For a complex manifold $X$, its sheaf of differential operators $\mathcal{D}_X$ is a sheaf of filtered algebras, and there is an isomorphism of sheaves of graded algebras
$$\text{gr } \mathcal{D}_X \xrightarrow{\sim}\mathcal{S}(\Omega_X)$$
(the right side is the symmetric algebra of the cotangent bundle). This is called the "symbol map".
One can also consider the sheaf of differential operators $\mathcal{E} \rightarrow \mathcal{F}$, for vector bundles $\mathcal{E}$ and $\mathcal{F}$. Is there a "symbol map" (which can be expressed in a way similar to the one above) in this situation too?