Applet: Nonlinear 2D change of variables map
The map $(x,y)=\cvarf(\cvarfv,\cvarsv) = (\cvarfv^2-\cvarsv^2,2\cvarfv \cvarsv)$ can be used to change variables in a double integral, yielding an integral in $d\cvarfv\,d\cvarsv$ from an integral in $dx\,dy$. To perform such a change of variables, one needs to calculate a region $\dlr^*$ in the $\cvarfv\cvarsv$-plane that is mapped onto the given region $\dlr$ in the $xy$-plane. This applet illustrates how a rectangle $\dlr^*$ in the $\cvarfv\cvarsv$-plane (left panel) is mapped by $\cvarf$ onto a region $\dlr$ in the $xy$-plane (right panel) whose boundaries are parabolas. You can change the regions $\dlr$ and $\dlr^*$ by dragging the purple and cyan points in either panel. To further visualize the action of the map $(x,y)=\cvarf(\cvarfv,\cvarsv)$, you can drag the labeled red and blue points anywhere inside the square $-2 \le \cvarfv \le 2$, $0 \le \cvarsv \le 4$ and the region in the $xy$-plane defined by $y^2/64-16 \le x \le 4-y^2/16$.
Applet file: nonlinear_2d_change_variables_map.ggb
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