Partial Derivative of a Multi-Index 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 $\alpha=(\alpha_1,\dots,\alpha_n)$ be a multi-index of length $n$, with the zero multi-index $0$, the standard basis multi-indices $e_i$, and the difference $\alpha-e_i$ as in that definition. Index ranges such as $1\le i\le n$ use the order on the natural numbers.

For $f:U\to\mathbb{R}$ we define, recursively in the order $|\alpha|$, what it means for the partial derivative of $f$ of multi-index $\alpha$ to exist on $U$, and in that case its value $\partial^{\alpha}f:U\to\mathbb{R}$.

1. (Zero multi-index) If $\alpha=0$, then $\partial^{\alpha}f$ exists on $U$, and $\partial^{\alpha}f=f$.

2. (Recursion) Suppose $\alpha\ne 0$, and let $i$ be the least natural number $j$ with $1\le j\le n$ and $1\le\alpha_j$. Then $\partial^{\alpha}f$ exists on $U$ if $\partial^{\alpha-e_i}f$ exists on $U$ and the partial derivative of $\partial^{\alpha-e_i}f$ with respect to the $i$th variable exists at every point of $U$; in that case $\partial^{\alpha}f$ is the function from $U$ to $\mathbb{R}$ whose value at $x\in U$ is that partial derivative at $x$.

For a map $F=(F_1,\dots,F_m):U\to\mathbb{R}^{m}$ with coordinate functions $F_j:U\to\mathbb{R}$, we say that $\partial^{\alpha}F$ exists on $U$ if $\partial^{\alpha}F_j$ exists on $U$ for every $j$ with $1\le j\le m$; in that case $\partial^{\alpha}F$ is the map from $U$ to $\mathbb{R}^{m}$ whose $j$th coordinate function is $\partial^{\alpha}F_j$.