# 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 . The -algebra of -functions from to is denoted by .

### Definition part

A linear differential operator is a map which has order for some integer , where an operator is said to be of order if can be written as a fintie linear combination of compositions of derivations (vector field operators) with each composition involving at most derivations.

Equivalently, is of order , if for any functions :

is an ordinary scalar function, where

## Particular cases

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