Applet: Area transformation of nonlinear 2D change of variables map
When using the map $(x,y)=\cvarf(\cvarfv,\cvarsv) = (\cvarfv^2-\cvarsv^2,2\cvarfv \cvarsv)$ to change variables in a double integral, one needs to calculate a region $\dlr^*$ in the $\cvarfv\cvarsv$-plane (rectangle in left panel) that is mapped onto the given region $\dlr$ in the $xy$-plane (such as the region in right panel). The transformation $\cvarf$ also maps each small rectangle in $\dlr^*$ to a “curvy rectangle” in $\dlr$. Although the small rectangles in $\dlr^*$ are the same size, the corresponding “curvy rectangles” vary greatly in size. Depending on the coordinates $(\cvarfv,\cvarsv)$, the map $\cvarf(\cvarfv,\cvarsv)$ shrinks or expands the area by different amounts. You can visualize the mapping of the small rectangles by dragging the yellow point in either panel; the corresponding small rectangle in $\dlr^*$ and its image in $\dlr$ are highlighted. You can also change the regions $\dlr^*$ and $\dlr$ by dragging the purple and cyan points in either panel.
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