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
Applet links
This applet is found in the pages
General information about Geogebra Web applets
This applet was created using Geogebra. In most Geogebra applets, you can move objects by dragging them with the mouse. In some, you can enter values with the keyboard. To reset the applet to its original view, click the icon in the upper right hand corner.
You can download the applet onto your own computer so you can use it outside this web page or even modify it to improve it. You simply need to download the above applet file and download the Geogebra program onto your own computer.