\begin{definition}[Definition:Partial Differential Operator]
Let $U \subseteq \R^n$ be a open set.
Let $\CC \subseteq \map {\CC^k} {U, \R}$ be a set of $k$-times continuously differentiable functions.
Let $\ds \partial_i = \frac {\partial} {\partial x_i}$ denote the partial derivative, $i = 1, \ldots, n$.
For a multiindex $\alpha = \tuple {\alpha_1, \ldots, \alpha_n}$ indexed by $\set {1, \ldots, n}$ let $\partial^\alpha = \partial_1^{\alpha_1} \cdots \partial_n^{\alpha_n}$.
A mapping $T : \CC \to \map {\CC^k} {U, \R}$ is a '''partial differential operator''' if there exist $r \in \N$ and functions $f_\alpha : \R^n \to \C$ for each multiindex $\alpha$ with $\cmod \alpha \le r$ such that for all $g \in \CC$:
:$\ds \map T g = \sum_{\cmod \alpha \mathop \le r} f_\alpha \partial^\alpha g$
\end{definition}