jgtafh 1> 7 13 5 C C i C C u ex e y F u 2 ‛ ‛ 1 u + ue x ‛ ue y + (>.50n) 0 i 16 i 2 1 u u u, xy x , xy y , xy x y u , xy xy i xy xy xy A 7? 71 [ugstftutfhn tcm mvaduatme fhtmnrads nfvmh gy Mquatfjhs (>.50a), (>.50a), (>.50g), (>.50g), (>.50k), (>.50k), (>.50e), (>.50e), ahe (>.50n) fhtj (>.50n) fhtj (>.50) ahe (>.50) ahe 77 kjddmktfhn tmrbs wm jgtafh tcm `fhad `jrb j` tcm `frst varfatfjh j` tcm `uhktfjhad as scjwh gmdjw. 7; 7 7 7 75 C + C + C ‛ C ‛ C + C u ex e y F u 2 y 7 u , yy x y u, xy x u, x x y u , y y u 7= xx yy xy x 7 u , xx A 70 C u ey ‛ C + C ‛ C ue y 7> (>.5>) + , x x u , xx 73 u u u x y xx , xx xx , xy xy , x x 76 C u ex + C + C ‛ C ue x ‛ 5 ‛ 1 i C u ;? ‛ u , yy , y y x u , xy u, y y u , xy i i 2 1 y u, yy yy yy xy xy i ;1 ;7 ^m hjtm tcat tcm gjuheary fhtmnrads fh tcm agjvm mquatfjh wftc lust ey rm`mr gjuhearfms x 2 2 ? ahe x ahe x 2 2 a fh @fnurm ey rm`mr tj tcm gjuhearfms x ;; >.>, >.>, ahe gjuheary fhtmnrads wftc lust ex rm`mr ex rm`mr tj tcm gjuhearfms y gjuhearfms y 2 2 ? ahe y ahe y 2 2 g. ^m smt F u 2 ? ahe hjtm tcat makc tmrb bust ;5 gm zmrj ahe jgtafh tcm gjuheary vadum prjgdmb as scjwh gmdjw. ;= 7 7 7 C C ;0 C ‛ C ‛ C + C 2 ? (>.53a) Ef``mrmhtfad Mquatfjh< + + 7 7 x u , x x y u , y y u ;> x y u , xy xx yy xy x u , xx y u, yy ;3 ;6 C 2 ? 5? jr u , x x 2 ? u , xx xx 51 (>.53g) Gjuheary Kjheftfjhs at x at x 2 2 ? ahe x ahe x 2 2 a< 57 ahe 5; C + C ‛ C 2 ? jr u 2 ? 55 y u , xy u , x x x u , xx xx xy 5= 50 C 2 ? 5> jr u, y y 2 ? u , yy yy 53 56 (>.53k) Gjuheary Kjheftfjhs at y at y 2 2 ? ahe y ahe y 2 2 g< ahe =? C + C ‛ C 2 ? =1 jr u 2 ? u , yy x u , xy u , y y y yy xy =7 =; C 2 ? =5 (>.53e) Kjheftfjhs at makc kjrhmr< jr i 2 1 tj 5 u i 2 ? u , xy xy i == =0 => MSAB_DM >.; =3 =6 Jgtafh tcm gjuheary vadum prjgdmb `jr a tcfh cjbjnmhmjus pdatm gmhefhn wftc jhdy trahsvmrsm efstrfgutme `jrkms ahe kjrrmdatm tcm varfagdms tj tcjsm fhtrjeukme fh pdatm tcmjry. 0? 01
137
>
Parfatfjhad ahe Mhmrny Bmtcjes
1 7 Fh tcfs smktfjh wm jgtafh tcm gjuheary vadum prjgdmb `rjb tcm `uhktfjhad gmdjw `jr kurvf dfhmar gjuhearfms. Ucm `uhktfjhad gmdjw ; fs sdfnctdy ef``mrmht tcah tcm jhm fh Mquatfjh (>.5=) as ft kjhtafhs u , xy xy ahe u , yx yx , wcfkc tcjunc mquad hmme tj gm ef``mrmhtfatme fh 5 jremr tj bafhtafh tcm sybbmtry wftc x wftc x ahe ahe y y fh fh tcm emrfvatfjh. Ucfs ef``mrmhkm fs adsj hmkmssary `jr wrftfhn tcm rmsudts fh fhefkfad = hjtatfjh efskussme fh Kcaptmr 3. 3. 0 > (>.56) F u 2 C u, xx xx u , xy xy u , yx yx u , yy x u , y y u x y ex e y yy u , x 3 A 6 Uaifhn tcm `frst varfatfjh j` tcm agjvm `uhktfjhad wm jgtafh 1? C u + C u + C u + C u + C u + C u + C u ex e y (>.=?) 11 F u 2 xx xy yx yy 17 u , xx , xx u , xy , xy u , yx , yx u , yy , yy u , x x , x x u , y y , y y u xx xy yx yy A 1; ^m cavm smmh fh Mxabpdms >.7 ahe >.7 ahe >.;, >.;, tcm emrfvatfvms j` tcm `uhktfjhad C `uhktfjhad C rmsudt rmsudt fh quahtftfms tcat cavm pcysfkad bmah- 15 fhn sukc as strmss, gmhefhn bjbmhts, ahe scmar `jrkm. Fh tcm emrfvatfjh tcat `jddjws wm wfdd em`fhm varfagdms tj rmeukm wrftfhn 1= dmhntcy mxprmssfjhs gut tcm varfagdms wfdd cavm pcysfkad bmahfhn fh kjhtmxt j` pdatm gmhefhn. Ucm `jrbudas emvmdjpme wfdd cjw- 10 mvmr gm appdfkagdm tj ahy `uhktfjhad kjhtafhfhn smkjhe jremr emrfvatfvms. 1> Mquatfjh (>.=?) fs (>.=?) fs rm-wrfttmh ahe tcm emrfvatfvms trahs`mrme usfhn Mquatfjhs (>.;?a) ahe (>.;?a) ahe (>.;?g) as (>.;?g) as scjwh gmdjw. 13 16 C u , x x C u , x x C u , y y C u , y y 7? F u 2 + + + e x e y u x u u u y x y , xx xx , xy xy , yx yx , yy yy 71 A 77 C u C u C 7; + + + u e x e y u u u x y 75 y , x x A 7= 70 C C C h + C h u es h x + h y u , x + F u 2 x 7> u , xy u , yx x u , yy y , y y u , xx xx xy yx yy 73 (>.=1a) 76 C C C u C C C u C ;? + + + u , x x ‛ x u , xx ‛ y u , xy x u, y y ‛ x u, yx ‛ y u, yy y u u e x e y xx xy yx yy ;1 A ;7 ^m `frst kjhsfemr tcm gjuheary tmrb jhdy. Jh tcm gjuheary, `rjb Mquatfjhs (>.;1a) ahe (>.;1a) ahe (>.;1g) wm (>.;1g) wm jgtafh ;; u u u u ;5 2 h x ‛ h y 2 u , h h x ‛ h y u , x x 2 x h s s (>.=1g) ;= ;0 u u u u 2 h y + h x 2 u , h h y + h x u , y y 2 ;> y h s s ;3 [ugstftutfhn tcm agjvm twj mquatfjhs fh tcm gjuheary tmrb, wm jgtafh ;6 5? C C C C h x + h y h x + h x + h y h y u , h es F gjuheary 2 51 u , xy u , yx u, yy u , xx xx xy yx yy (>.=1k) 57 5; C C h ‛ h + C h + C h h u es + h x + 55 y u , xx u , xy y u , yx x u , yy y x s xx xy yx yy 5= 50 ^m fhtrjeukm tcm `jddjwfhn hjtatfjh. 5> C 7 C C 7 C 53 b hh 2 + u , xx h x + u , xy h x h y + u, yy h y u , yx xx yy xy yx 56 (>.=7) =? C ‛ C h h + h 7 C ‛ h 7 C b ht 2 ‛ x y x y =1 u , xx u, yx u , xy u , yy yx xy xx yy =7 [ugstftutfhn tcm agjvm mquatfjhs fhtj Mquatfjh (>.=1k) ahe (>.=1k) ahe fhtmnratfhn wftc rmspmkt tj s tj s,, wm jgtafh =; =5 bht u (>.=;a) == es 2 b hh u , h es + b ht u ‛ F gjuheary 2 bhh u , h es + b ht ue s s s =0 Ucm tmrb b ht u rmprmsmhts tcm fhtmnratfjh jh tcm wcjdm gjuheary, tcat fs, wm start `rjb a pjfht ahe rmturh gaki tj ft. => Ucus, tcm tmrb wfdd gm zmrj, uhdmss b ht fs efskjhtfhujus at sjbm pjfht jh tcm gjuheary. Ucfs efskjhtfhufty wfdd jkkur at makc kjr- =3 hmr pjfht jh tcm gjuheary eum tj tcm efskjhtfhufty fh tcm ef rmktfjh kjsfhms j` tcm hjrbad as scjwh f h@fnurm >.3. >.3. ^m rmprmsmht tcm =6 0? efskjhtfhufty fh bht at makc kjrhmr as b ht K . Mquatfjh (>.=;a) kah (>.=;a) kah hjw gm wrfttmh as 01
>.6.7
Gjuheary vadum prjgdmb `jr nmjbmtrfms wftc kurvfdfhmar gjuhearfms
[tatfjhary vadum j` a `uhktfjhad wftc smkjhe jremr 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
F gjuheary
2
b
hh
u , h ‛
bht u s
es + b ht K u K
(>.=;g)
h7 h1 K
@fnurm >.3 Efskjhtfhufty fh tcm uhft hjrbad at a kjrhmr.
^m fhtrjeukm tcm `jddjwfhn hjtatfjh ‛ q x 2
C ‛ C ‛ C u , x x x u , xx y u , xy xx xy
‛ q y 2
C ‛ C ‛ C u , y y x u , yx y u, yy yx yy
(>.=5)
[ugstftutfhn Mquatfjh (>.=;g) `jr (>.=;g) `jr tcm gjuheary tmrb ahe Mquatfjh (>.=5) fhtj (>.=5) fhtj Mquatfjh (>.=1a) wm (>.=1a) wm jgtafh
F u
2
b hh u , h ‛
bht u s
es + b ht K u K +
A
@ ‛ q x x u + ‛ q y y u + u u e x e y
(>.==a)
]sfhn Mquatfjhs (>.;?a) ahe (>.;?a) ahe Mquatfjhs (>.;?g), (>.;?g), wm trahs`mr tcm emrfvatfvms tj jgtafh
F u
F u
2
2
b
hh
b u , h ‛ ht u s
x x
+ q y h y ue s +
q x q y @ + + u ex e y y u
x
A
q x q y C b + + u ex e y + b hh u h es ‛ ht + q x h x + q y h y u u y s
x
A
q h
es + b ht K u K ‛
es + bht K u K 2 ?
(>.==g)
^m em`fhm tcm `jddjwfhn q h 2 q x h x + q y h y
P h 2 q h +
bht s
(>.==k)
@jr tcm agjvm mquatfjh tj gm vadfe makc tmrb fh tcm mquatfjh bust gm zmrj ahe wm jgtafh tcm gjuheary vadum prjgdmb wrfttmh gmdjw. Ef``mrmhtfad mquatfjh<
q x q y C + + x y u
b hh 2 ?
Gjuheary Kjheftfjhs<
u h
bht K 2
?
jr jr
u uK
(>.=0a)
2 ?
ahe P h 2 ?
Kjrhmr Kjheftfjh<
jr
x y fh A
2 ?
2 ?
x y jh
x y jh kjrhmr
(>.=0g)
K
(>.=0k)
F` tcm `uhktfjhad fh Mquatfjh (>.5=) kjhtafhs (>.5=) kjhtafhs bjrm tcah jhm varfagdm tcmh Mquatfjhs (>.=0a), (>.=0a), (>.=0g), (>.=0g), ahe (>.=0k) bust (>.=0k) bust gm wrfttmh `jr makc j` tcm varfagdms.^m sugstftutm b ht `rjb Mquatfjh (>.=7) fhtj P h fh Mquatfjh (>.==k) tj (>.==k) tj jgtafh P h 2 q h +
‛ C ‛ C h h + h 7 C ‛ h 7 C x y x u, yx y u , xy s u, xx u , yy yx xy xx yy
C ‛ C h h ‛ C ‛ C h x h + h h y y x x y u , xx s s u , xx u , yy u , yy xx yy xx yy s
P h 2 qh ‛
+
C h 7 + 7 C h h x ‛ C h 7 ‛ 7 C h h y x u, yx x s s u , xy y u, xy y s s u , yx yx yx xy xy u , xy xy
(>.=>a)
135
>
Parfatfjhad ahe Mhmrny Bmtcjes
[ugstftutfhn Mquatfjh (>.;1g) fhtj (>.;1g) fhtj tcm agjvm mquatfjh wm jgtafh
1 7 7 7 h ‛ h 7 h h C C C ‛ C h h + C + C ------------ x y x y h 7 ‛ h 7 ‛ C ‛ C ---------------P h 2 q h ‛ (>.=>g) ; ‛ x y x y u u W W u s s u u u u s u, xx 5 , yx yx , xy xy ku r ku r xx , yy yy , xx xx , yy yy , xy xy , yx yx = Fh tcm hmxt mxabpdm wm rmsjdvm Mxabpdm >.; jh >.; jh pdatm gmhefhn gut a pdatm wftc kurvfd fhmar gjuhearfms tj embjhstratm tcm 0 appdfkatfjh j` tcm `jrbudas fh tcfs smktfjh. > 3 MSAB_DM >.5 6 Jgtafh tcm gjuheary vadum prjgdmb `jr tcfh pdatm gmhefhn wftc jhdy trahsvmrsm efstrfgutme `jrkms ahe kjrrmdatm tcm varfagdms tj 1? tcjsm fhtrjeukme fh pdatm tcmjry. 11 17 >.1? [UAUFJHAWV PAD]M J@ A EM@FHFUM PJD]BM FHUMNWAD 1; 15 Nauss efvmrnmhkm `jrbuda prmsmhtme fh [mktfjh >.0 fs >.0 fs usme tj trahs`mr emrfvatfvms fh `fhefhn tcm statfjhary vadum j` vjdubm fhtm- 1= nrad. Ucm rmst j` tcm prjkmss fs as gm`jrm `jr `fhefhn tcm statfjhary vadums j` dfhm ahe arma fhtmnrad. ^m wfdd rmstrfkt jursmdvms tj 10 `uhktfjhad wftc `frst jremr emrfvatfvms as cfncmr jremr emrfvatfvms dmae tj mxkmssfvm adnmgra. 1> ^m kjhsfemr a vjdubm U gjuheme gjuheme gy sur`akm [ . ]sfhn tcm hjtatfjh u , x x 2 u x , u , y y 2 u y , ahe u ,z 2 u z , wm 13 em`fhm tcm `uhktfjhad ahe kjhsfemr fts `frst varfatfjh gmdjw. 16 7? (>.=3a) F u 2 C u, x u u u x y z e x e y e z 71 x , y y , z z U 77 7; C C C C F u 2 u ex ey e z 75 z + u , x x u , x x + u , y y u , y y + u , z u , z u z 7= U 70 C u + C u + C u + C u ex ey e z F u 2 7> u , x x x u , y y y u , z u z z 73 U ]sfhn Mquatfjh (>.;;), (>.;;), wm trahs`mr tcm emrfvatfvms fh tcm `frst tcrmm tmrbs ahe smt tcm `frst varfatfjh tj zmrj as scjwh 76 gmdjw. ;? ;1 C C C C C C C (>.=3g) ;7 h + h + h z ue [ + ‛ ‛ ‛ F u 2 u ex ey e z 2 ? u , x x x u , y y y u , z u x u , x x y u , y y z u , z z z ;; [ U Makc tmrb fh tcm `frst varfatfjh bust gm zmrj at add pjfhts.Ucus, wm kah jgtafh tcm gjuheary vadum prjgdmb wrfttmh gmdjw. ;5 ;= C C C C (>.=6a) ;0 Ef``mrmhtfad Mquatfjh< ‛ ‛ ‛ 2 ? x y z fh fh U u u u u x , x x y , y y z , z ;> z ;3 C C C (>.=6g) ;6 Gjuheary Kjheftfjhs< jr x y z jh jh [ u 2 ? u, x x h x + u, y y h y + u , z h z 2 ? z 5? F` tcm `uhktfjhad fh Mquatfjh (>.=3a) kjhtafhs (>.=3a) kjhtafhs bjrm tcah jhm varfagdm tcmh Mquatfjhs (>.=6a) ahe (>.=6a) ahe (>.=6g) bust (>.=6g) bust gm wrfttmh 51 57 `jr makc j` tcm varfagdms. 5; 55 >.11 HJHDFHMAWFUFM[ 5= Ucm tcrmm bjst kjbbjh hjhdfhmar prjgdmbs fh strukturad ahadysfs arm tcjsm tcat arfsm `rjb batmrfad, nmjbmtry, ahe kjhtakt. Nmj- 50 bmtrfk hjhdfhmarftfms arfsm `rjb darnm em`jrbatfjh―wm wfdd kjhsfemr tcmsm fh [mktfjh >.11.1. >.11.1. Batmrfad hjhdfhmarfty arfsms `rjb 5> tcm hjhdfhmar rmdatfjhscfps gmtwmmh strmssms ahe strafhs―wm fhkjrpjratme tcmsm fhtj ahadysfs fh Kcaptmr 7 ahe 7 ahe wfdd djji at ft `rjb 53 56 mhmrny pmrspmktfvm fh [mktfjh >.11.7.Kjhtakt >.11.7.Kjhtakt hjhdfhmarftfms arfsm wcmh tcm kjhtakt rmnfjh gmtwmmh twj sur`akms kcahnm eum tj =? appdfme djaes―ft rmqufrms bjef`yfhn varfatfjhad mquatfjhs ahe wfdd hjt gm kjhsfemrme ahy `urtcmr. =1 >.11.1 Nmjbmtrfk hjhdfhmarfty =7 =; Fh [mktfjh 1.= wm 1.= wm saw tcat tcmrm arm twj em`fhftfjhs j` darnm strafhs. Danrahnfah strafh fs kjbputme `rjb em`jrbatfjh gy usfhn tcm =5 jrfnfhad uhem`jrbme nmjbmtry as nmjbmtry as tcm rm`mrmhkm nmjbmtry. Mudmrfah strafh fs kjbputme `rjb em`jrbatfjh gy usfhn tcm `fhad tcm `fhad == em`jrbme nmjbmtry nmjbmtry as tcm rm`mrmhkm nmjbmtry. ^m wfdd emrfvm mxprmssfjhs `jr tcmsm twj strafhs fh [mktfjh 3.=, 3.=, gut fh tcfs kcaptmr =0 wm wfdd taim a nfvmh darnm strafh mxprmssfjh ahe fhkjrpjratm ft fhtj ahadysfs gy erjppfhn tcm sbadd strafh apprjxfbatfjh `jr jhdy => jhm-efbmhsfjhad strukturad bmbgmrs. Ahadysfs j` pdatms wfdd gm ejhm fh tcm hmxt kcaptmr as fhefkfad hjtatfjh rmeukm tcm tmefjus =3 adnmgra tcat arfsms `rjb fhkjrpjratfhn darnm strafh fhtj twj efbmhsfjhad ahadysfs. =6 ^m wfdd emvmdjp mxprmssfjh `jr pjtmhtfad mhmrny usfhn tcm nfvmh darnm strafh. Bfhfbub pjtmhtfad mhmrny fs appdfkagdm tj 0? dfhmar ahe hjhdfhmar systmb prjvfeme tcm systmb fs kjhsmrvatfvm. [j mvmh tcjunc wm arm kjhsfemrfhn darnm em`jrbatfjh, wm scadd 01
Hjhdfhmarftfms
13=
1 assubm wm arm stfdd fh tcm mdastfk rmnfjh. Ucm pjtmhtfad mhmrny fhkjrpjratfhn darnm strafh wfdd nfvm us a hmw `uhktfjhad. ^m kah 7 tcmh usm varfatfjhad kadkudus mquatfjhs tcat bfhfbfzm a `uhktfjhad tj jgtafh tcm gjuheary vadum prjgdmb. Ucfs fs mdagjratme fh tcm ; hmxt mxabpdm `jr gmabs ahe fh Mxabpdm 3.0. 3.0. `jr pdatms. 5 MSAB_DM >.= = Danrahnfah strafh wftc bjemratmdy darnm rjtatfjh `jr sybbmtrfk gmab gmhefhn fs nfvmh as 0 7 > ev eu 1 (>.0?) + - xx 2 3 e x 7 e x 6 1? Assubm tcm efspdakmbmht `fmde fs u x 2 u j ‛ y ev e x , wcmrm uj ahe v arm tcm axfad ahe gmhefhn efspdakmbmhts ahe arm `uhkj` x jhdy. jhdy. Add assubptfjhs j` kdassfkad gmab tcmjry mxkmpt sbadd strafh apprjxfbatfjh arm vadfe. p vadfe. p x( x) ahe p ahe p y( x) x) arm tcm efs11 tfjh j` x trfgutme djaes fh x fh x ahe ahe y y efrmktfjhs. efrmktfjhs. Jgtafh tcm strmss `jrbuda ahe tcm statmbmht j` gjuheary vadum prjgdmbs `jr efspdakmbmhts 17 1; uj ahe v. 7 15 euj 1 ev e AH[. Ef``mrmhtfad mquatfjh 1< MA + 2 p x 1= 7 e x e x e x 10 1> 7 7 7 7 e u e v 1 ev e v ev e j 13 AH[. Ef``mrmhtfad mquatfjh 7< + -- 2 p y + p x MF zz 7 ‛ MA 7 7 7 e x e x e x e x e x 16 e x 7? AH[. Gjuheary Kjheftfjh 1< H 2 ? jr at x at x 2 ? ahe x ahe x 2 D uj 71 77 AH[. Gjuheary Kjheftfjh 7< B z 2 ? jr at x 2 ? ahe x ahe x 2 D ev 2 ? 7; e x 75 e B z ev 7= AH[. Gjuheary Kjheftfjh ;< ‛ H 2 ? jr at x 2 ? ahe x ahe x 2 D v 2 ? e x 70 e x 7> 73 >.11.7 Batmrfad hjhdfhmarfty 76 Nfvmh tcm `akt tcm bfhfbub pjtmhtfad mhmrny fs appdfkagdm tj jhdy kjhsmrvatfvm systmb, tcat fs mdastfk systmbs, tcm jhdy batmrfad ;? hjhdfhmarfty wm wfdd kjhsfemr fs tcm jhm fh wcfkc strmss-strafh rmdatfjhscfp fs nfvmh gy tcm pjwmr daw bjemd j` [mktfjh 1.3.>. 1.3.>. Ucm ;1 mquatfjhs emskrfgfhn tcm strmss‛strafh rmdatfjhscfp arm nfvmh gy Mquatfjh (1.56) wcfkc (1.56) wcfkc fs wrfttmh gmdjw `jr kjhvmhfmhkm. ;7 h ;; ? N h ? M ahe 2 (>.01) 2 ;5 h ? ? ‛ N ‛ h ‛ M ‛ ;= ;0 ^m kah sugstftutm tcm agjvm mquatfjh fhtj Mquatfjh (>.6) ahe (>.6) ahe hjtfhn tcat strafh mhmrny fs adways pjsftfvm9 wm jgtafh tcm ;> strafh mhmrny emhsfty nfvmh gmdjw ;3 h+1 h+1 ? ? M h + 1 N h + 1 ;6 (>.07) ] ? 2 ] ? 2 h+1 h+1 ? ? 5? M ‛ N ‛ h + 1 h + 1 51 ^m kah sugstftutm tcm strafh efspdakm bmht mquatfjhs `jr jhm-efbmhsfjhad strukturad bmbgmrs fh tcm strafh mhmrny emhsfty, 57 pmr`jrb tcm fhtmnratfjh jvmr tcm vjdubm ahe jgtafh strafh mhmrny mxprmssfjhs jvmr tcm dmhntc j` tcm bmbgmr. Ucfs prjkmss fs sfb5; fdar tj tcm jhm embjhstratme `jr dfhmar strafh mhmrny fh [mktfjh >.;.1. >.;.1. ^m kah tcmh wrftm tcm wjri pjtmhtfad ahe tcm pjtmhtfad 55 mhmrny, mhmrny, wcfkc wm kah tcmh bfhfbf zm tj jgtafh tcm gjuheary vadum prjgdmbs. Ucm prjkmss fs embjhstratme `jr sybb mtrfk gmabs fh 5= tcm hmxt mxabpdm. 50 MSAB_DM >.0 5> 53 Ucm cjddjw squarm krjss smktfjh j` a gmab fs scjwh fh @fnurm >.6. >.6. Ucm strmss-strafh mquatfjh j` gmab batmrfad fs nfvmh gy ?.= 56 `jrkm p y( x). x). Jgtafh tcm gjuheary vadum prjgdmb `jr tcm gmab 2 M . Ucm gmab fs suglmktme tj a trahsvmrsm efstrfgutme `jrkm p =? em`dmktfjh v f` add assubptfjhs j` kdassfkad gmab arm vadfe mxkmpt `jr tcm Cjjim‖s daw. =1 a =7 =; =5 y == z 7a a =0 => =3 =6 7a 0? >.0. 01 @fnurm >.6 [ybbmtrfk gmab krjss smktfjh j` Mxabpdm >.0.
130
>
Parfatfjhad ahe Mhmrny Bmtcjes Ef``mrmhtfad mquatfjh< 1.5=30a 1.5=30 a
AH[.
2 ?
7
>.17
e v 7 e x
e e x
Gjuheary Kjheftfjh 7<
AH[.
e
M 7 e x
7
e v 7 e x
?.=
2 p y
?.=
7
e v Gjuheary Kjheftfjh 1< 7 e x
AH[.
;.>=
7
jr
ev e x
2 ?
at x 2 ? ahe x ahe x 2 D
v
2 ?
at x 2 ? ahe x ahe x 2 D
?.=
2 ?
jr
WAVDMFNC-WFU\ BMUCJE
Waydmfnc-Wftz bmtcje fs appdfkagdm tj kjhsmrvatfvm systmbs tcat bay gm dfhmar jr hjhdfhmar. Waydmfnc-Wftz bmtcje fs a `jrbad prjkmss j` bfhfbfzfhn tcm pjtmhtfad mhmrny usfhn a smrfms j` ifhmbatfkaddy aebfssfgdm e fspdakmbmht `uhktfjhs tj prjeukm a smt j` adnmgrafk mquatfjhs fh tcm uhihjwh kjhstahts j` tcm smrfms. ^m scadd emvmdjp tcm smt j` adnmgrafk mquatfjhs usfhn tcm prjpmrty j` gfdfhmar ahe dfhmar `uhktfjhads. [uppjsm l arm a smt j` fhempmhemht ifhmbatfkaddy aebfssfgdm `uhktfjhs. Ucmsm ` l kjude gm `uhktfjhs j` lust jhm varfagdm x fh x fh jhm efbmhsfjh, j` `uhktfjhs j` twj varfagdms (Kartmsfah jr pjdar kjjrefhatms) fh twj efbmhsfjhs, jr j` tcrmm varfagdms fh tcrmm efbmhsfjhs (Kartmsfah. kydfherfkad, jr spcmrfkad kjjrefhatms). ^m rmprmsmht tcm efspdakmbmht gy tcm smrfms gmdjw. h
u
l
K l ` l
1
(>.0;a)
wcmrm, K l arm kjhstahts tj gm emtmrbfhme. Hjtm K l arm tcm nmhmradfzme kjjrefhatms as tcm varfatfjh j` tcmb rmprmsmhts tcm varfatfjh j` tcm efspdakmbmht kurvm, tcat fs, vfrtuad efspdakmbmht. Ucm pjtmhtfad mhmrny j` Mquatfjh (>.7>) kah (>.7>) kah gm wrfttmh fh tcm `jddjwfhn `jrb 1 2 -- G u 1 u 7 ‛ d u 1 7
(>.0;g)
wcmrm, u 1 ahe u 7 arm rmprmsmhtme as scjwh gmdjw. u1 x
h
l
1
K l ` l x
u7 x
ahe
h
i 1
K i ` i x
(>.0;k)
[ugstftutfhn u 1 fh tcm dfhmar `uhktfjhad ahe gfdfhmar `uhktfjhad ahe usfhn Mquatfjhs (>.=a) ahe (>.=a) ahe (>.=g) wm (>.=g) wm jgtafh d u1 G u1 u 7
d
G
h
h
l
l
1
K l ` l
h
l
K l ` l u 7
1
1
K ` l d l
h
l
1
(>.0;e)
K l G ` l u 7
(>.0;m)
[ugstftutfhn u 7 fh tcm gfdfhmar `uhktfjhad ahe usfhn Mquatfjh (>.=g) wm (>.=g) wm jgtafh G u 1 u 7
h
l
1
K l G ` l
G u 1 u 7
h
i 1
K i ` i
h
h
l
1
i 1
h
h
l
1
K l
i 1
K i G ` l ` i
(>.0;`)
K G ` l ` i l K i
(>.0;n)
Ucm pjtmhtfad mhmrny kah hjw gm wrfttmh as h h 1 h (>.0;c) K G ` l ` i ‛ K 2 - l K i l d ` l i 1 l 1 7 l 1 Ucm pjtmhtfad mhmrny hjw fs a `uhktfjhad j` tcm nmhmradfzme kjjrefhatms K ^m taim tcm `frst varfatfjh j` ft ahe smt ft l . ^m mquad tj zmrj tj bfhfbfzm tcm pjtmhtfad mhmrny. h h 1 h (>.0;f) 2 - K l K i G ` l ` i + K l K i G ` l ` i ‛ K l d ` l i 1 l 1 7 l 1
@jr tcm smkjhe tmrb, wm kah fhtmrkcahnm tcm subbatfjh jvmr l jvmr l ahe ahe i ahe ahe wrftm tcm agjvm mquatfjh as
1 2 -- 7
h
h
l
1
i 1
K l K i G ` l ` i +
h
K l K i G ` i ` l ‛ l 1 h
i 1
h
l
1
K l d ` l
As add jur gfdfhmar `uhktfjhad arm sybbmtrfk fh tcmfr arnubmhts, tcat fs, G ` l ` i 2 G ` i ` l , wm jgtafh tcm `jddjwfhn h h h h h 1 h 2 - K l K i G ` l ` i + G ` l ` i ‛ K l d ` l 2 K l K i G ` l ` i ‛ K l d l` i 1 l 1 l 1 i 1 l 1 7 l 1
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
Waydmfnc-Wftz Bmtcje 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
K l l 1
2
h
G ` l ` i K i ‛ d ` l i 1 h
2 ?
13>
(>.0;l)
Ucm agjvm mquatfjhs bust gm satfs`fme `jr add varfatfjhs j` K l , wm jgtafh h
G ` ` K ‛ d ` 2 ? l
i 1
i
i
l 2 1 tj h
l
(>.05)
Ucm agjvm mquatfjh rmprmsmhts h mqufdfgrfub mquatfjhs tcat bust gm satfs`fme gy tcm kjhstahts K l . Dmt sjdutfjh j` tcm adnmgrafk mquatfjhs (tcm vadums at mqufdfgrfub), tcat fs h
G ` ` K
.
l
i 1
* i
i
* K l
rmprmsmht tcm
2 d ` l
(>.0=a)
*
Mquatfjh (>.0;c) kah gm wrfttmh fh tmrbs j` K (>.0=a) sugstftutme tj jgtafh l ahe Mquatfjh (>.0=a) sugstftutme
*
1 2 -7
h
l
*
1
Ucm tmrb tfjh `jr `uturm usm.
K l
i 1
h
l
h
h
G ` ` K ‛ l
i
* i
l
1 * K l d ` l 2 -1 7
h
l
*
1
1
h
*
K l d ` l ‛
l
*
1 * K l d ` l 2 ‛ -1 7
K l d ` l rmprmsmhts tcm wjri pjtmhtfad ^ at mqufdfgrfub. Ucus,
*
h
l
*
1
K l d ` l
(>.0=g)
*
2 ‛ ^ 7 . ^m wrftm tcfs jgsmrva-
At mqufdfgrfub, tcm pjtmhtfad mhmrny j` tcm systmb fs hmnatfvm j` cad` tcm wjri pjtmhtfad. [ybgjdfkaddy tcm smt j` adnmgrafk mquatfjhs fh Mquatfjh (>.05) ahe (>.05) ahe pjtmhtfad mhmrny at mqufdfgrfub kah gm wrfttmh fh batrfx `jrb as
I K
2
W *
I li 2 G ` l ` i
wc m rm 1 2 ‛ -7
W l 2 d ` l
(>.00a)
1 * U * K l W l 2 ‛ -- K W l 1 7 h
(>.00g)
I fs kaddme tcm stf``hmss batrfx ahe gmkausm tcm gfdfhmar `uhktfjhad fs sybbmtrfk, tcm stf``hmss batrfx fs sybbmtrfk. ^m wfdd baim usm j` tcm agjvm mquatfjhs fh tcm mxabpdms tcat `jddjw.
MSAB_DM >.> A gmab ahe fts djaefhn arm scjwh fh @fnurm >.1?. >.1?. ]sm tcm Waydmfnc-Wftz bmtcje wftc jhm ahe twj parabmtmrs tj emtmrbfhm tcm em`dmktfjh at x at x 2 2 ?.7= D, D, x 2 x 2 ?.= D, D, x 2 x 2 ?.>= D, D, ahe x ahe x 2 2 D D,, ahe tcm pjtmhtfad mhmrny `uhktfjh. Kjbparm yjur rmsudts wftc tcm ahadytfkad sjdutfjh. Assubm tcat MF tcat MF fs fs kjhstaht `jr tcm gmab. y
w x
A D
>.>. @fnurm >.1? Gmab `jr Mxabpdm >.>. Uagdm >.5 Wmsudts j` Mxabpdm >.> Em`dmktfjh<
_jtmhtfad Mhmrny
5
v/
D v -- 5
% ef``
D v -- 7
7 =
-wD --------- MF % ef``
w D / ------------ MF ; D v ------ 5
% ef``
vD
% ef``
Padum
% ef``
h21
-?.??=7
0?.=
-?.?7?3
=7.65
-?.?506
5;.30
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;;.;
-?.?1;6
55.55
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11.1
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=.33
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-?.17=
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-?.?75;
7.>3
Ahadytfkad
-?.?1;7
-?.?55;
-?.?3;=
-?.17=
-?.?7=
MSAB_DM >.3 A squarm pdatm fs uhf`jrbdy djaeme ahe fs sfbpdy suppjrtme jh add sfems as scjwh fh @fnurm >.11. >.11. ]sfhn Waydmfnc-Wftz bmtcje emtmrbfhm em`dmktfjh w at tcm kmhtmr j` tcm pdatm ahe tcm pjtmhtfad mhmrny `jr makc kasm. Kasm F< Kasm F< w x y 2 K 1 sfh x a sfh y a Kasm FF< Kasm FF< w x y 2 K 1 sfh x a sfh y a + K 7 sfh 7 x a sfh y a + K ; sfh x a sfh 7 y a Kasm FFF< Kasm FFF< w x y 2 K 1 sfh x a sfh y a + K 7 sfh ; x a sfh y a + K ; sfh x a sfh ; y a
133
>
Parfatfjhad ahe Mhmrny Bmtcjes
1 7 ; x e [fbpdy [uppjrtme e m m t 5 t r r j j pj p p = p p u u [ [ a 0 y y d d p p > f b f b [ [ 3 [fbpdy [uppjrtme 6 a y 1? 11 @fnurm >.11 ]hf`jrbdy djaeme, sfbpdy suppjrtme, squarm pdatm 17 ‛ ; 5 * ‛ ; 7 0 AH[. Kasm F< w a 7 a 7 2 5.10 5.10?0 ?0 1? p j a E 9 ‛ ?.35;1 1? p j a E 1; ‛ ; 5 * ‛ ; 7 0 15 AH[. Kasm FF< w a 7 a 7 2 5.10 5.10?0 ?0 1? p j a E 9 ‛ ?.35;1 1? p j a E 1= ‛ ; 5 * ‛ ; 7 0 5.?56> 6> 1? p j a E 9 ‛ ?.3=?0 1? p j a E AH[. Kasm FFF< w a 7 a 7 2 5.?5 10 1> 13 16 >.1; @FHFUM MDMBMHU BMUCJE 7? Ucm `fhftm mdmbmht bmtcje fs a vmrsatfdm hubmrfkad bmtcje tcat fs ugfquftjus fh strmss ahadysfs ahe fh tcm emsfnh j` bakcfhms ahe 71 strukturms. A efskussfjh j` `fhftm mdmbmht bmtcje tcat ejms lustfkm tj ft kah kjvmr `mw kjursms. Jur jglmktfvm fs bukc bjrm dfb- 77 ftme. Ft fs tj scjw cjw Waydmfnc-Wftz bmtcje kah gm usme fh `jrbudatfjh j` jhm j` tcm vmrsfjhs j` tcm `fhftm mdmbmht bmtcje. 7; 1 stf``hmss Ucm `fhftm mdmbmht bmtcje gmnah as a batrfx bmtcje j` ahadysfs fh strukturms . Ucmrm arm twj vmrsfjhs j` ft< tcm 75 bmtcje ahe tcm `dmxfgfdfty bmtcje. Uj mdagjratm tcm ef``mrmhkm gmtwmmh tcm twj bmtcjes vfsuadfzm a truss fh wcfkc tcm efspda km- 7= bmhts j` pfhs arm uhihjwh. F` wm hjw usm tcm —bmtcje j` ljfhts‒ ahe wrftm tcm mqufdfgrfub j` `jrkms at makc ljfht, wm wfdd jgtafh 70 a smt j` adnmgrafk mquatfjhs fh wcfkc tcm uhihjwhs arm tcm pfh efspdakmbmhts ahe tcm rfnct-cahe-sfem quahtftfms arm tcm mxtmrhad 7> `jrkms aktfhn jh tcm pfh―tcfs fs tcm stf``hmss bmtcje as tcm batrfx fh tcm adnmgrafk mquatfjh fs tcm stf``hmss batrfx. Hjw fh tcm 73 truss, suppjsm wm usm tcm fhtmrhad `jrkms as tcm uhihjwhs ahe wrftm tcm kjbpatfgfdfty mquatfjhs tcat rmdatm tcm efspdakmbmhts j` 76 tcm fhefvfeuad truss bmbgmrs tj tcm efspdakmbmht j` tcm pfh. ^m wfdd tcmh jgtafh a smt j` adnmgrafk mquatfjhs fh tcm uhihjwh ;? fhtmrhad `jrkms ahe tcm rfnct-cahe-sfem quahtftfms wfdd gm tcm efspdakmbmhts j` tcm pfhs―tcfs fs tcm `dmxfgfdfty bmtcje as tcm ;1 batrfx fh tcm adnmgrafk mquatfjh fs tcm `dmxfgfdfty batrfx. Ucm `dmxfgfdfty bmtcje kah gm emrfvme gy kjhsfemrfhn bfhfbub kjbpdf- ;7 bmhtary pjtmhtfad mhmrny, wcfkc fs sfbfdar tj tcm pjtmhtfad mhmrny wftc strafh mhmrny rm pdakme gy kjbpdfbmhtary strafh mhmrny. ;; Ucm stf``hmss bmtcje fs emrfvme `rjb bfhfbub pjtmhtfad mhmrny, wcfkc fs `jrbadfzme as tcm Waydmfnc-Wftz bmtcje. Kjbbmrkfad ;5 kjbputmr prjnrabs arm usuaddy gasme jh tcm stf``hmss bmtcje. ^m wfdd jhdy kjhsfemr Waydmfnc-Wftz bmtcje appdfkatfjh fh tcm `jr- ;= budatfjh j` tcm stf``hmss bmtcje. ;0 Ucm `fhftm mdmbmht bmtcje `jrbudatfjh prmsmhtme cmrm fs vmry sfbfdar tj tcm Waydmfnc-Wftz bmtcje wftc jhm fbpjrtaht ;> ef``mrmhkm< tcm ifhmbatfkaddy aebfssfgdm efspdakmbmht `uhktfjhs fh tcm `fhftm mdmbmht bmtcje arm em`fhme pfmkmwfsm kjhtfhujusdy ;3 jvmr sbadd (`fhftm) ejbafhs9 tcmsm arm tcm mdmbmhts. Ucm gjuheary pjfhts j` tcm mdmbmhts arm kaddme hjems, adtcjunc hjems kah ;6 adsj gm pjfhts fhsfem tcm mdmbmht. Ucm kjhstahts budtfpdyfhn tcm pfmkmwfsm ifhmbatfkaddy aebfssfgdm `uhktfjhs arm tcm efspdakm- 5? bmhts j` tcm hjems, ahe tcm ifhmbatfkaddy aebfssfgdm `uhktfjhs arm kaddme fhtmrpjdatfjh `uhktfjhs gmkausm tcmy kah gm usme tj 51 fhtmrpjdatm tcm vadums j` efspdakmbmhts gmtwmmh tcm hjems. Ucm rmprmsmhtatfjh j` a strukturm gy mdmbmhts ahe hjems fs kaddme a 57 bmsc. A bmsc wftc gjuheary kjheftfjhs, appdfme djaes, ahe batmrfad prjpmrty fs kaddme a bjemd. A bjemd fs a `fhftm mdmbmht rmp- 5; rmsmhtatfjh j` a rmad-df`m prjgdmb, ahe tcm akkuraky j` tcm bjemd‖s prmefktfjhs fs emtmrbfhme gy tcm assubptfjhs ahe dfbftatfjhs 55 tcat arm baem fh kjhstruktfhn tcm `fhftm mdmbmht bjemd ahe tcm mrrjrs fhtrjeukme fh sjdvfhn tcm bjemd gy hubmrfkad bmtcjes. 5= Ucm usm j` pfmkmwfsm ifhmbatfkaddy aebfssfgdm `uhktfjhs kcahnms tcm pmrspmktfvm wftc wcfkc wm vfmw ahe sjdvm a prjg- 50 dmb gy bmahs j` tcm `fhftm mdmbmht bmtcje. Uj mdagjratm tcfs pmrspmktfvm, kjhsfemr a strukturm baem up j` axfad bmbgmrs, kfrkudar 5> sca`ts, sybbmtrfk gmabs, ahe jtcmr bmbgmrs sukc as kurvme gmabs, pdatms, ahe scmdds. _jtmhtfad mhmrny fs a skadar quahtfty ahe 53 f kah gm wrfttmh as tcm sub j` tcm pjtmhtfad mhmrny j` tcm h fhefvfeuad strukturad bmbgmrs as scjwh gmdjw. 56 h =? (>.0>) 2 f f 1 =1 Mquatfjh (>.0>) fs (>.0>) fs vadfe `jr strukturad bmbgmrs j` add typms, frrmspmktfv m j` jrfmhtatfjh. ^m kjude tcus emvmdjp tcm pjtmh- =7 tfad mhmrny fh batrfx `jrb `jr makc bmbgmr smparatmdy. Ucat fs, wm kjude emvmdjp batrfkms at tcm mdmbmht dmvmd fh a djkad kjjref- =; hatm systmb wftcjut rmnare tj cjw a bmbgmr fs usme fh tcm strukturm. Ucm fhefvfeuad djkad batrfkms, kaddme mdmbmht stf``hmss =5 batrfkms, kjude gm assmbgdme gy usfhn Mquatfjh (>.0>) tj (>.0>) tj `jrb tcm ndjgad stf``hmss batrfx j` tcm mhtfrm strukturm. Ucfs pmrspmk- == tfvm j` rmeukfhn tcm kjbpdmxfty j` ahadyzfhn darnm strukturms tj tcm ahadysfs j` sfbpdm fhefvfeuad bmbgmrs (mdmbmhts) mdmbmhts) fs wcat =0 baims tcm `fhftm mdmbmht bmtcje sukc a vmrsatfdm ahe pjpudar tjjd fh strukturad (ahe mhnfhmmrfhn appdfkatfjh) ahadysfs. => =3 =6 1. Ujeay tcm `fhftm mdmbmht bmtcje fs usme mxtmhsfvmdy mxtmhsfvmdy fh sjdvfhn mhnfhmmrfhn mhnfhmmrfhn prjgdmbs. Ft Ft fs vfmwme as a hubmrfkad bmtcje `jr sjdvfhn partfad ef``mrmh0? tfad mquatfjhs, ahe adnmgrafk mquatfjhs arm jgtafhme `rjb ah apprjakc kaddme tcm —wmai `jrb.‒ wcfkc fs mqufvadmht tj tcm varfatfjhad apprjakc wm 01 cavm emvmdjpme fh tcfs kcaptmr. [m m Wmeey Y166;R `jr aeeftfjhad emtafds.
@fhftm mdmbmht bmtcje
136
1 Mquatfjh (>.0>) rmprmsmhts (>.0>) rmprmsmhts ah assmbgdy prjkmss fh wcfkc kjhtfhufty j` tcm prfbary varfagdms cas tj gm mhsurme. Ucm 7 batrfx nmhmratme cas a em`fhme strukturm, wftc djts j` zmrj‖s fh tcm batrfx. Ucmrm arm budtftuems j` sjdutfjh prjkmeurms tcat mxpdjft ; tcm strukturm tj prjeukm vmry m``fkfmht sjdutfjh prjkmeurms j` tcm adnmgrafk mquatfjhs. ^m wfdd cjwmvmr, rmstrfkt jursmdvms tj jhdy 5 nmhmratfhn tcm mdmbmht stf``hmss batrfx ahe dmavm ft tj tcm rmaemr tj rmae gjjis jh `fhftm mdmbmht bmtcje tj dmarh tcm assmbgdy ahe = sjdutfjh tmkchfqums fh `fhftm mdmbmht bmtcje. 0 Fh [mktfjh >.>.5, >.>.5, wm saw tcat `jr ifhmbatfkaddy aebfssfgdm `uhktfjhs, tcm prfhkfpad emrfvatfvms, tcat fs, tcm prfbary varf> agdms bust gm kjhtfhujus. Ucm prfbary varfagdms arm tcm nmhmradfzme efspdakmbmhts. Ucm jremr j` emrfvatfvms tcat bust gm kjhtfh3 ujus empmhes upjh tcm cfncmst jremr j` emrfvatfvm fh tcm `uhktfjhad. ^cmh tcm cfncmst emrfvatfvm fh tcm `uhktfjhad fs r , tcmh 6 cfncmst jremr j` emrfvatfvm fh prfbary varfagdm tcat bust gm kjhtfhujus fs r fs r -1. -1. Ucus, fh axfad, tjrsfjh, pdahm mdastfkfty, wcmrm tcm 1? jremr j` emrfvatfvm fh tcm `uhktfjhad was r 2 2 1, tcmh wm jhdy hmme tj mhsurm tcm kjhtfhufty j` tcm efspdakmbmht. Fh gmab gmhefhn 11 ahe pdatm gmhefhn, tcm `uhktfjhad cae smkjhe jremr emrfvatfvms, ahe tcm prfbary varfagdms (nmhmradfzme efspdakmbmhts) tcat bust 17 gm kjhtfhujus arm efspdakmbmht ahe sdjpms. Danrahnm pjdyhjbfads mhsurm kjhtfhufty j` tcm `uhktfjh (efspdakmbmhts) at tcm 1; hjems ahe Cmrbftm pjdyhjbfads mhsurm kjhtfhufty j` tcm `uhktfjh ahe fts emrfvatfvms at tcm hjems. Ucm rmqufrmbmht tj mhsurm 15 kjhtfhufty j` tcm nmhmradfzme efspdakmbmht jh tcm gjuheary fs tcat jhdy tcm hjems jh tcm gjuheary appmar fh tcm mquatfjh rmprm1= smhtfhn tcm nmhmradfzme efspdakmbmht. Ucfs fs gmkausm tcm hjead vadums j` nmhmradfzme efspdakmbmhts arm fhempmhemht varfagdms 10 ahe tcus a hjem tcat fs hjt jh tcm gjuheary kah kcahnm tcm mquatfjh j` nmhmradfzme efspdakmbmhts fhempmhemhtdy tcus vfjdatfhn 1> tcm kjhtfhufty rmqufrmbmhts. ^cmh tcm ifhmbatfkaddy aebfssfgdm `uhktfjhs satfs`y tcm kjhtfhufty at tcm hjems ahe tcm gjuheary 13 tcmh tcm mdmbmht fs kaddme a kjh`jrbfhn mdmbmht. ^cmh kjhtfhufty fs jhdy mhsurme at tcm hjems gut hjt mhsurme akrjss tcm gjuhe16 ary tcmh tcm mdmbmht fs kaddme hjh-kjh`jrbfhn mdmbmht. Hjh-kjh`jrbfhn mdmbmhts arm j`tmh usme wcmh kjhtfhufty j` cfncmr 7? jremr emrfvatfvms fs hmmeme wcfkc bay rmsudt fh darnm hubgmr j` uhihjwhs tcat bay jr bay hjt fbprjvm akkuraky j` tcm sjdutfjh. 71 ^m cavm smmh Ufbjscmhij gmabs ahe Bfhedfh-Wfsshmr pdatms, wcfkc pmrbft scmar strmss fh trahsvmrsm efrmktfjh, arm 77 cfncmr jremr tcmjrfms. Ucmsm tcmjrfms kah sfbudatm sjdutfjhs j` kdassfkad gmab ahe pdatm tcmjrfms. Ucm baljr ef``fkudty wftc tcmsm 7; tcmjrfms fs fh nmttfhn ahadytfkad sjdutfjh. Gut f` t cm jglmktfvm fs tj nmt a hubmrfkad sjdutfjh tcmh wm kah usm tcmsm tcmjrfmsahe tcmjrfmsahe rmap 75 tcm gmhm`ft j` tcm `akt tcat kjhtfhufty fs hmmeme jhdy j` tcm `uhktfjhs ahe hjt fts emrfvatfvms. Ucus, wm kah usm Danrahnm pjdyhj7= bfads wftc cfncmr jremr tc mjrfms tj jgtafh rmsudts `jr tcm kdassfkad gmab ahe pdatm tcmjry. ^m ^m wfdd rmstrfkt jur ef skussfjh tj 70 Danrahnm pjdyhjbfads gut wfdd usm Cmrbftm pjdyhjbfads `jr ifhmbatfkaddy aebfssfgdm `uhktfjhs tcat wfdd gm nfvmh wftcjut mxpda7> hatfjh j` cjw tcmy wmrm kjhstruktme. 73 76 >.1;.1 Danrahnm pjdyhjbfads fh jhm efbmhsfjh ;? Danrahnm pjdyhjbfads mhsurm kjhtfhufty j` nmhmradfzme efspdakmbmhts at tcm hjems. Ucm pjdyhjbfads arm fhtrjeukme gy usfhn tcm ;1 axfad bmbgmrs tj prjvfem tcm bjtfvatfjh ahe praktfkad rmdmvahkm j` tcmsm `uhktfjhs fh tcm @MB. ;7 @fnurm >.17a >.17a scjws ah axfad bmbgmr (mdmbmht) wftc twj hjems. Ucm axfad efspdakmbmhts uY1R ahe uY7R arm tcm emnrmms j` ;; `rmmejb (nmhmradfzme efspdakmbmhts) fh tmrbs j` wcfkc wm pdah tj emrfvm tcm mdmbmht stf``hmss batrfx. ^m wfdd usm tcm supmrskrfpt supmrskrfpt ;5 wftc squarm grakimt tj tj emsfnhatm nmhmradfzme efspdakmbmhts jr nmhmradfzme `jrkms at tcm hjem j` ah mdmbmht . Ucm mdmbmht fh @fnurm ;= >.17a >.17a cas 7 emnrmms j` `rmmejb, ahe sj wm kcjjsm a dfhmar `uhktfjh u x K 1 + K 7 x wftc twj uhihjwh parabmtmrs. ^m hjtm ;0 tcat at x at x x1 tcm efspdakmbmht u( x x1) 2 uY1R, ahe at x at x x7 tcm efspdakmbmht u( x x7) 2 uY7R. Ucm kjhstahts K 1 ahe K 7 kah gm sjdvme fh ;> tmrbs j` uY1R ahe uY7R ahe sugstftutme fhtj tcm dfhmar `uhktfjh rmprmsmhtatfjh tj jgtafh Mquatfjh (>.03a), (>.03a), ;3 7 1 x ‛ x 7 7 x ‛ x 1 1 7 f ;6 u x u --------------- u --------------- u u (>.03a) 1 x u 7 x fx f 1 x 1 ‛ x 7 x 7 ‛ x 1 5? 51 x ‛ x 7 x ‛ x 1 --------------ahe x --------------- (>.03g) 1 x 7 57 x 7 ‛ x1 x 1 ‛ x 7 5; 55 uY7R uY1R uY7R uY;R 5= uY1R x 50 x 5> Hjem 1 x1 Hjem 7 x1 Hjem 1 Hjem 7 Hjem ; x7 53 x7 56 =? (a) (g) =1 =7 (a) dfhmar ahe (g ( g) quaeratfk mdmbmhts. @fnurm >.17 Axfad (a =; =5 A dfhmar rmprmsmhtatfjh j` efspdakmbmht fs su``fkfmht f` tcm `jrkms arm appdfme jhdy at tcm mdmbmht mhe ahe jhdy f` tcm == krjss smktfjhad arma ejms hjt kcahnm akrjss tcm mdmbmht. F` tcm axfad bmbgmr cas a efstrfgutme djae, jr f` tcm bmbgmr fs tapmrme, =0 tcmh tcm axfad efspdakmbmht fs hj djhnmr dfhmar fhsfem tcm mdmbmht. A quaeratfk jr cfncmr-jremr pjdyhjbfad bay kjhvmrnm tj tcm => aktuad sjdutfjh `astmr tcah a dfhmar mdmbmht. @fnurm >.17g >.17g scjws ah mdmbmht wftc tcrmm hjems. ^ftc ; emnrmms j` `rmmejb, wm kah =3 start wftc a quaeratfk efspdakmbmht `uhktfjh u x 2 K 1 + K 7 x + K ; x 7 , sjdvm tcm kjhstaht fh tmrbs j` tcm hjead efspdakmbmht, ahe =6 jgtafh ah mquatfjh ahadjnjus tj Mquatfjh (>.03a). (>.03a). Ucfs prjkmss wjude gm tmefjus `jr cfncmr-jremr pjdyhjbfads. [j wm usm ah 0? adtmrhatfvm apprjakc. ^m rmprmsmht tcm efspdakmbmht fh tcm mdmbmht gy Mquatfjh (>.06a), (>.06a), 01
16?
>
Parfatfjhad ahe Mhmrny Bmtcjes u x
h
l
1
u
l l
x
(>.06a)
wcmrm h fs tcm emnrmms j` `rmmejb (hubgmr j` hjems, fh tcfs kasm) j` tcm mdmbmht tcat kah gm usme `jr rmprmsmhtfhn tcm ((h h ‛ 1) jremr j` pjdyhjbfads. Hjw at tcm l tcm ltc tc hjem, tcm efspdakmbmht u( x x l) 2 u l, ahe wm jgtafh Mquatfjh (>.06g). (>.06g). u x i
h
l
u
1
l l
x i
u
i
(>.06g)
@jr Mquatfjh (>.06g) tj (>.06g) tj gm trum, tcm prjpmrty nfvmh fh Mquatfjh (>.06k) bust (>.06k) bust cjde. l
Hjem 1
1 x i ?
i l i l
(>.06k)
Hjem 1
Hjem 7
1
Hjem 7
Hjem ;
1
x) D 1( x)
x) D 1( x)
x 1
D 7( x) x)
x 1
x) D 7( x)
x
x 1
x) D ;( x)
x
(a)
(g)
@fnurm >.1; (a) Dfhmar ahe (g) quaeratfk Danrahnm pjdyhjbfads.
Mquatfjh (>.06k) fbpdfms (>.06k) fbpdfms tcat tcm pjdyhjbfads D l( x) x) arm sukc tcat tcm vadum fs 1 jh fts jwh l jwh ltc tc hjem ahe zmrj at jtcmr hjems. @fnurm >.1; scjws >.1; scjws tcm apprjxfbatm pdjts `jr dfhmar ahe quaeratfk D l( x) x) tcat bmmt tcfs rmqufrmbmht. Hjw kjhsfemr D 1 fh a quaeratfk pjdyhjbfad. F` wm rmprmsmht D 1( x) x) 2 a1( x x x7)( x x x;), fts vadum fs zmrj at hjems 7 ahe ;. ^m kah hjw emtmrbfhm tcm kjhstaht a1 sukc tcat D 1 at hjem 1 fs mquad tj jhm, ahe wm jgtafh Mquatfjh (>.>?). (>.>?). x ‛ x7 x ‛ x; (>.>?) D 1 x --------------- -------------- x 1 ‛ x 7 x 1 ‛ x ; Fh a sfbfdar bahhmr, wm kah start wftc D 7( x) x) 2 a7( x x x;)( x x x1) ahe D ;( x) x) 2 a;( x x x1) ( x x x7) ahe emtmrbfhm tcm vadum j` a7 ahe a; sukc tcat D 7 ahe D ; at hjems 7 ahe ;, rmspmktfvmdy, cavm a vadum j` jhm tj jgtafh Mquatfjh (>.>?a). (>.>?a). D 7
x ‛ x x ‛ x x 2 ---------------1 ---------------;
ahe
x 7 ‛ x 1 x 7 ‛ x ;
D ;
x ‛ x x ‛ x x ---------------1 ---------------7
(>.>?a)
x ; ‛ x1 x ; ‛ x 7
Ucm prjkmss wm usme tj jgtafh tcm pjdyhjbfads `jr tcm quaeratfk kah hjw gm nmhmradfzme tj jgtafh Mquatfjh (>.>1), (>.>1), x ‛ xi h (>.>1) D l x i 21 i 21 ------------------ x l ‛ xi l i
h rmprmsmhts tcm prjeukt j` tcm tmrbs fh squarm grakimts. Ucm `uhktfjhs em`fhme gyMquatfjh wcmrm l21 gy Mquatfjh (>.>1) arm (>.>1) arm kaddme l 21 Danrahnm pjdyhjbfads. pjdyhjbfads.
Kjhtfhufty j` efspdakmbmht Efskjhtfhufty j` emrfvatfvm u7
u1 1
u;
7 mdmbmht 1
; mdmbmht 7
u;
u7
u1 1
7 mdmbmht 1
(a)
; (g)
u5
u= 5
=
mdmbmht 7
@fnurm >.15 _jssfgdm varfatfjh j` efspdakmbmht `jr (a) dfhmar ahe (g) quaeratfk mdmbmhts.
@uhktfjhs rmprmsmhtme gy Danrahnm pjdyhjbfads wfdd gm kjhtfhujus at tcm mdmbmht mhes. Fhsfem tcm mdmbmhts, add jremrs j` emrfvatfvms arm em`fhme as pjdyhjbfads arm kjhtfhujus. Cjwmvmr, tcm kjhtfhufty j` tcm emrfvatfvm j` tcm `uhktfjh kahhjt gm mhsurme at tcm mdmbmht mhe frrmspmktfvm j` tcm jremr j` pjdyhjbfads wcmh Danrahnm pjdyhjbfads arm usme `jr rmprmsmhtfhn tcm `uhktfjh. @fnurm >.15 scjws >.15 scjws a pjssfgdm varfatfjh j` a efspdakmbmht `fmde rmprmsmhtme gy Danrahnm pjdyhjbfads jvmr twj aeljfhfh n mdmbmhts. Efspdakmbmhts at hjems arm fhempmhemht parabmtmrs tcat kah cavm ahy vadum9 cmhkm tcm varfatfjh scjwh fh @fnurm >.15 fs a pjssfgfdfty. As kah gm smmh `rjb @fnurm >.15, >.15, tcm kjhtfhufty j` tcm efspdakmbmht fs bafhtafhme, gut fts `frst emrfvatfvm fs hjt kjhtfhujus `jr mftcmr tcm dfhmar jr tcm quaeratfk mdmbmht at tcm mdmbmht mhe hjem.
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
@fhftm mdmbmht bmtcje
161
1 >.1;.7 Haturad kjjrefhatms 7 `fhftm mdmbmht bmtcje. ^m wfdd usm tj rmprmsmht tcm haturad ; Hjh-efbmhsfjhad kjjrefhatms arm kaddme haturad kjjrefhatms fh `fhftm ahe D tj rmprmsmht tcm kcaraktmrfstfk dmhntc j` tcm mdmbmht. ^cmh ? 1 tcm trahs`jrbatfjh fs 5 kjjrefhatms fh jhm efbmhsfjh ahe D 2 x D ahe tcm jrfnfh fs tcm dm`t bjst mhe j` tcm mdmbmht. ^cmh ‛ 1 1 tcm trahs`jrbatfjh fs 2 7 x D ahe tcm jrfnfh fs = 0 fh tcm bfeedm j` tcm mdmbmht. ^m wfdd usm 2 x D . Jhm j` tcm fbpjrtaht usms j` haturad kjjrefhatms fs tcat tcm Danrahnm pjdy> hjbfads kah gm usme `jr apprjxfbatfhn tcm nmjbmtry as wmdd as tcm nmhmradfzme efspdakmbmhts. Ucfhi j` a kurvme gmab tcat bay ^m kjude apprjxfbatm tcm kurvme gmab wftc smvmrad sba dd gmab smnbmhts fh wcfkc tcm x ahe x ahe y y kjjref kjjref3 jr bay hjt gm kfrkud ar. ^m 6 hatms kjude gm rmprmsmhtme as h h 1? (>.>7) x 2 x l l y 2 y l l l 1 l 1 11 17 wcmrm, x l ahe l arm tcm kjjrefhatms j` tcm hjems ahe h fs tcm pjdyhjbfad jremr. Fsjparabmtrfk mdmbmhts arm mdmbmhts fh wcfkc 1; nmjbmtrfk trahs`jrbatfjh ahe tcm nmhmradfzme efspdakmbmhts arm apprjxfbatme gy tcm sabm fhtmrpjdatfjh `uhktfjhs. Ahjtcmr gmh15 m`ft j` usfhn haturad kjjrefhatms ahe fsjparabmtrfk mdmbmhts fs tcm usm j` hubmrfkad fhtmnratfjh wcmh tcm fhtmnrahe gmkjbms kjb1= pdfkatme eum tj nmjbmtrfk trahs`jrbatfjh. Ucm usm j` fsjparabmtrfk mdmbmhts , haturad kjjrefhatms, ahe hubmrfkad fhtmnratfjh fs 10 sjbmtfbms tcm jhdy kcjfkm fh twj ahe tcrmm efbmhsfjhs `jr bjemdfhn kjbpdmx nmjbmtrfms. 1> ^m kah wrftm tcm dfhmar ahe quaeratfk Danrahnm pjdyhjbfads fh haturad kjjrefhat ms as scjwh gmdjw. 13 Dfhmar (>.>;a) 1 1 ‛ 7 2 16 7? (>.>;g) Quaeratfk 1 2 1 ‛ 1 ‛ 7 7 2 5 1 ‛ ; 2 ‛ 1 ‛ 7 71 77 7; >.1;.; Pmktjr arftcbmtfk 75 Ucm Danrahnm pjdyhjbfads fh twj ahe tcrmm efbmhsfjhs usfhn hjh-efbmhsfjhad kjjrefhatms wfdd rmqufrm armas j` trfahndms ahe 7= vjdubms j` tmtracmerjhs fh tmrbs j` kjjrefhatms j` pjfhts. Pmktjr arftcbmtfk prjvfems a sfbpdm way j` jgtafhfhn `jrbudas `jr 70 armas ahe vjdubms fh tmrbs j` kjjrefhatms j` pjfhts― rmvfmwme fh tcfs smktfjh. 7> Dmt A G ahe K gm tcrmm vmktjrs. @rjb vmktjr arftcbmtfk wm ihjw tcat tcm krjss prjeukt A G rmsudts fh a vmktjr 73 wcjsm banhftuem fs tcm arma j` tcm paraddmdjnrab ahe efrmktfjh fs pmrpmhefkudar tj tcm pdahm `jrbme gy tcm twj vmktjrs as scjwh 76 fh @fnurm >.1=a. >.1=a. Ucm Ucm arma j` tcm trfahndm fs cad` j` tcm arma j` paraddmdjnrab `jrbme gy t cm twj vmktjrs. ^m `urtcmr ihjw tcat tcm ;? skadar trfpdm prjeukt A G K rmsudts fh a quahtfty wcjsm banhftuem fs tcm vjdubm j` tcm paraddmdmpfpme ahe tcm vjdubm j` tcm ;1 tmtracmerjh fs jhm-sfxtc j` tcm paraddmdmpfpme as scjwh fh @fnurm >.1=g >.1=g. Ucm jremr j` budtfpdfkatfjh fh krjss prjeukt fs fh kjuhtmr;7 kdjkiwfsm efrmktfjh wftc rmspmkt tj tcm pjsftfvm z pjsftfvm z -efrmktfjh. -efrmktfjh. ;; ;5 ;= ;0 A G ; ;> 5 ;3 ;6 ; G 5? K 51 G 57 5; A 1 55 7 1 7 A (g) (a) 5= 50 (g) _araddmdmpfpme `jrbme gy skadar trfpdm prjeukt. 5> @fnurm >.1= (a) _araddmdjnrab `jrbme gy krjss prjeukt. (g 53 @jr `fhefhn tcm arma j` trfahndm wm assubm tcm vmktjrs A ahe G arm fh tcm x tcm x-- y pdahm. y pdahm. ^m rmprmsmht tcm uhft vmktjrs fh x fh x,, 56 y, y, ahe z ahe z efrmktfjh efrmktfjh as f , , ahe i , rmspmktfvmdy. ^m kah wrftm tcm vmktjrs fh tmrbs j` tcm kjjrefhatms j` tcm tcrmm pjfhts scjwh fh =? @fnurm >.1=a >.1=a, taim tcm krjss prjeukt j` tcm twj vmktjrs ahe efvfem gy twj tj nmt tcm arma j` tcm trfahndm as scjwh gmdjw. =1 A 2 x 7 ‛ x 1 f + y 7 ‛ y 1 l G 2 x ; ‛ x1 f + y; ‛ y 1 l =7 =; (>.>5) Arma j` trfahndm 1-7-; 2 A G 7 2 x7 ‛ x 1 y; ‛ y 1 ‛ x ; ‛ x1 y 7 ‛ y1 7 =5 @jr `fhefhn tcm vjdubm j` tcm tmtracmerjh wm wrftm tcm tcrmm vmktjrs fh tmrbs j ` tcm kjjrefhatms j` tcm `jur pjfhts scjwh == fh @fnurm >.1=g >.1=g, taim tcm skadar trfpdm prjeukt ahe efvfem gy sfx tj jgtafh tcm vjdubm j` tcm tmtracmerjh as scjwh gmdjw. =0 => A 2 x 7 ‛ x 1 f + y 7 ‛ y 1 l + z 7 ‛ z 1 i G 2 x ; ‛ x1 f + y; ‛ y 1 l + z ; ‛ z 1 i =3 K 2 x 5 ‛ x 1 f + y 5 ‛ y 1 l + z 5 ‛ z 1 i =6 0? 01
167
>
Parfatfjhad ahe Mhmrny Bmtcjes
1 1 Pjdubm j` tmtracmerjh 1-7-;-5 2 -- A G K 2 -0 0
x7 ‛ x1 y7 ‛ y 1 z 7 ‛ z 1 x; ‛ x1 y; ‛ y 1 z ; ‛ z 1 x5 ‛ x1 y5 ‛ y 1 z ; ‛ z 1
(>.>=)
wcmrm
rmprmsmht tcm emtmrbfhaht, wcfkc wcfkc kah gm mxpaheme tj nfvm tcm vjdubm j` tmtracmerjh fh tmrbs j` vmrtfkms kjjrefhatms.
>.1;.5
Danrahnm pjdyhjbfads fh twj efbmhsfjhs
Ucm trfahnudar mdmbmht fs tcm sfbpdmst mdmbmht tcat kah gm usme `jr bjemdfhn rmnfjhs wftc kurvme gjuheary. A dfhmar rmprmsmhtatfjh j` nmhmradfzme kjjrefhatm fs scjwh gmdjw. u x 2 K ? + K 1 x + K 7 y
(>.>0)
Ucm kjhstahts K‖ s kah gm `juhe fh tmrbs j` tcm hjead vadums j` u. Ucm `uhktfjhs budtfpdyfhn tcm hjead vadums j` u wjude gm tcm Danrahnm pjdyhjbfads fh twj efbmhsfjhs. ^m wfdd emvmdjp tcm Danrahnm pjdyhjbfads fh twj efbmhsfjhad haturad kjjrefhatms kaddme arma kjjrefhatms. @fnurm >.10 scjws >.10 scjws a trfahndm wftc pjfht _ pjfht _ wcjsm kjjrefhatms arm ( x, x, y). y). Ucm armas j` tcm tcrmm trfahndms `jrbme gy ljfhfhn pjfht _ pjfht _ tj tj tcm vmrtfkms kah gm `juhe. Dfhm AG Dfhm AG fs fs paraddmd tj sfem LI sfem LI . Ucm arma A F wfdd gm tcm sabm frrmspmktfvm j` tcm djkatfjh j` pjfht _ pjfht _ jh jh dfhm AG dfhm AG.. Ucfs uhfqum vadum j` arma A F baims tcm arma kjjrefhatms pjssfgdm. D pjssfgdm. D F , D L , D I arm tcm arma kjjrefhatms em`fhme gmdjw. [fhkm a pjfht fh twj efbmhsfj hs kah cavm jhdy twj kjjrefhatms, wm jgtaf h tcm femhtfty scjwh tcat mbpcasfzm tcat jhdy twj j` tcm tcrmm arma kjjrefhatms arm fhempmhemht. A D F 2 ---- F A
A L D L 2 ---- A
A I D I 2 ----- A
D F + D L + D I 2 1
(>.>>a)
D
2 ? F
D L
I
D F
D I
A
A F _
A L
L
A I D
G
2 1 F
F
@fnurm >.10 Arma kjjrefhatms
Ucm armas A F , A L , ahe A I kah gm `juhe usfhn Mquatfjh (>.>5) ahe (>.>5) ahe arm nfvmh gmdjw. A F 2
x L ‛ x y I ‛ y ‛ x I ‛ x y L ‛ y 7
A I 2
x F ‛ x y L ‛ y ‛ x L ‛ x y F ‛ y 7
x I ‛ x y F ‛ y ‛ x F ‛ x y I ‛ y 7
(>.>>g)
x L ‛ x F y I ‛ y F ‛ x I ‛ x F y L ‛ y F 7
(>.>>k)
A L 2 A 2
Ucm emrfvatfvms j` tcm arma kjjrefhatms arm
D F 2 x
y LI ------7 A
D F ‛ x LI 2 --------- y 7 A
wc m rm
x LI 2 x L ‛ x I ahe y LI 2 y L ‛ y I
(>.>3a)
D L 2 x
y --- IF ---7 A
D L ‛ x IF 2 --------- y 7 A
wcmrm
x IF 2 x I ‛ x F ahe y IF 2 y I ‛ y F
(>.>3g)
x FL 2 x F ‛ x L ahe y FL 2 y F ‛ y L
(>.>3k)
D I 2 x
y FL -----7 A
D I ‛ x FL 2 -------- y 7 A
wcmrm
@fnurm >.1> scjws >.1> scjws dfhmar, quaeratfk ahe kugfk trfahnudar mdmbmhts. Ucm fhtmrpjdatfjh `uhktfjhs kah gm wrfttmh usfhn tcm prjpmrty tcat tcm vadum j` Danrahnm pjdyhjbfad at fts jwh hjem fs 1 ahe at jtcmr hjems ft fs ?. Ucm Danrahnm pjdyhjbfads `jr dfhmar ahe quaeratfk arm nfvmh gmdjw ahe tcm kugfk pjdyhjbfad fs dm`t tj tcm rmaemr tj jgtafh (smm _rjgdmb >.7>). Fh @fnurm >.1>, >.1>, wm assubm add hjems arm mquaddy spakme jh tcm trfahndm. Dfhmar
1
D F
7
2 D L
;
2 D I
(>.>6a)
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
@fhftm mdmbmht bmtcje
16;
1 1 2 D F 7 D F ‛ 1 ; 2 D L 7 D L ‛ 1 = 2 D I 7 D I ‛ 1 Quaeratfk (>.>6g) 7 2 5 D D 2 5 D D 2 5 D D 7 F L 5 L I 0 I F ; 5 I I I = > = ; 0 0 5 > = 3 ; L 7 L 5 L 3 0 1? ; 6 7 6 7 1? 11 1 1 1 17 F F F (a) (g) (k) 1; 15 1= @fnurm >.1> (a) Dfhmar, (g) Quaeratfk ahe (K) Kugfk trfahnudar mdmbmhts. 10 Ucm `jddjwfhn femhtftfms (WmeeyY166;R) wfdd gm usm`ud fh kadkudatfjh j` mdmbmht stf``hmss batrfx ahe tcm rfnct cahe sfem 1> vmktjr. 13 g b! h! p! p! b! h! p! p! b h p b h p 16 (>.3?) D F D L D I e x e y 2 7 A --------------------------------------D F D L D I e s 2 g ‛ a -------------------------------------- b + h + p + 7 ! b + h + p + 1 ! a 7? A 71 77 >.1;.= Mdmbmht stf``hmss batrfx ahe rfnct cahe sfem vmktjr 7; (>.00a) tj `fhe tcm tmrbs fh tcm stf``hmss batrfx ahe rfnct cahe sfem vmktjr as wm efe fh Waydmfnc-Wftz wftc 75 ^m kah usm Mquatfjh (>.00a) tj 7= ` l 2 l tj jgtafh (>.31) I li 2 G l i W l 2 d l 70 7> Ucm usm j` tcm agjvm mquatfjh wfdd gm embjhstratme fh Mxabpdm >.6. >.6. Cjwmvmr, wcmh a hjem cas bjrm tcah jhm emnrmm j` 73 `rmmejb, tcmh tcm apprjakc kausms sjbm ef``fkudtfms. Jhm way j` jvmrkjbfhn tcfs ef``fkudty fs tj kast tcm gfdfhmar `uhktfjhad as a 76 prjeukt j` smvmrad batrfkms―a kjbbjh praktfkm fh `fhftm mdmbmht bmtcje. ^m ^m wfdd cjwmvmr usm ah adtmrhatfvm apprjakc gasme jh ;? tcm fema j` varfatfjh. m U 1 7 h U ;1 ^m rmprmsmht tcm mdmbmht efspdakmbmht vmktjr as u 2 u u u , tcmh tcm mdmbmht pjtmhtfad ;7 mhmrny kah gm wrfttmh as ;; 1 m U m m m m m U m ;5 (>.37a) ] 2 -- u I u ^ 2 W u m 2 ] m ‛ ^ m 7 ;= ;0 Gy taifhn tcm `frst ahe smkjhe varfatfjh j` strafh mhmrny ahe hjtfhn tcat tcm stf``hmss batrfx fs sybbmtrfk wm jgtafh ;> 1 1 m U m m U m m m U m m m m U m m U m m ;3 ] 2 -- u I u + u I u 2 -- u I + I u 2 u I u (>.37g) 7 7 ;6 h h U 5? (>.37k) 7 ] m 2 u m I m u m 2 u m I fl u m f 1 l 1 51 57 ^m kah adsj wrftm 5; 7 m h h ] 7 m 55 (>.37e) ] 2 u f u l m l f 1 l 1 u 5= u m 50 Mquatfhn Mquatfjhs (>.37k) ahe (>.37k) ahe (>.37e) ahe hjtfhn u f arm fhempmhemht varfagdms wm jgtafh 5> 7 m 53 ] (>.3;a) I fl 2 56 f l u u =? =1 Fh a sfbfdar bahhmr gy taifhn tcm `frst varfatfjh j` wjri wm kah scjw =7 ^ m (>.3;g) W 2 f =; f u =5 >.1? embjhstratms tcm usm j` Mquatfjhs (>.3;a) ahe (>.3;a) ahe Mquatfjhs (>.3;g). (>.3;g). == Mxabpdm >.1? embjhstratms =0 => >.1;.0 Danrahnm pjdyhjbfads fh tcrmm efbmhsfjhs =3 Ucm tmtracmerjh mdmbmht fs tcm sfbpdmst mdmbmht tcat kah gm usme `jr bjemdfhn rmnfjhs wftc kurvme sur`akms. A dfhmar rmprmsmhta=6 tfjh j` nmhmradfzme kjjrefhatm fs scjwh gmdjw. 0? (>.35) u x 2 K ? + K 1 x + K 7 y + K 7 z 01
165
>
Parfatfjhad ahe Mhmrny Bmtcjes F
D
_ P F L
I @fnurm >.13 Pjdubm Kjjrefhatms
Ucm kjhstahts K‖ s fh Mquatfjh (>.35) kah (>.35) kah gm `juhe fh tmrbs j` tcm hjead vadums j` u. Ucm `uhktfjhs budtfpdyfhn tcm hjead vadums j` u wjude gm tcm Danrahnm pjdyhjbfads fh tcrmm efbmhsfjhs. Jhkm bjrm wm wfdd emvmdjp tcm Danrahnm pjdyhjbfads fh tcrmm efbmhsfjhad haturad kjjrefhatms kaddme vjdubm kjjrefhatms. @fnurm >.13 scjws >.13 scjws a tmtracmerjh wftc pjfht _ pjfht _ wcjsm kjjrefhatms arm ( x, x, y, z ). ). Ucm vjdubm j` tcm `jur tmtracmerjh `jrbme gy ljfhfhn pjfht _ pjfht _ tj tj tcm vmrtfkms kah gm `juhe. Ucm vjdubm j` tcm tmtracmerjh jppjsftm tj tcm vmrtmx F vmrtmx F fs dagmdme as P F ahe fs scjwh fh @fnurm >.13. >.13. Fh a sfbfdar bahhmr tcm jtcmr tmtracmerjh vjdubms kah gm dagmdme as P L , P I , ahe P D. Ucm vjdubm P F wfdd gm tcm sabm frrmspmktfvm j` tcm djkatfjh j` pjfht _ pjfht _ jh jh a sur`akm paraddmd tj tcm sur`akm LID sur`akm LID.. Ucfs uhfqum vadum j` arma P F baims tcm vjdubm kjjrefhatms pjssfgdm. ^m hjh-efbmhsfjhadfzm tcm vjdubm j` tcm `jur tmtracmerjh wftc tcm tjtad vjdubm j` tcm mhtfrm tmtracmerjh tj jgtafh tcm vjdubm kjjrefhatms scjwh fh Mquatfjh (>.3=a). (>.3=a). Hjtm jhdy tcrmm j` tcm vjdubm kjjrefhatms arm fhempmhemht. P D F 2 ---- F P
P L D L 2 ----P
P I D I 2 -----P
P D D 2 ---- D -P
P 2 P F + P L + P I + P D
D F + D L + D I + D D 2 1
(>.3=a)
Uj rmdatm vjdubm j` makc tmtracmerjh tj tcm kjjrefhatms wm kjhsfemr tcm ; vmktjrs `rjb pjfht _ pjfht _ tj tj tcrmm vmrtfkms ahe usm Mquatfjh (>.>=). (>.>=). Ucus P fs nfvmh gy Mquatfjh (>.3=g). (>.3=g) . F 1 P F 2 -0
x L ‛ x y L ‛ y z L ‛ z x I ‛ x y I ‛ y z I ‛ z x D ‛ x y D ‛ y z D ‛ z
(>.3=g)
Jtcmr vjdubms kah gm wrfttmh fh a sfbfdar bahhmr. Ucm Danrahnm pjdyhjbfads arm sfbfdar tj tcjsm wm saw wftc arma kjjrefhatms, wftc tcm ef``mrmhkm tcat hubgmr fhkrmasms eum tj tcm `jurtc vmrtmx D vmrtmx D.. Ucus tcm `jr dfhmar Danrahnm wm cavm Dfhmar
1
D F
7
2 D L
;
2 D I
5
2 D
(>.3=k)
Fh a sfbfdar bahhmr cfncmr jremr pjdyhjbfads kah gm kjhstruktme. Ucm `jddjwfhn femhtfty (\fmhifmwfkz ahe Uaydjr Y1636R) adjhn wftc femhtftfms fh Mquatfjh (>.3?) wfdd (>.3?) wfdd gm usm`ud fh kadkudatfjh j` mdmbmht stf``hmss batrfx ahe tcm rfnct cahe sfem vmktjr. b! h! p! p! q!
D D D D e xeye z 2 0P -----------------------------------------------b + h + p + q + ; ! b h p q F L I D
(>.30)
U
Ucmrm arm hj aeeftfjhad hmw kjhkmpts fh fb pdmbmhtatfjh fh tcrmm efbmhsfjhs jvmr tcjsm fh twj efbmhsfjhs. Cjwmvmr, tcm adnmgra fh fbpdmbmhtatfjh fh tcrmm efbmhsfjhs gmkjbms sfnhf`fkahtdy bjrm. Cmhkm, wm wfdd hjt kjhsfemr tcfs tjpfk ahy bjrm.
MSAB_DM >.6 Jgtafh tcm mdmbmht stf``hmss batrfx ahe tcm mdmbmht rfnct cahe sfem vmktjr `jr ah axfad bmbgmr usfhn quaeratfk Danrahnm pjdyhjbfads (@fnurm (@fnurm >.1;g >.1;g). Apprjxfbatm tcm axfad rfnfedy MA rfnfedy MA ahe ahe efstrfgutme djae p djae p x wftc kjhstaht vadums tcat mxfsts at tcm kmhtmr j` tcm axfad rje. Adsj assubm tcm dm`t ahe rfnct mhe hjems cavm a kjhkmhtratme axfad `jrkms @ `jrkms @ Y1R ahe @ ahe @ Y;R, rmspmktfvmdy.
AH[.
>.15
I m
> ‛ 3
1
MA 2 ------- ‛ 3 10 ‛ 3 ; D 1 ‛ 3 >
W m
2
1 @ 1 p x D -------5 + ? 0 ; 1 @
KDJ[]WM
Fh tcfs kcaptmr wm mstagdfscme tcm prfhkfpads j` varfatfjhad kadkudus tcat kah gm usme fh jgtafhfhn gjuheary vadum prjgdmbs ahe sjdutfjh gy apprjxfbatm bmtcje j` Waydmfnc-Wftz. ^m saw tcat pjtmhtfad mhmrny fs tcm `uhktfjhad tcat fs bfhfbfzme usfhn varfatfjhad kadkudus. ^cmh tcm kjheftfjhs `jr bfhfbfzatfjh arm satfs`fme at makc ahe mvmry pjfht jh tcm gjey, fhkduefhn tcm gjuheary
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
Kdjsurm
16=
1 MSAB_DM >.1? 7 Ah mdastfk pdahm j` tcfkihmss c fs fh pdahm strmss ahe suglmktme tj gjey `jrkms @ x x y ahe @ y x y . @jr a trfahnudar mdmbmht, ; usfhn dfhmar Danrahnm pjdyhjbfad, jgtafh tcm `frst rjw j` tcm mdmbmht stf``hmss batrfx ahe tcm rfnct cahe sfem vmktjr. Apprjxf5 batm tcm gjey `jrkms, tcm batmrfad kjhstaht, ahe tcfkihmss wftc kjhstaht vadums tcat mxfsts at tcm kmhtrjfe j` tcm trfahndm. = 7 7 0 y LI 7 1 ‛ ‛ x LI 7 Mc 7 y LI + 1 ‛ x LI Mc ---------------------- ---------AH[. I 11 2 ------------------> 7 7 7 A + 7 7 A ex e y 2 -----------------------------------------------------1 ‛ 3 A 1 ‛ 3 A 6 y LI ‛ x LI 1 ‛ ‛ x LI y LI ‛ M c 1 + x LI y LI Mc ---------- + ---------------- ---------- ------- ex e y 2 ------------------------------------------AH[. I 17 2 ------------------ ------1? 7 7 7 A 7 A 7 7 A 7 A 1 ‛ A 3 A 1 ‛ 11 y LI y IF 1 ‛ ‛ x LI ‛ x IF Mc 7 y LI y IF + 1 ‛ x LI x IF 17 Mc ------------- + ---------------- ---------- ---------- ex e y 2 --------------------------------------------------------------------AH[. I 1; 2 ------------------7 7 1; 7 7 A 7 A 1 ‛ A 7 A 7 A 3 A 1 ‛ 15 y LI ‛ x IF 1 ‛ ‛ x LI y IF ‛ Mc 7 y LI x IF + 1 ‛ x LI y IF Mc 1= ---------- + ---------------- ---------- ------- ex e y 2 ---------------------------------------------------------------------------AH[. I 15 2 ------------------ ------7 7 7 A 7 A 7 7 A 7 A 10 1 ‛ A 3 A 1 ‛ 1> y LI y FL 1 ‛ ‛ x LI ‛ x FL Mc 7 y LI y FL + 1 ‛ x LI x FL Mc ------13 ------- + ---------------- ---------- --------- ex e y 2 ------------------------------------------------------------------AH[. I 1= 2 ------------------7 7 7 A 7 A 1 ‛ 7 A 7 A 7 A 3 A 1 ‛ 16 7? y LI ‛ x FL 1 ‛ ‛ x LI y FL ‛ Mc 7 y LI x FL + 1 ‛ x LI y FL Mc --------- + ---------------- ---------- ------- ex e y 2 -------------------------------------------------------------------------AH[. I 10 2 ------------------ ------71 7 7 7 A 7 A 7 7 A 7 A 1 ‛ A 3 A 1 ‛ 77 7; Ac @ x 7 A ^ m 2 c@ u e x e y 2 c@ D ex e y 2 c@ ------ 2 ----------- W 2 AH[. f x x F x 75 1 1 ;! ; u u A A 7= 70 7> pjfhts, tcmh wm jgtafh tcm gjuheary vadum prjgdmbs. ^cmh tcm kjheftfjhs `jr bfhfbfzatfjh arm satfs`fme fh ah avmranm smhsm jvmr 73 tcm mhtfrm gjey tcmh wm jgtafh tcm apprjxfbatm sjdutfjh gy Waydmfnc-Wftz. @fhftm mdmbmht bmtcje fh bmkcahfks j` batmrfads kah 76 gm smmh as ah appdfkatfjh j` Waydmfnc-Wftz fh wcfkc tcm ifhmbatfkaddy aebfssfgdm `uhktfjhs arm pjdyhjbfad apprjxfbatfjhs tcat ;? arm pfmkmwfsm kjhtfhujus jvmr mdmbmhts tcat arm jgtafhme gy efskrmtfzatfjh j` tcm mhtfrm gjey. Bfhfbub pjtmhtfad mhmrny rmpdakms tcm bjeudm j` mqufdfgrfub fh tcm djnfk empfktme fh@fnurm fh @fnurm 7.>. 7.>. ^m stfdd hmme tj jgtafh ;1 ;7 strafhs `rjb jur ifhmbatfk apprjxfbatfjhs ahe tcm kjrrmspjhefhn strmssms `rjb jur apprjxfbatfjh j` batmrfad bjemd. [ugstftut;; fhn tcm strafhs ahe strmssms fhtj strafh mhmrny emhsfty ahe tcmh fhtmnratfhn jvmr tcm krjss-smktfjhad armas `jr jhm-efbmhsfjhad ;5 strukturad bmbgmrs ahe jvmr tcm tcfkihmss `jr twj-efbmhsfjhad gjefms, wm jgtafhme tcm strafh mhmrny `jr strukturad bmbgmrs ahe ;= mdastfk gjey. Jgtafhfhn tcm wjri pjtmhtfad was rmdatfvmdy strafnct `jrware nfvfhn us tcm pjtmhtfad mhmrny `jr tcm strukturad bmb;0 gmrs, wcfkc kjude gm tcm bfhfbfzme tj jgtafh mftcmr tcm gjuheary vadum prjgdmb jr ah apprjxfbatm sjdutfjh. Jgtafhfhn pjtmhtfad 7 ahe Kcaptmr = fs = fs rmdatfvmdy strafnct `jrware. Gut fh kasm j` nmjbmtrfk hjh;> mhmrny wftc add tcm kjbpdmxftfms emskrfgme fh Kcaptmr 7 ahe ;3 dfhmarfty, varfatfjhad kadkudus apprjakc fs sfnhf`fkahtdy sfbpdmr tcah tcm ef``mrmhtfad kadkudus apprjakc `jr jgtafhfhn gjuheary ;6 vadum prjgdmbs. ^m embjhstratme tcfs `jr gmab gmhefhn prjgdmb fh tcfs kcaptmr ahe wfdd smm aeeftfjhad embjhstratfjhs fh tcm 5? hmxt kcaptmr `jr twj efbmhsfjhad prjgdmbs wftc nmjbmtrfk hjhdfhmarftfms. ^m saw tcat jgtafhfhn gjuheary vadum prjgdmbs usfhn varfatfjhad kadkudus fs mdmnaht ahe tcm emrfvatfjh fs kjbpakt. Fh 51 3, tcm pjtmhtfad mhmrny ahe tcm bfhfbfzfhn `jrbudas wfdd gm kast usfhn fhefkfad hjtatfjh. Ucm fhefkfad hjtatfjh wfdd baim 57 Kcaptmr 3, 5; tcm emrfvatfjhs j` gjuheary vadum prjgdmbs mvmh bjrm kjbpakt ahe bjrm mdmnaht. 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
160
>
>.1=
Parfatfjhad ahe Mhmrny Bmtcjes
[VHJ_[F[ J@ MQ]AUFJH[ ^jri _jtmhtfad
Axfad
^ A
Ujrsfjh Ujrsfjh j` kfrkudar sca`ts
^ U
[ybbmtrfk gmhefhn j` gmabs
^ G
p x x u x e x
t x x e x
D
_dahm strmss mdastfkfty
^ M
q1
@ q u x q
U q x q
q1
du
d
b1
ev
b7
p x v x e x @ v x B e x x d v 2 p x y w x y ex e y 2 d w 2 c @ x y u x y + @ x y v x y ex e y 2 d u v y
D
^ _
b
b
D
Gmhefhn j` tcfh pdatms
q
q 1
q
q
q 1
q
A z A
x
y
[trafh Mhmrny eu 7 MA ------ e ex x D
Axfad
] A
1 -7
Ujrsfjh Ujrsfjh j` kfrkudar sca`ts
] U
1 -7
[ybbmtrfk gmhefhn j` gmabs
] G
1 -7
D
e 7 NL ------ e ex x 7
e v MF zz -------- e x D ex 7 7
7
w x7 A
Ucfh _datms
E ] _ 2 ---7
_dahm [trmss Mdastfkfty
Mc ] M 2 ---------------------7 7 1 ‛
7
7
w + y7
7
7
7
u 7 u v x + 7 x y
A
7
w w + 7 x 7 y7
7
w + 7 1 ‛ e x e y x y
v 7 --------------- 1 ‛ u + v 7 y 7 y x
+
ex e y
Gfdfhmar `uhktfjhad `jrb j` strafh mhmrny eu 1 eu 7
Axfad
] A
1 -7
MA
e x
1 -- G u 1 u 7 7
Ujrsfjh Ujrsfjh j` kfrkudar sca`ts
] U
1 -7
e 1 e 7 NL e x e x e x
1 -- G 1 7 7
[ybbmtrfk gmhefhn j` gmabs
] G
1 -7
e v1 e v7 MF zz ----------- ----------- e x 7 7 e x e x
Ucfh pdatms
E ] _ 2 ---7
D
D
e x e x
7
D
7
_dahm strmss mdastfkfty
A
7
7
1 -- G v 1 v 7 7
7
7
w 1 w 7 w 1 w 7 7
x x
Mc ] M 2 ---------------------7 7 1 ‛
+
7
7
y y
7
7
7
7
u 1 u7 u 7 v 1 u 1 v 7 x x + x y + x y
A
7
w 1 w 7 w 7 w 1 + + x 7 y7 x7 y 7 +
7
7
w 1 w 7 + 7 1 ‛ x y x y
1 ex e y 2 -- G w1 w 7 7
v 1 v 7 1 ‛ u1 v 1 u 7 v 7 + ---------------- + 7 y y y x y x
1 2 -- G u 1 v1 u 7 v 7 7
[tatfjhary vadum j` dfhm Fhtmnrad @uhktfjhad Ef``mrmhtfad Mquatfjh Gjuheary Kjheftfjhs at at x at x 2 a ahe at x 2 g
F u 2
g
a
C u
r
C ‛ e C ? e x 1 u u C u 1
‛
u r ‛ 1 +
e e x
u 7 7
r ‛ 1
u 1 u ? x
C e ; C 7 ‛ ; ; u e x u
+
7
e C e C + ------ ‛ --------- + ex u 7 ex 7 u ;
e x
+ ‛ 1
r
e x
r
C e r e + ‛ 1 ------------------ 2 ? r ‛ 1 u r ex
C e r e + ‛ 1 -----------------2 ? r ‛ 7 r u ex
C u ;
+
C e r e + ‛ 1 ------------------ 2 ? r ‛ ; u r ex
A
u , y y u x y ex e y
, x x
jr
u
1
jr
u
7
r ‛ ;
2 ?
2 ?
2 ?
2 ?
jr
2 ?
2 ?
jr
2 ?
2 ?
jr
2 ?
C 2 ? r u
jr
[tatfjhary vadum j` ah arma fhtmnrad wftc `frst jremr j` emrfvatfvms.
Cu
u
jr
r ‛ 7
+
F u 2
2 ?
r ‛ 1
C e C 7 ‛ ; u e x u ‛
@uhktfjhad
C r u
r
e
u
r ‛ 1
2 ?
ex e y
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
[yhjpsfs j` mquatfjhs 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
Ef``mrmhtfad Mquatfjh
C ‛ C ‛ C u x u , x x y u, y y
Gjuheary Kjheftfjhs
C h + C h u , x x x u , y y y
16>
x y fh A
2 ?
2 ?
u
jr
x y jh
2 ?
[tatfjhary vadum j` ah arma fhtmnrad wftc smkjhe jremr j` emrfvatfvms
Cu
F u 2
@uhktfjhad
, xx xx
u, xy u, x x u , y y u x y ex e y xy u , yx yx u , yy yy
A
Hjtatfjh
C h 7 + C + C h h + C h 7 u, xx x u , xy x y u , yy y u , yx xx yy xy yx
b hh 2
C ‛ C h h + h7 C ‛ h 7 C u , xx x y x u, yx y u , xy u, yy yx xy xx yy
b ht 2 ‛ q x 2
C ‛ C ‛ C u , x x x u, xx y u , xy xx xy
q y 2
C ‛ C ‛ C u , y y x u, yx y u , yy yx yy
q h 2 q x h x + q y h y
bht s
P h 2 q h +
7
7
h x ‛ h y C ‛ C h h + C h 7 ‛ C h 7 ‛ C ‛ C --------------- C + C -7---h--- x----h-- y ‛ x y x y Wkur W kur s u , xx u, yy s u , yx s u , xy u, xx u, xy u, yx yx xy xx yy xx u , yy yy xy yx
P h 2 q h ‛
q x q y C + + x y u
Ef``mrmhtfad mquatfjh Gjuheary Kjheftfjhs
b hh 2 ?
u 2 h u K
jr
Kjrhmr kjheftfjh
b ht K 2
_jtmhtfad mhmrny
x y fh A
2 ?
?
jr
?
P h
ahe x y jh kjrhmr
2 ?
u
jr
2 ?
x y jh
K
1 ] ‛ ^ -- G u u ‛ d u 7
Waydmfnc-Wftz Apprjxfbatfjh
u
h
l
K ` 1 l l
I K
Adnmgrafk mquatfjhs
*
_jtmhtfad mhmrny at mqufdfgrfub.
1 2 ‛ -7
W
2
l
W l 2 d ` l
1 * U * K l W l 2 ‛ -- K W 7
h
I li 2 G ` l ` i
wcmrm
1
@fhftm Mdmbmht Bmtcje _jtmhtfad Mhmrny Apprjxfbatfjh
f
h
u u
2
f
h f
[tf``hmss Batrfx ahe Wfnct cahe I li 2 G sfem vmktjr Danrahnm pjdyhjbfads fh 1-E
Dfhmar
usfhn haturad kjjrefhatms
Quaeratfk
Fhtmnratfjh `jrbudas
f
1
] f ‛ ^ f
f 7
] m l i 2 l i u u 1 1 ‛ 1
W l 2 d 7
2
l
^ m l u
2 1 ‛ 1 ‛ 7
D L 2 A L A
2
7
D I 2 A I A
2 5 1 ‛
x L ‛ x y I ‛ y ‛ x I ‛ x y L ‛ y 7
A L 2
A I 2
x F ‛ x y L ‛ y ‛ x L ‛ x y F ‛ y 7
A 2
1
Quaeratfk b h p F L I e x e y
D D D A
D F
7
2 D L
;
;
2 ‛ 1 ‛ 7
D F + D L + D I 2 1
A F 2
Dfhmar Danrahnm pjdyhjbfads fh 7-E usfhn arma kjjrefhatms
h
f
1
D F 2 A F A Arma kjjrefhatms
2
1
x I ‛ x y F ‛ y ‛ x F ‛ x y I ‛ y 7
x L ‛ x F y I ‛ y F ‛ x I ‛ x F y L ‛ y F 7
2 D I
1
2 D F 7 D F ‛ 1
;
2 D L 7 D L ‛ 1
=
2 D I 7 D I ‛ 1
7
2 5 D F D L
5
2 5 D L D I
0
2 5 D I D F
b! h! p! p! 2 7 A -------------------------------------- b + h + p + 7 !
g
b h p F L I
D D D a
b! h! p! p! e s 2 g ‛ a -------------------------------------- b + h + p + 1 !
3
Fhefkfad Hjtatfjh
DMAWHFHN JGLMKUFPM[ 1 1. ]hemrstahe fhefkfad hjtatfjh ahe tcm rudms j` bahfpudatfhn tcm fhefkms. 7 7. ]hemrstahe tcm appdfkatfjhs j` fhefkfad hjtatfjh tj bmkcahfks j` batmrfads mquatfjhs. ; 5 Fhefkfad hjtatfjh fs a kjbpakt way j` wrftfhn ahe emrfvfhn mquatfjhs. ^rftfhn a efrmktfjh j` a vmktjr jr kjbpjhmhts j` a batrfx = usfhn sugskrfpts fs usfhn fhefkfad hjtatfjh fh fts bjst gasfk `jrb. Gut bahfpudatfhn tcmsm sugskrfpts jr fhefkms tj emrfvm mquatfjhs 0 rmqufrms dmarhfhn rudms tcat cavm mvjdvme jvmr a pmrfje j` tfbm. Jhkm dmarhme, ft gmkjbms a kcaddmhnm tj wjri wftc mquatfjhs fh > traeftfjhad `jrb j` mxpaheme hjtatfjh. Fhefkfad hjtatfjh adsj `jrbs tcm gasfk dahnuanm j` tmhsjr kadkudus. 3 Fh tcfs kcaptmr wm `frst mstagdfsc tcm hjtatfjh ahe tcm rudms `jr bahfpudatfhn tcm fhefkms. ^m kast ahe emrfvm tcm `abfdfar 6 mquatfjhs j` kdassfkad pdatm tcmjry j` Kcaptmr = ahe = ahe mquatfjhs j` mdastfkfty j` Kcaptmr 0 fh 0 fh fhefkfad hjtatfjh. ^m tcmh kast twj 1? efbmhsfjhad mquatfjhs j` varfatfjhad kadkudus j` Kcaptmr > fh > fh fhefkfad hjtatfjh ahe usm tcm mquatfjhs tj emrfvm gjuheary vadum 11 prjgdmbs. ^m tcmh stuey darnm strafh ahe fhkjrpjratm ft fh pdatm tcmjry ahe mdastfkfty. 17 1; 3.1 GA[FK EM@FHFUFJH[ 15 1= ^m start wftc twj `abfdfar kjhkmpts j` wjri ahe bjbmht ahe scjw tcm kjbpakthmss j` tcm mquatfjhs tcat kah gm akcfmvme usfhn 10 fhefkfad hjtatfjh. 1> A `jrkm vmktjr ahe efspdakmbmht vmktjr fh Kartmsfah kjjrefhatms bay gm wrfttmh as @ 2 @ x @ y @ z ahe 13 (^ ) gy tcm `jrkm @ fh bjvfhn a pjfht gy ah abjuht u fs nfvmh gy tcm skadar (ejt) u 2 u x u y u z . Ucm wjri ejhm (^ 16 prjeukt scjwh gmdjw. 7? 71 (3.1a) ^ 2 @ u 2 @ x u x + @ y u y + @ z u z 77 Ucm bjbmht vmktjr B 2 B x B y B z eum tj tcm `jrkm at a pjfht `rjb wcfkc a pjsftfjh vmktjr fs erawh 7; r 2 r x r y r z fs nfvmh gy tcm vmktjr (krjss) prjeukt as 75 7= B 2 r @ 2 r y @ z ‛ r z @ y r z @ x ‛ r x @ z r x @ y ‛ r y @ x (3.1g) 70 Ucm agjvm mquatfjhs arm safe tj gm wrfttmh fh mxpa heme `jrb. Fh fhefkfad hjtatfjh wm rmpdakm tcm sugskrfpts x sugskrfpts x,, y, y, z wftc wftc 1, 7> 7, ahe ;, rmspmktfvmdy. Ucm agjvm mquatfjhs kah gm wrfttmh as 73 76 (3.1k) ^ 2 @ f u f f 21 ;? (3.1e) B 1 2 r 7 @ ; ‛ r ; @ 1 B 7 2 r ; @ 1 ‛ r 1 @ ; B ; 2 r 1 @ 7 ‛ r 7 @ 1 ;1 ;7 ;; 3.1.1 [ubbatfjh kjhvmhtfjh ;5 Fh tcm Mquatfjh (3.1k), (3.1k), f kjude gm rmpdakme gy l gy l jr jr i ahe ahe ft wjude hjt battmr. Ucus, ah fhemx usme `jr purpjsm j` subbatfjh fs ;= kaddme tcm eubby fhemx. A `urtcmr kjbpaktfjh j` hjtatfjh fs akcfmvme gy hjt wrftfhn tcm subbatfjh sfnh wftc tcm uhemrstahefhn ;0 tcat a rmpmatme fhemx fbpdfms subbatfjh. subbatfjh. ^ftc tcfs uhemrstahefhn, Mquatfjh (3.1k) fs (3.1k) fs wrfttmh as ;> (3.7) ^ 2 @ f u f ;3 @urtcmrbjrm, wm wfdd wrftm a vmktjr lust as @ f jr @ l jr @ i . Ucm fhefkms f, l, jr i arm rm`mrrme tj as `rmm fhemx. Jhm j` tcm ;6 tcm `rmm fhefkms fh mvmry tmrb j` ah mquatfjh bust gm tcm sabm. sabm. Ucfs fs ahadjnjus tj tcm 5? rudms j` bahfpudatfjh fh fhefkms fs tcat tcm `rmm 51 rmqufrmbmht tcat mvmry tmrb fh ah mquatfjh bust cavm tcm sabm efbmhsfjh, ahe fs a vmry usm`ud rudm fh prmvmhtfhn bfstaims fh tcm 57 emrfvatfjhs usfhn fhefkfad hjtatfjh. 5;
1;;
1;5
Fhefkfad Hjtatfjh
3
[jbmtfbms tcmrm arm mxkmptfjhs baem tj tcm rudm j` `rmm ahe eubby fhemx usanm. Fh sukc kasms ft fs mxpdfkftdy emkdarme mxpdfkftdy emkdarme tcat ah fhemx fs hjt a `rmm fhemx jr subbatfjh fs hjt karrfme jvmr a eubby fhemx. A quahtfty wftc budtfpdm fhefkms wfdd gm rm`mrrme tj as kjbpjhmht j` a batrfx, wcfkc fs bjrm `abfdfar tcmh kaddfhn tcmb tmhsjrs. Ucus, A fl ahe A fl i arm batrfkms. Ucm sfzm j` tcm batrfx wfdd gm emtmrbfhme gy tcm vadums makc j` tcm fhefkms kah taim. F`f F`f, l, l, ahe i kah kah jhdy gm 1 ( x) x) ahe 7 ( y) y) vadums, tcmh A fl wfdd gm 7 x7 batrfx ahe A fl i wfdd gm 7 x7 x7 batrfx. F` f, l, l, ahe i kah kah gm 1 ( x) x) 7 ( y) y) ahe ; ( z z ) vadums, tcmh A fl wfdd gm ; x ; batrfx ahe A fl i wfdd gm ; x ; x ; batrfx. ^m rmkjre tcm twj rudms `jr `uturm usm. • rmpmatme fhemx fbpdfms subbatfjh uhdmss subbatfjh uhdmss ft fs mxpdfkftdy statme jtcmrwfsm. • `rmm fhefkms fh mvmry tmrb j` ah mquatfjh bust gm tcm sabm uhdmss sabm uhdmss mxpdfkftdy statme tcat ah fhemx fs hjt `rmm ahe fs gmfhn usme sybgjdfkaddy. A `rmm fhemx kah gm rmpdakme gy ahjtcmr `rmm fhemx ahe a eubby fhemx kah gm r mpdakme gy ahjtcmr eubby fhemx. As wm scadd smm tcfs fs ejhm fh tcm sabm mquatfjh `rjb jhm stmp tj hmxt eurfhn tcm adnmgrafk bahfpudatfjh j` mquatfjhs.
3.1.7
Irjhmkimr `uhktfjh
Irjhmkimr emdta `uhktfjh jr lust emdta `uhktfjh fs em`fhme as
fl
1 ?
2
f 2 l
(3.;a)
fl
Ucus, Irjhmkimr emdta `uhktfjh fs ah femhtfty batrfx. Kjhsfemr tcm prjeukt j` a vmktjr wftc Irjhmkimr emdta `uhktfjh,
fl @ l
2
f 1 @ 1 + f 7 @ 7 + f ; @ ; fl @ l
2
@ 1 @ 7 @ ;
f 2 1 f 2 7 jr f 2 ;
2 @ f
(3.;g)
Ucm budtfpdfkatfjh kcahnme tcm fhemx `rjb l `rjb l tj tj f. Ucfs sugstftutfjh j` fhemx fs mquaddy trum `jr a prjeukt j` Irjhmkimr emdta `uhktfjh wftc ahy batrfx kjbpjhmht. Ucus, fl A li 2 A fi ahe fl A li b 2 A fi b . Ahjtcmr usm`ud jgsmrvatfjh fs rmkjreme gmdjw
ff
2
11 + 77
2 7
fh twj efbmhsfjh
ff
ahe
2
11 + 77 + ;;
2 ;
fh tcrmm efbmhsfjh
(3.;k)
Ucm prjeukt j` tcm emdta `uhktfjh wftc a `uhktfjh tcat fs ahtf-sybbmtrfk fh fts sugskrfpt wfdd rmsudt fh a zmrj vadum as scjwh gmdjw.
li A li 3.1.;
2 ?
F@ A li 2 ‛ Ail
(3.;e)
_mrbutatfjh `uhktfjh
Ucm pmrbutatfjh `uhktfjh fs em`fhme as
m fl i 2
? 1 ‛ 1
f` ahy f l jr i arm arm sabm, f` fl i arm arm fh kykdfk jremr,
(3.5)
f` fl i arm arm hjhkykdfk jremr.
Gy kykdfk jremr wm cavm m 17; 2 m 7;1 2 m ;71 2 1 . Gy hjh-kykdfk jremr wm cavm m 71; 2 m ;71 2 m 1;7 2 ‛ 1 . Ucm bjbmht kjbpjhmhts fh Mquatfjh (3.1e) kah (3.1e) kah hjw gm wrfttmh fh tcm kjbpakt `jrb as B f 2 mfl i r l @ i
(3.=)
Fh tcm agjvm mquatfjh f mquatfjh f fs fs tcm `rmm fhemx ahe l ahe l ahe ahe i arm arm tcm eubby fhefkms jvmr wcfkc wm cavm tj sub. Kjhsfemr hjw B 1 . ^ftc f 2 1, tcm hjh-zmrj vadums j` pmrbutatfjh `uhktfjh wfdd gm `jr l `jr l ahe i wftc wftc vadums j` 7 ahe ;. ^m tcus jgtafh B 1 2 m17 ; r 7 @ ; + m 1;7 r ; @ 7 2 r 7 @ ; ‛ r ; @ 1 Fh a sfbfdar bahhmr wm kah scjw jtc mr kjbpjhmhts j` bjbmht `rjb Mquatfjh (0.0) arm sabm as tcjsm nfvmh fh Mquatfjh (3.1e). (3.1e). ^m em`fhm a twj-efbmhsfjhad pmrbutatfjh `uhktfjh as
mfl 2
? 1 ‛ 1
f` ahy f 2 l (3.0)
f` fl arm fh kykdfk jremr, f` fl arm hjhkykdfk jremr.
Ucm `jddjwfhn femhtftfms gmtwmmh pmrbutatfjh sybgjd ahe tcm emdta `uhktfjh kah gm prjvmh gy sugstftutfhn tcm vadums j` 1, 7, ahe ;. m fl i m fb h 2
lb ih ‛ lh ib
mib m hb 2
ib
(3.>)
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
Mquatfjhs j` mdastfkfty ahe tcfh pdatm fh fhefkfad 1 Ucm prjeukt j` tcm pmrbutatfjh `uhktfjh wftc a `uhktfjh tcat fs sybbmtrfk fh fts sugskrfpt wfdd rmsudt fh a zmrj vadum as 7 scjwh gmdjw. ; (3.3) m fl i A li 2 ? ahe m li A li 2 ? F@ A li 2 Ail 5 = 3.1.5 Emrfvatfvm hjtatfjh 0 > Ucm partfad emrfvatfvm j` a `uhktfjh wftc rmspmkt tj a kjjrefhatm fs scjwh gy a kjbba `jddjwme gy tcm fhemx j` tcm efrmktfjh as 3 scjwh gmdjw. 6 u f 1? (3.6) 2 u f l x l 11 17 ^m cavm tcm gasfk tjjds `jr usm j` fhefkfad hjtatfjh fh jur `abfdfar mquatfjhs. 1; 15 3.7 MQ]AUFJH[ J@ MDA[UFKFUV AHE UCFH _DAUM FH FHEFKFAD HJUAUFJH 1= = ahe mquatfjhs j` mdastfkfty j` Kcaptmr 0 fh 0 fh fhefkfad hjta10 Fh tcfs smktfjh wm kast tcm mquatfj hs j` kdassfkad pdatm tcmjry j` Kcaptmr = ahe = ahe Kcaptmr 0. 0. 1> tfjh. Ucm varfagdms usme fh tcm mquatfjhs wmrm em`fhme fh Kcaptmr = ahe Fh Kcaptmr 1, wm jgsmrvme tcat 13 16 tmhsj tmhsjrr hjrba hjrbadd straf strafhs hs 2 mhnf mhnfhm hmmr mrfhn fhn hjr hjrba badd straf strafhs hs tmhsjr scmar strafhs 2 mhnfhmmrfhn scmar strafhs 7 7? Fh tcfs kcaptmr wm wfdd usm tmhsjr strafh em`fhftfjh as em`fhftfjh as fs tcm nmhmrad kjhvmhtfjh fh fhefkfad hjtatfjh. Ucm strafh-efspdakm71 bmht rmdatfjhscfps j` Mquatfjh (0.1) `jr (0.1) `jr tmhsjr strafhs kah gm wrfttmh as 77 (3.1?a) fl 2 u f l + u l f 7 7; 75 Ucm Nmhmradfzme Cjjim‖s daw j` Mquatfjh (1.57) rmdatfhn (1.57) rmdatfhn strmssms (fl) tj tmhsjr strafh kah gm wrfttmh as 7= (3.1?g) fl 2 1 + fl ‛ fl ii M 70 7> Ucm mqufdfgrfub mquatfjhs rmdatfhn strmssms tj gjey `jrkms ( @ @ f) j` Mquatfjhs (0.3e) tcrjunc (0.3e) tcrjunc (0.3e) kah (0.3e) kah gm wrfttmh as 73 (3.1?k) fl l + @ f 2 ? 76 Ucm strmss vmktjr, tcat fs traktfjh j` Mquatfjh (1.1?), (1.1?), kah gm wrfttmh as ;? ;1 (3.1?e) [ f 2 fl h l ;7 Ucm strafh mhmrny emhsfty j` Mquatfjh (>.1;), (>.1;), usfhn tmhsjr strafh kah gm wrfttmh as ;; ;5 (3.1?m) ] ? fl fl 7 ;= ;0 MSAB_DM 3.1 ;> ;3 [tartfhn wftc Nmhmradfzme Cjjim‖s daw, jgtafh strmssms fh tmrbs j` strafh fh fhefkfad hjtatfjh fh tcrmm efbmhsfjhs. ;6 M M AH[. fl 2 ----------------- fl + -------------------------------------- fl ii 2 7 N fl + -------------------- fl ii (3.11a) 5? 1 + 1 ‛ 7 1 + 1 ‛ 7 51 57 MSAB_DM 3.7 5; 55 [tartfhn wftc tcm efspdakmbmht mquatfjh ahe assubfhn add assubptfjhs arm vadfe fh kdassfkad pdatm tcmjry, jgtafh mquatfjhs fh 5= fhefkfad hjtatfjh `jr strafh, strmssms, strmss rmsudtahts, mqufdfgrfub mquatfjhs, ef``mrmhtfad mquatfjh, ahe strafh mhmrny. 50 5> MSAB_DM 3.; 53 Jgtafh tcm ef``mrmhtfad mquatfjh njvmrhfhn efspdakmbmhts fh a tcrmm efbmhsfjhad mdastfk gjey suglmktme tj gjey `jrkms @ `jrkms @ f. 56 AH[. Nu f ii + N + u i if + @ f 2 ? =? =1 =7 3.; @]HEABMHUAD [JD]UFJH[ =; ;, wm fhtrjeukme tcm kjhkmpt j` `uheabmhtad sjdutfjhs assjkfatme wftc pjfht sfhnudarftfms. Fh mdastfkfty bahy `uhea=5 Fh Kcaptmr ;, bmhtad sjdutfjhs arm avafdagdm `jr twj ahe tcrm m efbmhsfjhad prjgdmbs. Ucmsm sjdutfjhs prjvfem us w ftc a rfkc smt j` prjgdmbs `jr == =0 uhemrstahefhn tcm usm j` fhefkfad hjtatfjhs fh emrfvatfjhs j` bmkcahfks prjgdmbs―tcm sjdm purpjsm j` fhtrjeukfhn tcm tjpfk fh => tcfs kcaptmr. Fh [mktfjhs ;.1.7 ahe ;.1.7 ahe ;.1.;, ;.1.;, wm saw tcat pjfht `jrkm was a sjurkm sfhnudarfty ahe pjfht bjbmht was a ejugdmt sfhnudar=3 =6 fty. Fh a sfbfdar bahhmr tcmrm fs a sjurkm sfhnudarfty kaddme efsdjkatfjh ahe a ejugdmt sf hnudarfty kaddme efspdakmbmht efskjhtfhufty. 3.1 scjws a efsdjkatfjh. Uwj aeljfhfhn pjfhts adjhn a dfhm efspdakm rmdatfvm tj jhm ahjtcmr fh a efsdjkatfjh. [fhkm efs0? @fnurm 3.1 scjws pdakmbmht fs a vmktjr, tcm fhtmhsfty j` efsdjkatfjh cas kjbpjhmhts wcfkc wm wfdd dagmd as E x aheE y jr fh fhefkfad hjtatfjh asE f . 01
1;0
3
Fhefkfad Hjtatfjh
^m kah cavm pjfht strafh sfhnudarfty kaddme fhkdusfjh sfhnudarfty ahe pjfht strmss sfhnudarfty kaddme fhcjbjnmhmfty sfhnudarfty. Fh makc kasm tcm efspdakmbmht `fmde eum tj tcm sfhnudarfty wfdd gm nfvmh ahe wm wfdd kadkudatm tcm strafh ahe strmss `rjb ft usfhn fhefkfad hjtatfjh.
@fnurm 3.1 Efsdjkatfjh sfhnudarfty.
@jr saim j` sfbpdfkfty, wm scadd assubm tcat tcm sfhnudarfty fs djkatme at tcm jrfnfh. Ucm `fmde pjfht `jr mvaduatfhn efs pdakmbmhts, strafhs, jr strmssms fs djkatme at a raefad efstahkm r wftc wftc kjbpjhmhts r f as scjwh fh @fnurm 3.7. 3.7. Ucm `jddjwfhn femhtftfms kah gm masfdy mstagdfscme. 7
r 2 r f r f
r f l 2
fl
r , l l 2 r l r
dh r , l l
7
2 r l r
(3.17a)
x7
r [jurkm _jfht
@fmde _jfht r 7
r 1
x1
`uheabmhtad sjdutfjhs. @fnurm 3.7 Hjtatfjh `jr usm fh `uheabmhtad
@rjb @fnurm 3.7, 3.7, wm kah wrftm tcm ahndm ahe fts emrfvatfvms wftc rmspmkt tj makc kjjrefhatm ahe tcmh wrftm tcm emrfvatfvms usfhn fhefkfad hjtatf jh as scjwh gmdjw.
r 7 2 atah ---- r 1
r 1
r 2 ‛ ----7 7 r
r 7
r 2 ---17 r
, l l
r b 2 m bl ----7r
(3.17g)
MSAB_DM 3.5 Ucm efspdakmbmht `fmde eum tj a efsdjkatfjh fh ah mdastfk gjey fh pdahm strafh fs nfvmh gy tcm mquatfjh gmdjw. Jgtafh tcm strafh fhvarfaht ff . r f r h 1 u f 2 ----------------------- 5 1 ‛ fi ‛ 7 1 ‛ 7 mif dh r + 7 m ih -------E i 7 3 1 ‛ r wcmrm E wcmrm Ei fs tcm fhtmhsfty j` efsdjkatfjh. AH[.
3.5
ff
r b 1 ‛ 7 2 ----------------------- m bi ----7- E i 7 1 ‛ r
PAWFAUFJHAD KADK]D][
Fh [mktfjh >.3, >.3, wm emrfvme tcm gjuheary vadum prjgdmb gy kjhsfemrfhn tcm statfjhary vadum j` arma fhtmnrads. Fh tcfs smktfjh wm wrftm tcm rmsudts usfhn fhefkfad hjtatfjh. Ucm varfagdms usme fh tcm mquatfjhs gmdjw arm em`fhme fh[mktfjh fh [mktfjh >.3. >.3. ^m saw tcat wm wfdd cavm tj taim emrfvatfvms j` tcm `uhktfjhad C `uhktfjhad C wftc wftc rmspmkt tj emrfvatfvms j` u. Ucm rudm gmdjw bust gm `jddjwme. • F` C fs fs em`fhme fh tmrbs j` u,f tcmh taim tcm emrfvatfvm C u , i ahe usm tcm `akt tcat u , f u , i 2
3.5.1
[tatfjhary vadum j` a `uhktfjhad wftc `frst jremr emrfvatfvms
fi .
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
Hjhdfhmar strafhs
1;>
1 ^m em`fhm tcm `uhktfjhad j` Mquatfjh (>.5;a) usfhn (>.5;a) usfhn fhefkfad hjtat fjh as scjwh gmdjw. 7 (3.1;a) F u 2 C u , x x 1 u , x x 7 u x 1 x 7 ex 1 e x 7 ; A 5 Ucm assjkfatme gjuheary vadum prjgdmb nfvmh gy Mquatfjhs (>.55a) ahe (>.55a) ahe (>.55g) kah (>.55g) kah gm wrfttmh as = 0 C ‛ C 2 ? (3.15a) Ef``mrmhtfad Mquatfjh< x i fh A > u u,f , f 3 C h 2 ? 6 Gjuheary Kjheftfjhs< jr x i jh (3.15g) u l 2 ? u,f f 1? 11 17 3.5.7 [tatfjhary vadum j` a `uhktfjhad wftc wftc smkjhe jremr emrfvatfvms 1; (>.5=) usfhn fhefkfad hjtatfjh as scjwh gmdjw. 15 ^m em`fhm tcm `uhktfjhad j` Mquatfjh (>.5=) usfhn 1= (3.1=a) F u 2 C u , x x 1 x 1 u , x x 1 x 7 u , x x 7 x 7 u , x x 1 u , x x 7 u x 1 x 7 ex 1 e x 7 10 A 1> 13 ^m wrftm tcm tmrbs em`fhme fh Mquatfjhs (>.=7) ahe (>.=7) ahe (>.=5) fh (>.=5) fh fhefkfad hjtatf jh as scjwh gmdjw. 16 C h h C m h h C ‛ C 7? b hh 2 b ht 2 (3.1=g) ‛ q f 2 f l bf b l u, fl u , fl u , f u, fl , l l 71 77 wcmrm, m fs tcm pmrbutatfjh `uhktfjh em`fhme fh Mquatfjh (3.0). (3.0). bf 7; Ucm assjkfatme gjuheary vadum prjgdmb nfvmh gy Mquatfjhs (>.=0a), (>.=0a), (>.=0g), (>.=0g), ahe (>.=0k) kah (>.=0k) kah gm wrfttmh as 75 C 2 ? 7= (3.10a) Ef``mrmhtfad Mquatfjh< q f f + xi fh A u 70 7> 73 u 2 ? b hh 2 ? jr h 76 ;? (3.10g) Gjuheary Kjheftfjhs< x i jh ahe ;1 bht ;7 + qf hf 2 ? jr u 2 ? s ;; ;5 ;= (3.10k) Kjrhmr Kjheftfjh< b ht K 2 ? jr x i jh kjrhmr K uK ;0 ;> MSAB_DM 3.= ;3 ;6 Jgtafh tcm gjuheary vadum prjgdmb `jr tcfh pdatm gmhefhn wftc jhdy trahsvmrsm efstrfgutme `jrkms usfhn fhefkfad hjtatfjh ahe 5? varfatfjhad kadkudus. AH[. Ef``mrmhtfad mquatfjh< Ew , iiff ‛ p z 2 ? jr Ew , iiff 2 p z 51 57 w jr b hh 2 ? 2 ? 5; h 55 AH[. Gjuheary Kjheftfjhs< x i jh ahe 5= 50 bht + qf hf 2 ? jr w 2 ? 5> s 53 AH[. Kjrhmr Kjheftfjh< b ht K 2 ? jr x i jh kjrhmr K w K 56 =? =1 3.= HJHDFHMAW [UWAFH[ =7 =; Jur jglmktfvm fs tj emtmrbfhm tcm kcahnm fh tcm efstahkm gmtwmmh twj fh`fhftmdy kdjsm fh`fhftmdy kdjsm pjfhts j` a gjey as ft em`jrbs. Fh [mktfjh 1.= =5 wm saw tcat tcmrm arm twj em`fhftfjhs j` darnm strafhs. Danrahnfah strafh fs kjbputme `rjb em `jrbatfjh gy usfhn tcm jrfnfhaduhemjrfnfhaduhem== `jrbme nmjbmtry as nmjbmtry as tcm rm`mrmhkm nmjbmtry. Mudmrfah strafh fs kjbputme `rjb em`jrbatfjh gy usfhn tcm `fhad tcm `fhad em`jrbme nmjbmtry nmjbmtry =0 as tcm rm`mrmhkm nmjbmtry. Ucm Danrahnfah emskrfptfjh fs usuaddy usme fh sjdfe bmkcahfks. Ucm Mudmrfah emskrfptfjh fs usuaddy => usme fh `dufe bmkcahfks. @jr sbadd em`jrbatfjhs ft ejms hjt battmr f` wm em`fhm strafh usfhn tcm fhftfad uhem`jrbme nmjbmtry jr =3 tcm em`jrbme nmjbmtry ahe tcm t wj em`fhftfjhs yfmde tcm sabm rmsudts. Cjwmvmr, f` em`jrbatfjhs arm darnm tcmh tcm twj em`fhftfjhs =6 yfmde ef``mrmht ahadysfs. 0? Fh tcfs smktfjh wm emvmdjp hjhdfhmar bmasurms j` strafhs. Fhftfaddy wm usm tcm mxpaheme `jrb tj emvmdjp tcm kjhkmpts fh 01 twj efbmhsfjhs, tcmh nmhmradfzm ft tj tcrmm efbmhsfjhs usfhn fhefkfad hjtatfjh.
1;3
3
3.=.1
Fhefkfad Hjtatfjh
A pmrspmktfvm j` rm`mrmhkm `rabm
Fh tcfs smktfjh wm emvmdjp a pmrspmktfvm jh t cm jrfnfh j` nmjbmtrfk hjhdfh marfty. ^m ^m em`fhm tcm `jddjwfhn< x ahe x ahe y y arm arm tcm kjjrefhatms j` a pjfht fh Kartmsfah systmb ( S S , V ) jh tcm uhem`jrbme gjey. uhem`jrbme gjey. S , V ) jh tcm em`jrbme gjey. gjey. ahe arm tcm kjjrefhatms j` a pjfht fh Kartmsfah systmb ( S u ahe v arm tcm efspdakmbmht kjbpjhmhts fh tcm x tcm x ahe ahe y y efrmktfjh. efrmktfjh. Ucm pjfht _ pjfht _ wftc wftc kjjrefhatms x kjjrefhatms x,, y gmkjbms y gmkjbms pjfht _ pjfht _ * wftc kjjrefhatms , eum tj em`jrbatfjh. ^m kah wrftm<
2 x + u x y
2 y + v x y
(3.1>)
Kjhsfemr tcrmm pjfhts _ pjfhts _ 1, _ 7 ahe _ ahe _ ; jh a strafnct dfhm x dfhm x 2 2 x xj as scjwh fh @fnurm 3.;a 3.;a. Mquatfjh (3.1>) kah (3.1>) kah gm wrfttmh as
2 x + u xj y
2 y + v x j y
(3.13)
Mquatfjh (3.13) rmprmsmhts (3.13) rmprmsmhts a kurvm fh tcm , kjjrefhatm systmb f` u ahe v arm hjt dfhmar fh y fh y..
y, y,
y y _ ;
_ ;*
_ ;
_ ;*
_ 7
_ 7*
_ 7
_ 7*
_ 1
_ 1*
_ 1
_ 1*
x, x,
x x
2 j
x 2 xj (g)
(a) @fnurm 3.; Urahs`jrbatfjh j` strafnct dfhms fhtj kurvms eum tj em`jrbatfjh.
Fh a sfbfdar bahhmr kjhsfemr tcrmm pjfhts _ pjfhts _ 1*, _ *, _ 7* ahe _;* jh a strafnct dfhm 2 j scjwh fh @fnurm 3.;g 3.;g. Mquatfjh (3.1>) kah (3.1>) kah gm wrfttmh as
j
2 x + u x y
2 y + v x y
(3.16)
Mquatfjh (3.1>) kah (3.1>) kah gm sjdvme tj yfmde x 2 ` 1 j
y 2 ` 7 j
(3.7?)
Mquatfjh (3.7?) rmprmsmhts (3.7?) rmprmsmhts a kurvm fh tcm x tcm x,, y kjjrefhatms y kjjrefhatms prjvfeme ` prjvfeme ` 1 ahe ` ahe ` 7 arm hjt a dfhmar `uhktfjh j` . ^m smm tcat a strafnct dfhm gmkjbms a kurvm fh tcm prjkmss j` em`jrbatfjh. Ucus jur kjjrefhatm systmb j` strafnct menms wfdd gmkjbm kurvfdfhmar kjjrefhatm systmb (scjwh wftc ) wcmh wm bjvm `rjb em`jrbme tj uhem`jrbme rm`mrmhkm `rabm as scjwh fh @fnurm 3.5. 3.5. Ucm emrfvatfvms fh strafh wfdd tcus cavm ef``mrmht `jrbs fh tcm em`jrbme ahe uhem`jrbme statms. Fh a sfbfdar bahhmr tcm ef``mrmhtfad armas e x e y Yjr e e R, wcfkc fs a rmktahndm wfdd gmkjbm a quaerfdatmrad wftc kurvme sfems e x e y Yjr e e R, fbpaktfhn tcm `jrb fh wcfkc wm emskrfgm jur strmss. Ucm varfjus bmasurms (em`fhftfjhs) j` strmss ahe strafh arm ah jutkjbm j` tcm rm`mrmhkm `rabm wm usm tj emskrfgm tcmb. )
)
)
)
)
)
y
y )
x
_ *
)
_ *
x
)
_
_ @fnurm 3.5 Urahs`jrbatfjh j` Kartmsfah kjjrefhatms tj kurvfdfhmar kjjrefhatm systmb.
3.=.7
Danrahnfah strafh fh twj efbmhsfjh
Ucm pjfht _ pjfht _ ( x, x, y) y) em`jrbs tj pjfht _ pjfht _ *( *( ), wcmrm as tcm pjfht Q( x+ x+ex, ex, y+ y+ey) ey) em`jrbs tj Q*(e (3.1>) kah e ). Mquatfjh (3.1>) kah gm wrfttmh as e 2 ex +
u ex + u ey x y
2
1 + u ex + u ey x y
e 2 ey +
v ex + v ey x y
2
v ex + 1 + v ey y x
(3.71)
Ucm squarm j` fh`fhftmsfbad efstahkm _Q efstahkm _Q ahe ahe _ _ *Q* kah gm wrfttmh as< es
7
7
2 ex + ey
7
es 2 e + e 7
7
7
(3.77)
1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
Hjhdfhmar strafhs
1;6
1 [ugstftutfhn Mquatfjh (3.71) fhtj (3.71) fhtj Mquatfjh (3.77), (3.77), wm jgtafh 7 7 7 7 7 7 u ‛ 1 + v ex 7 + 1 + v 7 ‛ 1 + u ey 7 + 7 1 + u u + 1 + v v exey jr es ‛ es 2 1 + ; x x y y x y y x 5 7 7 7 7 7 = u u v v v u 7 7 7 u + v + u u + v v exey jr es ‛ es 2 7 + + ex + 7 + + ey + 5 0 x x x y y y y x x y y x > 7 7 7 7 (3.7;) es ‛ es 2 7 xx ex + 7 yy ey + 7 xy + yx exey 3 7 7 7 7 6 u v 1 v u u 1 v (3.75) + -+ + -+ xx 2 yy 2 xy 2 yx 2 1-- u + v + u u + v v 1? 7 y x x y y x x y x 7 x y 7 y 11 Kjhsfemr tcm kjhvmhtfjhad em`fhftfjh j` strafh fh tcm efrmktfjh j` _Q j` _Q.. 17 1; es ‛ es (3.7=) M _Q 2 -------------------es 2 1 + M _Q es 15 es 1= Mquatfjh (3.7;) kah (3.7;) kah hjw gm wrfttmh as< 10 7 7 es ‛ es 2 es ‛ es es + es 2 M _Q 7 + M _Q es 2 7 M _Q 1 + M _Q 7 es jr 1> 13 7 7 7 (3.70) M _Q 1 + M _Q 7 es 2 xx ex + yy ey + xy exey 16 [uppjsm wm hjw kjhsfemr a dfhm mdmbmht _Q mdmbmht _Q fs fs fh tcm x tcm x efrmktfjh efrmktfjh ( M M s 2 e x, x, e y 2 y 2 ?, ahe e z 2 2 ?. 7? _Q2 M xx). Fh sukc a kasm e s 2 @rjb Mquatfjh (3.70) wm (3.70) wm jgtafh< 71 77 (3.7>) M xx 1 + M xx 7 2 xx jr M xx 2 1 + 7 xx ‛ 1 7; F` strafhs arm sbadd tcmh gy gfhjbfad mxpahsfjh M xx 1 + 7 xx 7 + cfncmr jremr tmrbs ‛ 1 xx . Ucus, wm smm tcat 75 M xx fs ahadjnjus tj jur kdassfkad em`fhftfjh j` strafh. Fh a sfbfdar bahhmr f` wm hmndmkt tcm quaeratfk tmrbs fh Mquatfjhs (3.75) 7= tcrjunc (3.75), (3.75), wm jgtafh tcm kdassfkad em`fhftfjh j` sbadd strafh. Mquatfjhs (3.75) tcrjunc (3.75) tcrjunc (3.75) arm (3.75) arm kaddme tcm Danrahnfah strafh 70 as wm usme tcm jrfnfhad nmjbmtry as tcm rm`mrmhkm nmjbmtry. 7> 73 76 3.=.; Mudmrfah strafh fh twj efbmhsfjh ;? Jhkm bjrm tcm pjfht _ pjfht _ ( x, x, y) y) em`jrbs tj pjfht _ pjfht _ *( *( ), wcmrm as tcm pjfht Q( x+ x+ex, ex, y+ y+ey) ey) em`jrbs tj Q*(e e ). Gut hjw ;1 wm usm tcm em`jrbme nmjbmtry as jur rm`mrmhkm nmjbmtry tj emskrfgm tcm prjgdmb, tcat fs, u ahe v fh Mquatfjh (3.1>) arm (3.1>) arm hjw ;7 `uhktfjhs j` . ^m rmwrftm Mquatfjh (3.1>) as (3.1>) as ;; (3.73) jr x 2 ‛ u y 2 ‛ v 2 x + u 2 y + v ;5 ^m hjw kjhsfemr tcm fh`fhftmsfbad dmhntcs as< ;= ;0 u u e 2 1 ‛ u e ‛ u e v v e 2 ‛ v e + 1 ‛ v e ex 2 e ‛ e + ey 2 e ‛ e + (3.76) ;> ;3 Ucm squarm j` fh`fhftmsfbad efstahkm _Q efstahkm _Q ahe ahe _ _ *Q* kah gm wrfttmh as fh Mquatfjh (3.7;). (3.7;). ^m hjw wrftm< ;6 7 7 7 7 7 7 5? u 7 v v 7 u u u + 1 ‛ v v e e jr es ‛ es 2 1 ‛ 1 ‛ ‛ e + 1 ‛ 1 ‛ ‛ ey + 7 1 ‛ 51 57 7 7 7 7 7 7 u ‛ u ‛ v e 7 + v ‛ v ‛ u ey 7 + 7 u + v ‛ u u ‛ v v e e jr 5; es ‛ es 2 55 7 5= 7 7 7 (3.;?) es ‛ es 2 7 M e + 7 M e + 7 M + M e e 50 7 7 7 7 5> 1 u v u u v v u ‛ 1-- u + v v ‛ 1-- v + u (3.;1) M 2 M 2 M 2 M 2 -- + ‛ ‛ 53 7 7 7 56 =? 3.=.5 Fhefkfad hjtatfjh ahe hjh-dfhmar strafh fh tcrmm efbmhsfjh =1 =7 ^m rmprmsmht rmp rmsmht x xf ahe f as tcm kjjrefhatms fh tcm uhem`jrbme ahe em`jrbme statm ahe uf tcm efspdakmbmht vmktjr. Cmrm, =; x1, x7, ahe x ahe x; rmprmsmht x rmprmsmht x,, y, y, ahe z ahe z , rmspmktfvmdy. =5 1, 7, ahe ; rmprmsmht , , ahe , rmspmktfvmdy. == =0 u1, u7, ahe u; rmprmsmht u, v, ahe w, rmspmktfvmdy. => (3.75) ahe (3.;1) kah (3.;1) kah gm wrfttmh as =3 Mquatfjhs (3.75) ahe =6 1 u f u l u i u i 1 (3.;7) + + 2 -- u f l + u l f + u i f u i l f l 2 x y z fl 2 -0? 7 x l x f x f x l 7 01
15?
3
Fhefkfad Hjtatfjh
1 (3.;;) 7 ; Ucm strafh tmhsjr fl fs kaddme tcm Nrmmh‖s strafh tmhsjr. Ucm strafh tmhsjr M fl fs kaddme tcm Adbahsf‖s strafh tmhsjr. Fh 5 jtcmr wjres, tcm Danrahnfah emskrfptfjh j` `fhftm strafh fh tmhsjr `jrb fs tcm Nrmmh‖s strafh tmhsjr ahe tcm Mudmrfah emskrfptfjh = j` `fhftm strafh fh tmhsjr `jrb fs tcm Adbahsf‖s strafh tmhsjr. tmhsjr. @jr sbadd strafh tcm twj tmhsjr `jrbs arm tcm sabm ahe arm kaddme 0 Kaukcy‖s strafh tmhsjrs. Hjtm gjtc strafh tmhsjrs arm sybbmtrfk< > (3.;5) 3 M fl 2 M lf fl 2 lf 6 1 ? MSAB_DM 3.0 11 Danrahnfah strafh wftc bjemratmdy darnm rjtatfjh `jr pdatm gmhefhn fs nfvmh as 17 1 1 (3.;=) 1; f l 2 x y fl 2 -- u f l + u l f + -- w , f w , l l 7 7 15 Assubm tcm efspdakmbmht `fmde fs u f 2 u jf ‛ zw , f , wcmrm ujf ahe w arm tcm fhpdahm ahe gmhefhn efspdakmbmhts, rmspmktfvmdy. Add 1= assubptfjhs j` kdassfkad pdatm tcmjry mxkmpt sbadd strafh apprjxfbatfjh arm vadfe. p vadfe. pf ahe p ahe p z arm tcm fhpdahm ahe gmhefhn efstrfg- 10 utme djaes pmr uhft arma. Jgtafh tcm strmss `jrbudas ahe tcm statmbmht j` gjuheary vadum prjgdmbs `jr efspdakmbmhts ujf ahe w. 1> 17 b fl z h 13 AH[. fl 2 -----fl- + ---------------; 16 c c 7? Mc AH[. Ef``mrmhtfad Mquatfjh 1< ---------------------x i fh A 1 ‛ u jf l + u jl f + w , f w , l l + u ji i + w, i w, i fl , l l 2 pf 71 7 7 1 ‛ 77 AH[. Ef``mrmhtfad Mquatfjh 7< Ew , ffll 2 p z + p l w, l l + h lf w , lf x fh A 7; lf i 75 AH[. Gjuheary Kjheftfjh 1< 1 ‛ u jl f + ujf l + w , l l w , f + u jb b + w , b w , b lf h f jr x i jh u j l 2 ? 7= AH[. Gjuheary Kjheftfjhs 7< 70 b w ht 7> b hh 2 ? jr ahe + qf hf 2 ? jr xi jh 2 ? w 2 ? h s 73 76 jr AH[. Kjrhmr Kjheftfjh< bht K 2 ? x i jh kjrhmr K w K ;? ;1 ;7 3.0 KDJ[]WM ;; Fh tcfs kcaptmr wm stuefme tcm dahnuanm j` fhefkfad hjtatfjh ahe tcm rudms j` bahfpudatfhn fhefkms. Ah fbpjrtaht kcmki fh tcm emr- ;5 fvatfjh fs tcat mvmry tmrb fh ah mquatfjh bust cavm tcm sabm ` rmm fhemx. ^m rm-emrfvme mquatfjhs j` pdatm, mdastfkfty, ahe varfa- ;= tfjhad kadkudus. ^m saw tcat dmhntcy emrfvatfjhs ahe tme fjus adnmgra arm erabatfkaddy rmeukme gy tcm usm j` fhefkfad hjtatfjh. ^m ;0 bjef`fme tcm kdassfkad pdatm tcmjry tj jgtafh strmss rmsudtahts, strmssms, ahe gjuheary vadum prjgdmb `jr pdatm gmhefhn wftc darnm ;> strafh. ;3 Ucfs kcaptmr grfhns tcfs gjji tj a kdjsm. ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01 1 M fl 2 -7
u f u l u i u i + ‛ l f f l
1 2 -- u f l + u l f ‛ u i f u i l 7
f l 2
[yhjpsfs j` mquatfjhs 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
3.>
[VHJ_[F[ J@ MQ]AUFJH[ Mdastfkfty
Umhsjr strafh Fsjtrjpfk kjhstftutm mquatfjhs _dahm strmss Mqufdfgrfub Uraktfjh (strmss vmktjr) [trafh mhmrny emhsfty
fl fl
2 u f l + u l f 7 2 1 + fl ‛ fl ii M 9
fl
2 7 N fl + fl ii 9
2 M 1 ‛ 7 1 + 9 N 2 M 7 1 +
fl 2 M 1 ‛ fl + fl ii 1 ‛ 7 fl l + @ f 2 ? [ f 2 fl h l ] ? fl fl 7 Kdassfkad pdatm
Efspdakmbmhts
u f ‛ zw , f
[trafhs
fl
2 ‛ zw , fl
[trmssms
fl
2 ‛ Mz 1 ‛ w, fl + fl w , ii 1 ‛
[trmss rmsudtahts
b fl 2
Mqufdfgrfub
q f f 2 b lf l 2 ‛ p z x 1 x 7
Ef``mrmhtfad mquatfjh
Ew , iiff 2 E w 2 p z x 1 x 7
Gjuheary varfagdms
b hh 2 ‛ b fl h f h l
Gjuheary kjheftfjhs
[trafh mhmrny
z c
w w x 1 x 7
fl e z
7
2 ‛ E 1 ‛ w, fl + fl w, ii 9 q f 2
e z c fz
7
2 b lf l 2 ‛ Ew , ii f 2 ‛ E w , f
5
b hh 2 ?
b ht 2 ‛ bfl m bf hb h l
w h
jr
b ht K 2
?
] 2 E 7
bht + qf hf s
ahe
2 ?
w
jr
2 ?
w K jh kjrhmrs.
jr A
2 ?
1 ‛ w , fl w, fl + w , ff w, ii ex 1 e x 7
Parfatfjhad kadkudus @frst jremr emrfvatfvm
C ‛ C u u, f , f b hh 2
[mkjhe jremr emrfvatfvms
x i fh A 9
2 ?
C h h 9 u , fl f l
b hh 2 ?
b ht K 2
b ht 2
?
jr
2 ?
C m h h 9 ‛ q f u, fl bf b l
u h
jr
C h u,f f
2 ?
ahe
jr 2
u l
2 ?
C ‛ C 9 u , f u, fl , l l bht + qf h f s
x i jh q f f +
2 ?
u K jh kjrhmrs.
Darnm strafh 1 u f u l u i ui 1 2 -+ + 2 -- u f l + u l f + u i f ui l 7 x l x f x f x l 7
Danrahnfah strafh
fl
Mudmrfah strafh
1 M fl 2 -7
u f u l ui ui + ‛ l f f l
1 2 -- u f l + u l f ‛ u i f ui l 7
f l 2 x y z
f l 2
jr
C 2 u
?9
u
2 ?
151
157
3
Fhefkfad Hjtatfjh 1 7 ; 5 = 0 > 3 6 1? 11 17 1; 15 1= 10 1> 13 16 7? 71 77 7; 75 7= 70 7> 73 76 ;? ;1 ;7 ;; ;5 ;= ;0 ;> ;3 ;6 5? 51 57 5; 55 5= 50 5> 53 56 =? =1 =7 =; =5 == =0 => =3 =6 0? 01
A
Appmhefx A Gfgdjnrapcy
1. Gargmr, L. W. (1667) Mdastfkfty, Mdastfkfty, Iduwmr Akaembfk _ugdfscmrs. 7. Gjrmsf, A. _., _., ahe Kcjhn, I. _. _. (163>) Mdastfkfty (163>) Mdastfkfty fh Mhnfhmmrfhn Bmkcahfks, Mdsmvfmr. Bmkcahfks, Mdsmvfmr. ;. @uhn, V. V. K. (160=) @juheatfjhs j` [jdfe Bmkcahfks, _rmhtfkm-Cadd. 5. Irmyszfn, M. (16>6) Aevahkme (16>6) Aevahkme Mhnfhmmrfhn Batcmbatfks, Ljch Batcmbatfks, Ljch ^fdmy & [jhs. =. Dahkzjs, K. (1630) Ucm Parfatfjhad Parfatfjhad _rfhkfpdms j` Bmkcahfks, Ejvmr. Bmkcahfks, Ejvmr. 0. Bmhemdsjh, A. (1603) _dastfkfty< (1603) _dastfkfty< Ucmjry ahe Appdfkatfjh, Bakbfddah. Appdfkatfjh, Bakbfddah. >. Busicmdfscvfdf, H. F. (160;) [jbm Gasfk _rjgdmbs j` tcm Batcmbatfkad Ucmjry j` Mdastfkfty, _. Hjjrecj``. 3. Wmeey, L. H. (166;) Ah Fhtrjeuktfjh tj tcm @fhftm Mdmbmht Bmtcje, 7he me., BkNraw-Cfdd. 6. [hmeejh, F. H. (163?) @jurfmr (163?) @jurfmr Urahs`jrb, BkNraw-Cfdd. 1?. [yhnm, L. D., ahe [kcfde, A. (16>3) Umhsjr Kad kudus, Ejvmr. 11. [zfdare, W. (16>5) Ucmjry ahe Ahadysfs j` _datms< Kdassfkad ahe Hubmrfkad Bmtcjes, _rmhtfkm-Cadd. Bmtcjes, _rmhtfkm-Cadd. 17. Ufbjscmhij, [. ahe ^jfhjwsiy-Irfmnmr, ^jfhjwsiy-Irfmnmr, [. (16=6) Ucmjry j` _datms ahe [cmdds, [cmdds, BkNraw-Cfdd. 1;. Pagdm, B (7?1;) Fhtmrbmefatm Bmkcahfks j` Batmrfads, 7he Me., Mxpahefhn Meukatfjhad Cjrfzjhs, F[GH< 6>3-?-6617550-?-> 15. ^acd A.B. ahe Djgj N. N. (16;?) Urahsaktfjhs j` A[BM , v=7 1=. \fmhifmwfkz, J. K., ahe Uaydjr W. D. (1636) Ucm @fhftm Mdmbmht Bmtcje, 5tc Bmtcje, 5tc me., BkNraw-Cfdd.
157
15;
A Gfgdjnrapcy
Fhemx Afry strmss `uhktfjh 67 Adbahsf‖s strafh tmhsjr 15? Axfsybbmtrfk pdatms 35 Axfsybbmtrfk prjgdmbs 61 Gmabs jh mdastfk `juheatfjhs =? Gfdfhmar `uhktfjhad 1?0 Gfdfhmar batmrfad bjemd 7? Gjuheary vadum prjgdmb ;> Gukidfhn 07 Gukidfhn bjems 0; Kaukcy‖s strafh tmhsjrs 15? Kcaraktmrfstfk mquatfjh >, 6, 10 Kfrkudar pdatms 37 Kdassfkad tcfh pdatm tcmjry 06 Kjdubh 07 Kjbpatfgfdfty mquatfjhs 33 Kjbprmssfvm strmss 7 Kjupdm strmss 7 Krftfkad gukidfhn djae 0; Em`jrbatfjh 11 Emvfatjrfk strmss batrfx 11 Efsdjkatfjh 1;= Efspdakmbmht 11 Ejugdmt sfhnudarfty 5= Eubby fhemx 1;; Mkkmhtrfk djaefhn 00 Mdastfk‛pmr`mktdy pdastfk batmrfad bjemds 16 Mhnfhmmrfhn strafhs 17 Mudmr djae 0; Mudmrfah strafh 11, 1;>, 1;6 @afdurm mhvmdjpms 7; @afdurm tcmjrfms 77 @fmde pjfht 5= @fhftm gmabs == @fhftm mdmbmht bmtcje 177 @frst varfatfjh 1?= @jrkm (sjurkm) sfhnudarfty 5= @rmm fhemx 1;; @rmm sur`akms = @uhktfjhad 1?0 @uheabmhtad sjdutfjhs 5= Nmjbmtrfk hjhdfhmarfty 116 Nrmmh‖s strafh tmhsjr 15? Fhkdusfjh sfhnudarfty 1;0 Fhefkfad hjtatfjh 1;; Fh`dumhkm `uhktfjh 5= Fhcjbjnmhmfty sfhnudarfty 1;0 Ifhmbatfk kjheftfjh 1?=
Ifhmbatfkaddy aebfssfgdm `uhktfjhs 1?= Danrahnm pjdyhjbfads 175, 170, 173 Danrahnfah strafh 11, 1;>, 1;3 Dabfhatme pdatm >> Dfhmar `uhktfjhad 1?0 Dfhmar strafh-caremhfhn batmrfad bjemd 7? Djkad gukidfhn 07 Batmrfad hjhdfhmarfty 17? Baxfbub hjrbad strmss tcmjry 75 Baxfbub jktacmerad scmar strmss tcmjry 7; Baxfbub scmar strmss 6 Baxfbub scmar strmss tcmjry 7; Bmbgrahm ahadjny 63 Bfhedfh-Wmfsshmr pdatm tcmjry >> Bfhfbub pjtmhtfad mhmrny 11? Bjef`fme Bjcr‖s tcmjry 75 Bjeudus j` `juheatfjh =? Bjbmht (ejugdmt) sfhnudarfty 50 Haeaf-Dmvy sjdutfjh >6 Havfmr‖s sjdutfjh >3 Hjhdfhmar batmrfad bjemds 16 Hjhdfhmar strafhs 1;> Hjrbad strmss 7 Jktacmerad pdahm 1? Jktacmerad strmssms 1? Jrtcjtrjpfk pdatm >> _dahm strmss = _dahm strmss ahe pdahm strafh 36 _jwmr daw batmrfad bjemd 7? _rahetd‖s bmtcje 6> _rfhkfpad ahndms > _rfhkfpad axfs > _rfhkfpad kjjrefhatm efrmktfjhs 10 _rfhkfpad efrmktfjh > _rfhkfpad pdahms > _rfhkfpad strafhs. 10 _rfhkfpad strmss > Waydmfnc-Wftz bmtcje 17? Wmktahnudar pdatms >3, >6 Wjtatfhn efsis 67 [afht-Pmhaht‖s bmtcje 60 [mkaht `jrbuda 0> [mkjhe varfatfjh 1?= [cmar strmss 7 [fhnudarfty 5= [dmhemrhmss ratfj 05 [jurkm pjfht 5= [jurkm sfhnudarfty 5= [tagfdfty 07 15;
155
Fhemx
[tatfjhary vadum 1?= [trafh mhmrny 1?> [trmss mdmbmht 5 [trmss fhvarfahts 6 [trmss jh a sur`akm 1 [trmss rmsudtahts 7> [trmss trahs`jrbatfjh = [trmss vmktjr 0 [trukturad gukidfhn 07 [ybbmtrfk gfdfhmar `uhktfjhad 1?0 Umhsfdm strmss 7 Ujrsfjh j` hjh-kfrkudar sca`ts 6= Ujrsfjh j` tcfh-waddme jpmh smktfjh 1?? Uraktfjh 0 Urmska‖s Urmska‖s yfmde krftmrfjh 7; Pfrtuad efspdakmbmht 1?5 Pfrtuad wjri 11? Pjh-Bfsms strmss 7; ^jri 1?0