$C^k$ Euclidean maps and diffeomorphisms

Let $k\in \mathbb{N}$, let $m,q\ge 1$, let $U\subseteq {\mathbb{R}}^m$ be open, and let $f:U\to {\mathbb{R}}^q$. A map $f:U\to {\mathbb{R}}^q$ is of class $C^k$ when each component is of class $C^k$. Each scalar component uses the word-derivative convention of $C^k$ maps and multi-index derivative notation in Euclidean space, whose source dimension is positive. The map is smooth, or $C^{\mathrm{\infty }}$, when it is $C^k$ for every $k\in \mathbb{N}$.

Let $n\ge 1$ and let $U,V\subseteq {\mathbb{R}}^n$ be open. A bijection $f:U\to V$ is a $C^k$ diffeomorphism when both $f$ and $f^{-1}$ are $C^k$. For $k\ge 1$, a local $C^k$ diffeomorphism at $a\in U$ is a restriction $f{\mathrm{\mid }}_P:P\to Q$ between open neighbourhoods of $a$ and $f(a)$ that is a $C^k$ diffeomorphism. At $k=1$ this agrees with Continuously differentiable maps, local inverses, and local diffeomorphisms: continuous first partial derivatives give the required total derivative by If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative, while continuity of the total derivative gives continuity of its matrix entries and hence of the first partial derivatives.