Past exam of the year 1 of the computer science course of the University of Oxford / 2014 / Discrete mathematics, continuous mathematics and probability / 6 / a

Differentiating the polynomial coordinate map givesIts Jacobian matrix isThe determinant identifies where the inverse function theorem applies.

On a branch with fixed signs of $u,v$ and $uv\ne 0$, use $u^2=x+y$ and $v^2=y-x$. Differentiating these identities, or inverting the Jacobian matrix, givesThe inverse Jacobian matrix is thereforeThese are local partial derivatives. Choosing the signs specifies an inverse function on $y>|x|$; the original map has four preimages there and has no single global inverse. At $u=0$ or $v=0$, the displayed inverse derivatives are not finite.