Change of Variables in Multiple Integrals

Multiple integrals are often used to evaluate areas, volumes, mass distributions, and other physical quantities over regions in two or three dimensions. In many problems, however, the original region may have complicated curved boundaries when expressed in Cartesian coordinates. These complex boundaries can make the limits of integration difficult to describe and the overall calculation cumbersome. To simplify the evaluation process, a change of variables is introduced that transforms the original coordinates into a new coordinate system better suited to the geometry of the region.

A change of variables replaces the original coordinates x and y with new variables u and v. Through this transformation, the original region R in the xy-plane is mapped onto a new region S in the uv-plane. The purpose of this transformation is to reshape the integration region into a simpler form, often converting curved boundaries into straight lines or rectangular regions.

Because the coordinate system changes, the small area elements in the two systems are not identical. A small rectangular region in the uv-plane may become stretched, compressed, or skewed when mapped into the xy-plane. The factor that measures this local geometric distortion is called the Jacobian.

The Jacobian determines how area changes under the transformation. It is obtained from the partial derivatives that describe how the original variables depend on the new variables. Geometrically, the Jacobian measures how much a small area element expands or contracts during the mapping process.

When rewriting the integral in the new variables, the integrand must be multiplied by the absolute value of the Jacobian. This adjustment ensures that the transformed integral correctly preserves the area of the original region. By incorporating the Jacobian, complicated integrals over irregular regions can often be converted into much simpler integrals in the transformed coordinate system.