## Change of Variables: A Nonlinear Example

In multivariable calculus, we often use a "change of variables" transformation to make our double integrals easier to evaluate. In some cases these are linear transformations; in others, they are well known transformations, such as polar to rectangular coordinates. This page shows you an example of a nonlinear transformation which is not a standard change of coordinates (except that it's often used as an example in textbooks). It is given by:

The following example shows this transformation applied to the unit square in uv-space. One small square in uv-space is highlighted in red; its image under the transformation is highlighted on the right. You can click and drag the rectangle in the domain to see how its image changes.

Here are two things for you to think about with this transformation:

• Notice that the grid lines are not straight; using the equations above, can you figure out algebraically what the grid lines are?
• The "area expansion factor" of this transformation is given by . As you move the rectangle around, can you see this geometrically?
 To move the rectangle in uv-space, move the cursor over the red rectangle until a square is highlighted; then click and drag the mouse.

 The views and opinions expressed in this page are strictly those of the page author. The contents of this page have not been reviewed or approved by the University of Minnesota. rogness@math.umn.edu