Smooth Map on a Euclidean Open Set

Let $n$ and $m$ be natural numbers, let $\mathbb{R}$ be the real numbers, let $U$ be an open subset of Euclidean space $\mathbb{R}^{n}$, and let $F:U\to\mathbb{R}^{m}$.

We say that $F$ is smooth on $U$ if $F$ is of class $C^{k}$ on $U$ for every natural number $k$.

A function $f:U\to\mathbb{R}$ is smooth on $U$ if it is smooth as a map into $\mathbb{R}^{1}$ with single coordinate function $f$, in accordance with the scalar convention of clause 3 of that definition.