Past exam of the options I for mathematics and philosophy course of the University of Oxford / 2013 / F / 1 / c

A $C^1$ diffeomorphism $F:U\to V$ is a bijection for which both $F$ and $F^{-1}$ are continuously differentiable. If a higher regularity class is specified, both maps must belong to that class. In particular, a diffeomorphism is a homeomorphism, and the chain rule gives $D(F^{-1})(F(x))=(DF(x){)}^{-1}$.

The inverse function theorem states: let $F:U\subseteq {\mathbb{R}}^n\to {\mathbb{R}}^n$ be $C^1$, with $U$ open, and suppose $DF(x_0)$ is invertible. Then there exist open neighbourhoods $U_0\subseteq U$ of $x_0$ and $V_0$ of $F(x_0)$ such that $F:U_0\to V_0$ is a $C^1$ diffeomorphism. Its inverse derivative satisfiesThe conclusion is local; invertibility of one Jacobian matrix does not assert global injectivity.

The map $G$ is $C^1$ on the open ball $B$. Its Jacobian matrix isAt the origin $G(0)=0$ and $\boxed{\det DG(0)=1/3\ne 0}$. The inverse function theorem therefore supplies the required open sets $V_1\subset B$ and $V_2\subset {\mathbb{R}}^3$, both containing zero, with $G:V_1\to V_2$ a diffeomorphism. The explicit inverse formula on its image,also checks the local conclusion directly.