# 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 $\R$-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).