How to use the inverse function theorem to find a local inverse? 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.
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,
$$
\frac{\partial(u,v)}{\partial(x,y)} =
\left|\matrix{2x&amp;y\cr-2y&amp;x}\right|=2(x^2+y^2)\ne 0\iff(x,y)\ne(0,0)
$$
and the inverse exists locally in any point except the origin.
For a concrete formula write, as you say, $u=x^2-y^2$ and $v=xy$. Then $y=v/x$, so that 
$$u=x^2-\frac{v^2}{x^2}$$
which implies that 
$$x^4-ux^2-v^2=0.$$
Solve the above equation for $x$ using the quadratic formula and substitute back into $y=v/x$ to find the equation for $y.$

Of course care must be taken when choosing signs for particular regions of $\mathbb{R}^2$.