Past exam of the options II course of the University of Oxford / 2012 / F / F2 / a / i / Solution

A $C^1$ diffeomorphism between open subsets of ${\mathbb{R}}^n$ is a bijection $f:U\to V$ for which both $f$ and $f^{-1}$ are continuously differentiable. More generally, a $C^k$ diffeomorphism requires that regularity for both maps; a smooth diffeomorphism has derivatives of every order. In the $C^1$ setting used here,