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

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,

The inverse function theorem states that if $f:U\to {\mathbb{R}}^n$ is $C^1$ on an open set and $Df(a)$ is invertible, then there are open neighbourhoods $U_0$ of $a$ and $V_0$ of $f(a)$ such that $f:U_0\to V_0$ is a $C^1$ diffeomorphism. Its inverse satisfiesIf $f$ is $C^k$ or smooth, the inverse has the same regularity.

The planar implicit function theorem states that if $F$ is $C^1$ near $(a,b)$, $F(a,b)=0$, and $F_y(a,b)\ne 0$, then in some product neighbourhood the equation $F(x,y)=0$ is equivalent to $y=h(x)$ for a unique $C^1$ function with $h(a)=b$. Moreover,If instead $F_x(a,b)\ne 0$, exchange the coordinates to solve locally for $x$ as a function of $y$.