Smooth Map on an Open Subset of Euclidean Space

Let $n,m\in\mathbb{N}$, let $U\subseteq \mathbb{R}^n$ be open, and let $F=(F_1,\dots,F_m):U\to\mathbb{R}^m$. We say that $F$ is smooth on $U$ if for every multi-index $\alpha$ of length $n$ and every index $j\in\{1,\dots,m\}$, the partial derivative of $F_j$ of order $\alpha$ exists on $U$ in the sense of Partial Derivative of Order $\alpha$, 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 smooth if it is smooth as a map from $U$ to $\mathbb{R}^1$.