TITLE: How to use the inverse function theorem to find a local inverse?
QUESTION [5 upvotes]: I have a function $F(x,y)=(x^2 - y^2, xy)$ and I need to show that it has an inverse. How do I find the inverse of this function using the inverse function theorem? I did this a while ago and now I can't find it in my notes. I recall having to make $u = x^2 - y^2$ and $v = xy$ but then I'm lost.
Thank you for your assistance.

REPLY [2 votes]: The IFT proves the local existence of inverse function, but is not useful for finding the explicit form of the inverse function.
In this case,
$$
\frac{\partial(u,v)}{\partial(x,y)} =
\left|\matrix{2x&y\cr-2y&x}\right|=2(x^2+y^2)\ne 0\iff(x,y)\ne(0,0)
$$
and the inverse exists locally in any point except the origin.