Changing variables to calculate integral.

Question

If $u = x^{2} - y^{2}$ and $v=2xy$ where $u,v,x,y$ are all variables, use this to evaluate $\int \int 4(x^{4} - y^{4}) e^{-xy}dxdy $

I understand this question and I can basically do almost all of it except I'm struggling to change $(x^{4} - y^{4})$ into the variables $u$ and $v$. Any ideas? Thanks.

Answer 1

$(x^{2}+y^{2})^{2}=u^{2}+v^{2}$. So $x^{4}-y^{4}=u \sqrt {u^{2}+v^{2}}$.