Changing variables in triple integrals works in exactly the same way. Cylindrical and spherical coordinate substitutions are special cases of this method, which we demonstrate here.
Suppose that
is a region in
and is mapped to
in
(
[link] ) by a one-to-one
transformation
where
and
A region
in
mapped to a region
in
Then any function
defined on
can be thought of as another function
that is defined on
Now we need to define the Jacobian for three variables.
Definition
The Jacobian determinant
in three variables is defined as follows:
This is also the same as
The Jacobian can also be simply denoted as
With the transformations and the Jacobian for three variables, we are ready to establish the theorem that describes change of variables for triple integrals.
Change of variables for triple integrals
Let
where
and
be a one-to-one
transformation, with a nonzero Jacobian, that maps the region
in the
into the region
in the
As in the two-dimensional case, if
is continuous on
then
Obtaining formulas in triple integrals for cylindrical and spherical coordinates
Derive the formula in triple integrals for
cylindrical and
spherical coordinates.
For cylindrical coordinates, the transformation is
from the Cartesian
to the Cartesian
(
[link] ). Here
and
The Jacobian for the transformation is
We know that
so
Then the triple integral is
The transformation from rectangular coordinates to cylindrical coordinates can be treated as a change of variables from region
in
to region
in
For spherical coordinates, the transformation is
from the Cartesian
to the Cartesian
(
[link] ). Here
and
The Jacobian for the transformation is
Expanding the determinant with respect to the third row:
Since
we must have
Thus
The transformation from rectangular coordinates to spherical coordinates can be treated as a change of variables from region
in
to region
in
Then the triple integral becomes