Calculus 2nd Edition Rogawski

936 C H A P T E R 16 MULTIPLE INTEGRATION
By Eq. (14),
Jac(G) = Jac(F ) −1 =
1
2(x 2 + y 2 )
Normally, the next step would be to express f (x, y) in terms of u and v. We can avoid
doing this in our case by observing that the Jacobian cancels with one factor of f (x, y):
∫∫
∫∫
xy(x 2 + y 2 )dxdy = f (x(u, v), y(u, v)) |Jac(G)| dudv
D
R
∫∫
= xy(x 2 + y 2 1
)
R
2(x 2 + y 2 ) dudv
= 1 ∫∫
xy du dv
2 R
= 1 ∫∫
vdudv (because v = xy)
2 R
= 1 ∫ 3 ∫ 4
vdvdu= 1 ( 1
2
2 (6) 2 42 − 1 )
2 12 = 45
2
−3
1
Change of Variables in Three Variables
The Change of Variables Formula has the same form in three (or more) variables as in two
variables. Let
G : W 0 → W
be a mapping from a three-dimensional region W 0 in (u, v, w)-space to a region W in
(x, y, z)-space, say,
x = x(u, v, w), y = y(u, v, w), z = z(u, v, w)
REMINDER 3 × 3-determinants are
defined in Eq. (2) of Section 13.4.
The Jacobian Jac(G) is the 3 × 3 determinant:
∣ ∣∣∣∣∣∣∣∣∣∣∣ ∂x
∂u
∂(x, y, z)
Jac(G) =
∂(u, v, w) = ∂y
∂u
∂z
∂u
∂x
∂v
∂y
∂v
∂z
∂v
∂x
∂w
∂y
∂w
∂z
∂w
∣
15
The Change of Variables Formula states
dx dy dz =
∂(x, y, z)
∣∂(u, v, w) ∣ dudv dw
More precisely, if G is C 1 and one-to-one on the interior of W 0 , and if f is continuous,
then
∫∫∫
f (x, y, z) dx dy dz
W
∫∫∫
= f (x(u, v, w), y(u, v, w), z(u, v, w))
∂(x, y, z)
∣
W 0
∂(u, v, w) ∣ dudv dw 16