Chapter 5.4, Problem 20P

Find the Jacobians $\partial \left(x,y\right)/\partial \left(u,v\right)$ of the given transformations from variables $x,y$ to variables $u,v$

In the integral $I={\displaystyle {\int }_{x=0}^{1/2}}{\displaystyle {\int }_{y=x}^{1-x}{\left(\frac{x-y}{x+y}\right)}^{2}dydx,}$ make the change of variables $\begin{array}{l}x=\frac{1}{2}\left(r-s\right),\\ y=\frac{1}{2}\left(r+s\right),\end{array}$ and evaluate I. Hints: See Problem 19. To find the r and s limits, sketch the area of integration in the (x,y) plane and sketch the r and s axes. Then show that to cover the same integration area, you may take the r and s limits to be: s from 0 to r, r from 0 to 1.