Problem $3(15$ points$)$

When $D$ is the square with vertices

$(0,0),(2,-2),(4,0),(2,2),$

in the $xy-$plane, use the transformation

$T$:$(u,v)(x(u,v),y(u,v))$

with

$x=u+v,y=v-u$

to express the integral

$I={}_{D}f(x,y)dxdy$

in the form

$I={\displaystyle \underset{c}{\overset{d}{}}}({\displaystyle \underset{a}{\overset{b}{}}}F(u,v)|\frac{\text{d}\text{e}\text{l}(x,y)}{\text{d}\text{e}\text{l}(u,v)}|du)dv$

where

$F(u,v)=f(u+v,v-u).$