Sheaf of differential operators

Definition

The sheaf of differential operators of a differential manifold can be defined in many ways:

• It is the sheaf-theoretic algebra of differential operators for the sheaf of differentiable functions (in other words, if one views the sheaf of differentiable functions abstractly as a sheaf of $\mathbb{R}$-algebras, then the sheaf of differential operators is the sheaf-theoretic analogues of its algebra of differential operators)
• It associates to every open set, the algebra of differential operators of the $\mathbb{R}$-algebra of differentiable functions on that open set. Restriction maps are defined in the usual fashion.
• It is the sheaf generated, on every open set, by the algebra of differentiable functions, and the derivations