Graphics Reference
In-Depth Information
z
Figure 4.23.
The iterated integral in
equation (4.23).
f(x,y)
b(x)
y
a(x)
a
x
b
x
z
z = y
y
y = 6 - (x - 4)
2
X
y = 1 + (x - 3)
2
x = 2
Figure 4.24.
Example 4.8.5.
x = 5
x
Finally, there is change of variable theorem for higher-dimensional integrals just
like for real-valued functions of one variable.
4.8.6. Theorem.
(The Generalized Change of Variables Theorem) Let
A
Ã
R
n
be an
open bounded subset and g :
A
Æ
R
n
a one-to-one and continuously differentiable
function. Assume that the Jacobian matrix g¢ has a nonzero determinant at all points
of
A
. If f : g (
A
) Æ
R
is an integrable function, then
Ú
Ú
(
)
f
=
f
o
g
det
g
¢
.
(4.24)
(
)
g
A
A
Proof.
See [Spiv65] or [Buck78].
Consider the map g :
R
2
Æ
R
2
defined by
4.8.7. Example.
xr
yr
=
=
cos
sin
q
q
