Find $$\int_0^{\infty} \int_0^{\infty} e^{-2xy} \, \mathrm d x \mathrm dy$$ using $u = x^2 - y^2$ and $v=2xy$.
I have tried using the Jacobian matrix to obtain the Jacobian of the transformation. However, confusion arises since I do not know what should be kept constant. Do I directly differentiate $u$ with respect to $x$ while keeping $y$ constant, or do I substitute $y$ from $v=2xy$, and then differentiate $u$ with respect to $x$ while keeping $v$ constant?
It is always the old, original, variables with respect the new ones...but you can do it the other way around and take the inverse matrix's determinant!:
$$J=J\frac{(x,y)}{(u,v)}=\begin{vmatrix}\frac{\partial u}{\partial x}&\frac{\partial u}{\partial y}\\\frac{\partial v}{\partial x}&\frac{\partial v}{\partial y}\end{vmatrix}^{-1}=\begin{vmatrix}2x&-2y\\2y&2x\end{vmatrix}^{-1}=\frac1{4(x^2+y^2)}$$
We took the inverse determinant since we differentiated the new variables wrt the original ones...and there exists that relation.