Linear differential operator

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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)


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).