I have the following question:
Using $u=xy$ and $v=\frac{x}{y}$, solve the integral $$I = \int_{x=0}^1\int_{y=x}^{\frac{1}{x}}\frac{x^3}{y}\exp\left((xy)^2+\frac{x^2}{y^2}\right)\,dy\,dx$$ I find computing the Jacobian and the integral ok, it’s finding the inverse of the $u=xy$ and $v=\frac{x}{y}$ (i.e. rearranging for $x=…$ and $y=…$) that I keep struggling with. Can someone explain the process, as solutions always skip the steps as it’s the ‘easy’ part of the problem.
Inverting the equations is not necessary. Instead using the $u$ and $v$ equations one can solve for the inverse Jacobian
$$J^{-1} = \left|\begin{vmatrix}y&x\\\frac{1}{y}&-\frac{x}{y^2}\end{vmatrix}\right| = \left|-\frac{x}{y}-\frac{x}{y}\right| = \frac{2x}{y}$$
then the Jacobian is simply
$$J = \frac{1}{J^{-1}} = \frac{1}{\frac{2x}{y}} = \frac{1}{2v}$$