differential operator
Roughly speaking, a differential operator is a mapping, typically understood to be linear, that transforms a function into another function by means of partial derivatives^{} and multiplication^{} by other functions.
On ${\mathbb{R}}^{n}$, a differential operator is commonly understood to be a linear transformation of ${\mathcal{C}}^{\mathrm{\infty}}({\mathbb{R}}^{n})$ having the form
$$f\mapsto \sum _{I}{a}^{I}{f}_{I},f\in {\mathcal{C}}^{\mathrm{\infty}}({\mathbb{R}}^{n}),$$ 
where the sum is taken over a finite number of multiindices $I=({i}^{1},\mathrm{\dots},{i}^{n})\in {\mathbb{N}}^{n}$, where ${a}^{I}\in {\mathcal{C}}^{\mathrm{\infty}}({\mathbb{R}}^{n})$, and where ${f}_{I}$ denotes a partial derivative of $f$ taken ${i}_{1}$ times with respect to the first variable, ${i}_{2}$ times with respect to the second variable, etc. The order of the operator is the maximum number of derivatives^{} taken in the above formula^{}, i.e. the maximum of ${i}_{1}+\mathrm{\dots}+{i}_{n}$ taken over all the $I$ involved in the above summation.
On a ${\mathcal{C}}^{\mathrm{\infty}}$ manifold $M$, a differential operator is commonly understood to be a linear transformation of ${\mathcal{C}}^{\mathrm{\infty}}(M)$ having the above form relative to some system of coordinates. Alternatively, one can equip ${\mathcal{C}}^{\mathrm{\infty}}(M)$ with the limitorder topology^{}, and define a differential operator as a continuous transformation of ${\mathcal{C}}^{\mathrm{\infty}}(M)$.
The order of a differential operator is a more subtle notion on a manifold than on ${\mathbb{R}}^{n}$. There are two complications. First, one would like a definition that is independent of any particular system of coordinates. Furthermore, the order of an operator is at best a local concept: it can change from point to point, and indeed be unbounded^{} if the manifold is noncompact. To address these issues, for a differential operator $T$ and $x\in M$, we define ${ord}_{x}(T)$ the order of $T$ at $x$, to be the smallest $k\in \mathbb{N}$ such that
$$T[{f}^{k+1}](x)=0$$ 
for all $f\in {\mathcal{C}}^{\mathrm{\infty}}(M)$ such that $f(x)=0$. For a fixed differential operator $T$, the function $ord(T):M\to \mathbb{N}$ defined by
$$x\mapsto {ord}_{x}(T)$$ 
is lower semicontinuous, meaning that
$${ord}_{y}(T)\ge {ord}_{x}(T)$$ 
for all $y\in M$ sufficiently close to $x$.
The global order of $T$ is defined to be the maximum of ${ord}_{x}(T)$ taken over all $x\in M$. This maximum may not exist if $M$ is noncompact, in which case one says that the order of $T$ is infinite^{}.
Let us conclude by making two remarks. The notion of a differential operator can be generalized even further by allowing the operator to act on sections^{} of a bundle.
A differential operator $T$ is a local operator, meaning that
$$T[f](x)=T[g](x),f,g\in {\mathcal{C}}^{\mathrm{\infty}}(M),x\in M,$$ 
if $f\equiv g$ in some neighborhood^{} of $x$. A theorem^{}, proved by Peetre states that the converse^{} is also true, namely that every local operator is necessarily a differential operator.

1.
Dieudonné, J.A., Foundations of modern analysis^{}

2.
Peetre, J. , “Une caractérisation abstraite des opérateurs différentiels”, Math. Scand., v. 7, 1959, p. 211
Title  differential operator 

Canonical name  DifferentialOperator 
Date of creation  20130322 12:20:29 
Last modified on  20130322 12:20:29 
Owner  rmilson (146) 
Last modified by  rmilson (146) 
Numerical id  10 
Author  rmilson (146) 
Entry type  Definition 
Classification  msc 5300 
Classification  msc 3500 
Classification  msc 47E05 
Classification  msc 47F05 
Related topic  Operator 