definition of linear differential operators

An importance of the sheaves of jets is that they govern linear differential operators.

DEFINITION 9.8   Let $\varphi:X\to S$ be a separated morphism of schemes. Let $\mathcal{F},\mathcal{G}$ be quasi coherent sheaves on $X$ . Then a linear differential operator of $n$ -th order from $\mathcal{F}$ to $\mathcal{G}$ on $X$ relative to $S$ is a composition of an element of

$$\operatorname{Hom}_{\mathcal{O}_X} (\mathcal J_n(\mathcal{F}),\mathcal{G}).$$

with the Taylor expansion.