Let $n$ be a \reftext{def:natural-numbers-2026a}{natural number}, let $U$ be an \reftext{def:open-subset-euclidean-space-2026a}{open} subset of \reftext{def:euclidean-space-rn-2026a}{Euclidean space} $\mathbb{R}^n$, and let $f:U\to\mathbb{R}^n$ be a \reftext{def:c1-map-euclidean-open-set-2026a}{$C^1$ map}. Let $a\in U$. We say that $f$ is a \textit{local $C^1$ diffeomorphism at $a$} if there exist \reftext{def:open-subset-euclidean-space-2026a}{open} sets $V,W\subseteq\mathbb{R}^n$ with $a\in V\subseteq U$ and $f(a)\in W$ such that $f(V)=W$, the restriction $f|_V:V\to W$ is \reftext{def:bijection-sets-2026a}{bijective}, and its inverse $(f|_V)^{-1}:W\to V$ is \reftext{def:c1-map-euclidean-open-set-2026a}{of class $C^1$}.

We say $f$ is a \textit{local $C^1$ diffeomorphism} (on $U$) if $f$ is a local $C^1$ diffeomorphism at every $a\in U$.