changing variables in multiple integrals

Double integrals in coordinates which are taken over circular regions or have integrands involving are often better done in polar coordinates:

where

More generally, double integrals can sometimes be simplified by changing variables from to a new coordinate system that is better adapted to the region or integrand.

When we change variables, the area of the integration changes. The scaling factor for this change is given by the Jacobian determinant:

Using the determinant, the formula for the area element in the -system is given by

This leads to the change of variables formula:

Change of variable formula

While determinant can be negative, we always use the positive value here

Alternative Jacobian convention

Some sources compute the Jacobian

However, , so so either convention is valid.

Example: Polar Coordinate

See: done in polar coordinates

We can verify the power-coordinate transformation using the above formula

Example: Diagonal Region

Compute , where is the square with vertices at

Since the region is bounded by the lines and , we make a change of variables:

Notice that the area of integration changes from 2 to 4. We can verify that by computing the Jacobian: . Thus, . Using either method 1 or method 2 we see the boundaries are given by the integral is .

Reference

• Session 53: Change of Variables | Multivariable Calculus | Mathematics | MIT OpenCourseWare