Linear differential operator
A connected differential manifold . The -algebra of -functions from to is denoted by .
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
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).