Linear differential operator

Definition

Given data

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

Definition part

A linear differential operator is a map $D:C^{\mathrm{\infty }}(M)\to C^{\mathrm{\infty }}(M)$ 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 $f_1,f_2,\dots ,f_k$:

$[[[\dots [D,f_1],f_2]\dots f_k]$

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