Linear differential operator

This article defines a property that can be evaluated for a differential operator on a differential manifold (viz a linear map from the space of differentiable functions to itself)

Definition

Given data

A connected differential manifold $M$. The $\mathbb {R}$-algebra of $C^{\infty }$-functions from $M$ to $\mathbb {R}$ is denoted by $C^{\infty }(M)$.

Definition part

A linear differential operator is a map $D:C^{\infty }(M)\to C^{\infty }(M)$ which has order $k$ for some integer $k$, where an operator $D$ is said to be of order $k$ if $D$ can be written as a fintie linear combination of compositions of derivations (vector field operators) with each composition involving at most $k$ derivations.

Equivalently, $D$ is of order $k$, if for any functions $f_{1},f_{2},\ldots ,f_{k}$:

$[[[\ldots [D,f_{1}],f_{2}]\ldots f_{k}]$

is an ordinary scalar function, where $[D,f](g)=D(fg)-f(Dg)$

Particular cases

It turns out that first-order linear differential operators can be expressed in the form $D+f$ where $D$ is a derivation and $f$ is a function (Acting by pointwise multiplication).