# Math Insight

### Applet: Nonlinear 2D change of variables animation

This animation illustrates the mapping of a rectangle by the change of variables function $(x,y)=\cvarf(\cvarfv,\cvarsv)=(\cvarfv^2-\cvarsv^2,2\cvarfv\cvarsv)$. You can change the function $\cvarf$ by typing in new formulas for its components using the boxes in right panel. The left panel shows a rectangle $\dlr^*$ in the $\cvarfv\cvarsv$-plane, which you can change by dragging the yellow points. The mapping of $\dlr^*$ by $\cvarf$ is illustrated by the animation in the right panel, where the region $\dlr^*$ is stretched into the region $\dlr$, which is the image of $\dlr^*$ under the mapping $\cvarf$. The progress of the animation is shown by the blue slider in the upper left corner. When the blue point moves from left to right, the region in the right panel changes from the reference rectangle $\dlr^*$ into its image $\dlr$. You can stop and start the animation by clicking the button that appears in the lower left of one of the panels. You can also step through the animation by dragging the blue point on the slider. Use the + and - buttons of each panel to zoom in and out.

For the transformation $(x,y)=\cvarf(\cvarfv,\cvarsv)=(\cvarfv^2-\cvarsv^2,2\cvarfv\cvarsv)$, you must keep the rectangle $\dlr^*$ completely in one half-plane for $\cvarf$ to be a valid change of variables. If both $\cvarfv$ and $\cvarsv$ can change sign in $\dlr^*$, then the mapping onto $\dlr$ overlaps itself and isn't one-to-one. If you change the transformation, then there may be other restrictions on $\dlr^*$ to keep the mapping of $\dlr^*$ by $\cvarf$ one-to-one.

The animation of $\cvarf(\cvarfv,\cvarsv)$ uses a linear homotopy between $\cvarf$ and the identity. The right panel shows the mapping of the function $(x,y)=a\cvarf(\cvarfv,\cvarsv) + (1-a)(\cvarfv,\cvarsv)$. The value of $a$ varies from 0 to 1 with the blue slider. When $a=0$, the map is the identity, and the right panel shows the same rectangle $\dlr^*$ as the left panel. When $a=1$, the map is $\cvarf$, and the right panel shows the image $\dlr$ of the rectangle $\dlr^*$ under the map $\cvarf$.

Applet file: nonlinear_2d_change_variables_map_animation.ggb