LFOFCFN OFCOWCWY CFDMWP
EDPGKXFC CEZFP @WF @FD OGKWDFFDUF
@EYWYWD BCGN= OGCBLZBO 68 DFLF=
Xgd`rferfwfd
DEL 8:68 686 6:;
Af`onbcec
DEL 8:68 686 666
Eslerfted
DEL 8:68 686 669
@bsgdZgdkfsun =Aghredfhe`fsfre, L.Z`
AFOWCPFY OGKWXWFD @FD ECLW ZGD@E@EOFD WDEQGXYEPFY ZKXE ZFCGLHFDK 8:69
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F. EDPGKXFC CEZFP @WF
Wdtuo edtgkrfc cepft `uf `fre audkse `gdkfd `uf pguhfn pglhftfsfddyf f`fcfn hfnwf audkse `uf pguhfn tgrsghut tgr`gaedese pf`f suftu `fgrfn tg rtutup `e X8.Ufdk X8.Ufdk `elfosu` `fgrfn tgrtutup `esede f`fcfn `fgrfn hgsgrtf `gdkfd hftfs hftfsdyf. Fpfhecf `eoftfofd `fgrfn, lfof yfdk `elfosu` f`fcfn `fgrfn tgrtutup. Oetf tedmfu audkse `uf pguhfn a yfdk `e`gaedeseofd pf`f sgkeglpft tgrtutup. Lesfcofd audkse z audkse z < a(x ,y) ,y) `e`gaedeseofd pf`f suftu `fgrfn tgrtutup X tgrtutup X `e `e he`fdk xby. he`fdk xby.
, - , - * | + * | +
@fd lucf-lucf oetf lesfcofd a lesfcofd a ( x, x, y) y) pgrsflffd
:. Krfaeo a Krfaeo a f`fcfn pgrluoffd `gdkfd
. Lesfcofd Y f`fcfn hgd`f pgmfc yfdk tgrcgtfo `e ftfs X `fd `e
hfwfn krfaeo a, yfode=
(Cenft kflhfr `ehfwfn) tumufd oetf f`fcfn lgdifre vbculg Y. Cfdkofn pgrtflf f`fcfn lglhfk esgkeglpft X lgdmf`e hghgrfpf hfkefd. Oetf cfouofd ede `gdkfd lglhfke sgcfdk
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`fd `gdkfd lglhfke
lgdmf`e l sgcfdk hfkefd
,,-
lgdmf`e d sgcfdk hfkefd
hgrcghfr sflf
hgrcghfr sflf
. @gdkfd lgdfreo kfres-kfres sgmfmfr
tgrnf`fp sulhu obbr`edft lgcfcue teteo umudk sgcfdk hfkefd `fcfl hgdtuo sgkeglpft hfkefd.
,-,*| | + <
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Lfsedk- lfsedk `gdkfd cufs
( )
`fcfl lfsedk- lfsedk
Meof oetf pecen sfcfn sftu teteo sflpgc
lfof oetf `fpft lgdknflpere hfkefd Y yfdk
tgrcgtfo `e ftfs lfsedk- lfsedk
lgdkkudfofd obtfo sgkeglpft tepes (obcbl) `gdkfd
fcfs
( )
`fd
tedkke
.
Lfof vbcglg obtfo f`fcfn tedkke obtfo ofce cufs sgkeglpft fcfs =
( ) ( ) <
@fpft `ecenft lfof udtuo sgluf sgkeglpft meof `etflhfnofd vbculg obtfo yfdk hgrofetfd , lfof vbculg tbtfc Y nflper`epgrbcgn.
∖∖ ∖∖
Edtuese oetf lglhgretfnu hfnwf nflperfd yfdk `ehgreofd lgdmf`e cghen hfeo hgketu l `fd d lgdmf`e
cghen
hgsfr,
sgnedkkf
`enfrfpofd
lgdmf`e =
∖∖
Meof
lfof vbculg Q `fre hgd`f pgmfc yfdk tgrcgtfo `e ftfs sgkeglpft
X `fd `e hfwfn pgrluoffd
f`fcfn=
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Yeaft - seaft Edtgkrfc Cepft @uf @u f = 6.
a ( x, y ) k ( x, y )`F a ( x, y )`F k ( x, y )`F X
8.
X
X
ia ( x, y )` F i a ( x, y )`F , `gdkfd i obdstfdtf. X
X
9. Meof a ( x , y ) k ( x , y ) udtuo ( x , y ) X , lfof x, y) `F k ( x , y ) `F a ( x,y X
X
Wdtuo lgdknetudk edtgkrfc cepft `uf `fpft `ekudfofd edtgkrfc hgrucfdk yfdk `etuces `fcfl hgdtuo =
f.
y a 8 ( y ) a ( x, y )`x`y a ( x, y ) `x `y f y a 6 ( y ) X h
a ( x, y)`F X
`elfdf edtgkrfc yfdk f`f `fcfl ourudk nfrus `enetudk tgrcghen `fnucu `gdkfd lgdkfdkkfp vfrefhcg y vfrefhcg y obdstfdtf, oglu`efd nfsecdyf `eedtgkrfc oglhfce tgrnf`fp y. tgrnf`fp y. h
h.
a ( x, y)`F a ( x, y)`y`x X
X
f
y a 8 ( y ) a ( x, y ) `y `x y a ( y ) 6
`elfdf edtgkrfc yfdk f`f `fcfl ourudk nfrus `enetudk tgrcghen `fnucu `gdkfd lgdkfdkkfp vfrefhcg x vfrefhcg x obdstfdtf, oglu`efd nfsecdyf `eedtgkrfc oglhfce tgrnf`fp x. tgrnf`fp x. Meof Meof edtgkrfc cepft `uf `e ftfs f`f, lfof (f) `fd (h) sgifrf ulul fofd lglhgreofd nfsec yfdk sflf.
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Ibdtbn =
6. Netudk cufs `fgrfn yfdk `ehftfse bcgn x + y < 8 `fd 8y < x + y Zgdygcgsfefd =
8 8- y
F
`F X
`x`y
: 8y- 0
8
8- y
:
8y - 0
x `t
8
8
:
:
(8 y 8 y 0)`y (1 9 y )`y (1y y (1
9 8
8
y ) (68 1) 1 8
:
8. Pgdtuofd vbculg Q suftu hgd`f pf`ft `e hfwfn pgrluoffd
+
`fd `e ftfs pgrsgke pfdmfdk
*
Zgdygcgsfefd =
1 ; _ \
_ _ \
9. Ifrecfn vbculg hgd`f pgmfc yfdk tgrcgtfo `e hfwfn pfrfhbcbe`
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Zgdygcgsfefd =
*| + ∣ ∯ ∯ ∯ : 6 ∯ : 6 ∯ . / : 6 Ofrgdf etu, vbculg `e hfwfn
`fd `e ftfs @ f`fcfn
H. Ogkudffd Edtgkrfc Cepft @uf
Edtgkrfc cepft `uf `fpft `etgrfdkofd `fd `ekudfofd `fcfl hfdyfo ifrf yfetu= f. Qbculg
∣ Meof
f`fcfn pgrsflffd pgrluoffd, lfof
f`fcfn vbculg fdtfrfpgrluoffd `fd he`fdk
Qbculg `eftfs he`fdk
Pgdtudofd vbculg `ebotre pgrtflf `ehftfse bcgn
`enetudk pbsetea `fd hfwfn
Ibdtbn =
.
`enetudk dgkftea.
`gdkfd edtgkrfc cepft `uf.
Zgdygcgsfefd=
∯ ∯
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h. Cufs
Meof
`epgrbcgn `epgrbcgn
∣
Ibdtbn = Netudkcfn cufs `fgrfn yfdk `ehftfse bcgn pfrfhbcf
Zgdygcgsfefd=
Wdtuo
Hgdtuo cufs `fgrfn selgtres tgrnf`fp sulhu x
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1 ; _ \ i. Lfssf `fd pusft Lfssf Meof a `epfd`fdk `epfd`fdk sghfkfe lfssf mgdes, yfetu lfssf pgrsftufd cufs, lfof=
Ibdtbn= Meof a lfsf mgdes, lfof pusft lfssf
̇
`fre cfledf yfdk `ehftfse
bcgn=
Zgdygcgsfefd=
.
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̇
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Yg`fdkofd lblgd pbcfr edgrsef tgrnf`fp teteo fsfc : `ehgreofd bcgn=
Ibdtbn= Pgdtuofd lblgd edgrsefc
∝
udtuo cfledf yfdk `ehftfse
`gdkfd ogigpftfd
Zgdygcgsfefd=
∝
∝ _ \ 0 7
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∝ 0 7 0 7 Ybfc cftenfd= 6. Netudkcfn edtgkrfc cepft hgreout
∯ ∯
8. Netudkcfn edtgkrfc kfd`f `uf yfdk `etudmuofd ftfs X.
8 9
9. Pgdtuofd vbculgdyf `gdkfd edtgkrfc cepft meof `fgrfn X e hftfse bcgn he`fdk-he`fdk obbr`edft `fd he`fdk he`fdk x < 7 `fd y +8z ’ 0 0 < :
