Coordinate Functions, the $C^k$ Hierarchy, and Partial Derivatives of a Smooth Map

Let $n$ and $m$ be natural numbers, let $\mathbb{R}$ be the real numbers, let $U$ be an open subset of Euclidean space $\mathbb{R}^{n}$, let $F=(F_1,\dots,F_m):U\to\mathbb{R}^{m}$ have coordinate functions $F_j:U\to\mathbb{R}$, and let $k$ be a natural number. Index ranges such as $1\le j\le m$ use the order on the natural numbers. Then the following hold.

1. (Coordinate functions) $F$ is of class $C^{k}$ on $U$ if and only if $F_j$ is of class $C^{k}$ on $U$ for every $j$ with $1\le j\le m$. Likewise, $F$ is smooth on $U$ if and only if $F_j$ is smooth on $U$ for every $j$ with $1\le j\le m$.

2. (Hierarchy) If $F$ is of class $C^{k+1}$ on $U$, then $F$ is of class $C^{k}$ on $U$.

3. (Partial derivatives of a smooth map) If $F$ is smooth on $U$, then for all $i$ and $j$ with $1\le i\le n$ and $1\le j\le m$ the partial derivative of $F_j$ with respect to the $i$th variable exists at every point of $U$, and the function $\partial_i F_j:U\to\mathbb{R}$ whose value at $x\in U$ is that partial derivative at $x$ is smooth on $U$.