A f : A
2
⊂ R → R
2
⊂ R
f
A
g :
(u, v)
(x, y )
f =
A
R = (u, v)
{
g : R2 R
2
→R
0
0
2
→R
R
(u, v)
A
◦ ◦
f g
∈ R | u ≤ u ≤ u + ∆
u
≤ v ≤ v + ∆ }
, v0
0
v
g(u, v) = (x, y ) = (x(u, v), y (u, v )) uv R xy α : [u0 , u0 + ∆ u ]
β : : [v0 , v0 + ∆v ]
2
R
2
R
2
→ R
2
→ R
α(t) = g (t, v0 ) = (x(t, v0 ), y(t, v0 ))
β (t) = g (u0 , t) = (x(u0 , t), y(u0 , t))
α (t) = β (t) =
∂x ∂ y (t, v0 ), (t, v0 ) ∂u ∂u
∂x ∂ y (u0 , t), (u0 , t) ∂v ∂v
u0 +∆u
LAB
u0
∆u
v0 +∆v
α (t)dt
LAD =
v0
β (t)dt
∆v u0 +∆u
u0
α (u∗ )∆u = u
u = α (u∗ )∆u
β (v∗)∆ = v
v = β (v ∗ )∆v
v
v0 +∆v
α(t)dt = α (u∗ )∆u
v =
(u0 , v0 )
v
v0
∂x ∂ y , ∂u ∂u
u =
∂x ∂ y , ∂v ∂v
β (t)dt = β (v∗)∆
∆u
∆v R u
∆u
u
= ∆u ∆v det
ˆi
jˆ
∂x ∂u ∂x ∂v
∂y ∂u ∂y ∂v
∆v
× v = (α (u∗ )∆ ) × (β (v∗ )∆ ) = ∆ v
ˆ k 0 0
= ∆ u ∆v
R
∂x ∂u ∂x ∂v
∂y ∂u ∂y ∂v
u ×
∂ (x, y ) AreaR = (AreaR ) ∂ (u, v)
(u∗ ) × β (v∗ ) =
u ∆v α
v
∂ (x, y ) v = ∆u ∆v ∂ (u, v)
v
B
R
⊂ R
∈ R
ηi
i
f
A
Ri
Ri
f (g(ηi ))m(g (Ri ))
∅
B=
f (g(ηi ))m(g (Ri )) =
∅
Ri
B=
f (g(ηi )) det(Dg (ηi )) m(Ri )
f g det(Dg )
B
Rn
|
|
f g det(Dg )
f
f :
|
∅
B=
≈ ◦ ◦ | ≈ ≈ ◦ ◦ | A
Ri
g (Ri )
B
≈ ≈
R
Q
B
|
→R n
⊂ R
A
f : A
n
g : B
⊂ R → R det(D (x)) = 0, ∀ x ∈ B g
f =
A
B
(f g )
◦ ◦ · | det(D )| g
n
n
⊂ R → R
C 1
f :
2
R
→R f : R
F : R
2
2
⊂ R → R (u, v)
C 1
2
⊂ R → R
F (u, v) = (x, y ) = (φ(u, v ), ψ(u, v )) R (x, y ) F (u, v)
f (x, y )dxdy =
R
y
− x = 1 , y − x = −1 ,
(u, v)
∈ R
∂ (φ, ψ ) f (φ(u, v), ψ(u, v)) dudv ∂ (u, v ) R
2
⊂ R
R
f (x, y) = x + y + 1 y + x = 1 , y + x = 2
verde
≤ x ≤ 12
0
∴
R
1 2
0
1 2
1 2
1+x
1
(x + y + 1) dydx =
−x
−
x(1
− x) +
2 x2 +
azul
R
1 2
1
1 2
x(2
xy +
y 2
2 x3 3
≤ x ≤ 1
1
(1
− x) 2
1 2
=
0
2
+ y
2
+ (1
1+x 1 x
−
− x)
dx = 1 2
=
4x + 2 x2 dx =
0
7 12
− x ≤ y ≤ 2 − x
f (x, y )dA
− 1
1 2
0
(1 + x)2 x(1 + x) + (1 + x) 2
∴
− x ≤ y ≤ 1 + x
f (x, y )dA
0
1
− x)
(2
2
(x + y + 1) dydx =
−x
2
+ (2
xy +
1 2
1
− x) 2
1
x
− x)
−
x(1
− x) +
(1
y 2
2 2
− x) 2
+ y
−− 2 1
+ (1
x x dx
= 1
− x)
=
1 2
5 5 dx = 2 4
roja
≤ x ≤ 32
1
∴
R
3 2
x(2
− x)
1
(2
3 2
− 2
3 2
x
(x + y + 1) dydx =
−1+x
2
− x) 2
+ (2
−
− x)
−
∴
R
1 2
1+x
0
1
2
− −2x
R
3
+
x =
v
−u 2
y =
v + u
2
9 x 2
3 2
1
=
−x
1
2
1
∂ (x, y) = ∂ (u, v)
x
1+x
dx =
−
∂x ∂u ∂x ∂v
∂y ∂u ∂y ∂v
=
9 2x2 + dx = 2
=
1
x
1 2 1 2
(x + y + 1) dydx =
7 5 2 5 + + = 12 4 3 2
f (x, y ) = x + y + 1 y + x = 1 , y + x = 2
− x = 1 , y − x = 1 ,
3 2
−
−1+x
∂ (x, y ) f (x(u, v), y (u, v)) dudv = ∂ (u, v ) R
2
2 3
(x + y + 1) dydx+
−−
−
3 2
− x , v = y + x
f (x, y )dxdy =
2
+ y
−
y u = y
y 2
−
x
(x + y + 1) dydx+
1 2
xy +
( 1 + x)2 x( 1 + x) + + ( 1 + x) 2
f (x, y )dA 1
−x
1
3
− 1 ≤ y ≤ 2 − x
f (x, y )dA
1
x
− −
1 2 1 2
v
R
=
u
2
1 = 2
+
v + u
2
+1
−
1 dudv 2
1 2
1
2
−1
1
1 (v + 1) dvdu = 2
1
−1
(v + 1) 2 2
2
5 du = 4 1
1
du =
5 5 (2) = 4 2
1 , 2
y = 3x
−1
f (x, y ) = x 2 y2 xy = 1,
F (u, v ) =
x =
xy = 2,
√ u , v
u , v
uv
y =
y =
do d onde
√ uv ⇒
x =
u = xy
u v
v =
y =
y x
√ uv
x =
1 3
y =
3
x = y =
1 2
√ √ ⇒ √ ⇒ √ ⇒ √ √ √ ⇒ 2
x = 2 y = 1
x =
2 3 2 3
y = 3 √
u = xy = 1 v = xy = 3
u = xy = 1 v = xy = 12
u = xy = 2 v = xy = 12
u = xy = 2 v = yx = 3
f (x, y) = x 2 y2
∂x ∂u ∂y ∂u
⇒ √ x=
√ = y= uv
u v
− −
∂x ∂v ∂x ∂v
1 2 1 2
=
2
1
1 2
(uv )
3
1 2
v u
1 2
1 2
1 2
1 2
u v u v
u2 2v
3 2
1 2
u2 7 dvdu = ln(6) 2v 6
=
1 2v
