Partial Derivative of Order $\alpha$

Let $n\in\mathbb{N}$, let $U\subseteq \mathbb{R}^n$ be open, let $f:U\to\mathbb{R}$, and let

$$\alpha=(\alpha_1,\dots,\alpha_n)$$

be a multi-index of length $n$. We define recursively what it means for the partial derivative of $f$ of order $\alpha$ to exist on $U$, and when it does exist we denote it by

$$\partial^\alpha f.$$

1. If $\alpha=0$, then the derivative of order $\alpha$ exists on $U$ and is defined by

$$\partial^\alpha f = f.$$

2. Suppose $\alpha\ne 0$. We say that the derivative of order $\alpha$ exists on $U$ if there exists an index $i\in\{1,\dots,n\}$ such that $e_i\le \alpha$, the derivative

$$\partial^{\alpha-e_i}f$$

exists on $U$, and the ordinary partial derivative

$$\frac{\partial}{\partial x_i}\bigl(\partial^{\alpha-e_i}f\bigr)$$

exists at every point of $U$ in the sense of Partial Derivative of a Coordinate Function. In that case we define

$$\partial^\alpha f = \frac{\partial}{\partial x_i}\bigl(\partial^{\alpha-e_i}f\bigr).$$

For a map $F=(F_1,\dots,F_m):U\to\mathbb{R}^m$, we say that the derivative of order $\alpha$ exists on $U$ if each component derivative $\partial^\alpha F_j$ exists on $U$, and then we define

$$\partial^\alpha F = (\partial^\alpha F_1,\dots,\partial^\alpha F_m).$$