$C^k$ Map on an Open Subset of Euclidean Space

Let $k,n,m$ be natural numbers with $k\ge 1$. Let $U$ be an open subset of Euclidean space $\mathbb{R}^n$, and let $f=(f_1,\dots,f_m):U\to\mathbb{R}^m$.

We say that $f$ is of class $C^k$ on $U$ if for every multi-index $\alpha$ of length $n$ with order $|\alpha|\le k$ and every index $j\in\{1,\dots,m\}$, the partial derivative $\partial^\alpha f_j$ of order $\alpha$ exists on $U$, and the resulting function

$$\partial^\alpha f_j:U\to\mathbb{R}$$

is continuous at every point of $U$.

A real-valued map $f:U\to\mathbb{R}$ is of class $C^k$ if it is of class $C^k$ as a map from $U$ to $\mathbb{R}^1$.