Local implicit-function parameterization

Local implicit-function parameterization: Let $U\subseteq\mathbb R^n$ be an open set, let $F:U\to\mathbb R^m$ be continuously differentiable with $m<n$, and let $p\in U$ satisfy $F(p)=0$. If the derivative $DF(p):\mathbb R^n\to\mathbb R^m$ has rank $m$ (equivalently, $p$ is a regular point of $F$), then after reordering coordinates in $\mathbb R^n$ there exist neighborhoods $W$ of $p$ and $V$ of some $u_0\in\mathbb R^{n-m}$ and a continuously differentiable map $\varphi:V\to\mathbb R^m$ such that, in these coordinates,