Differential operators as sections of a vector bundle

There is a standard way to obtain differential operators, even those acting on sections of a vector bundle, as sections of a vector bundle. This goes via jet bundles. Given a vector bundle $E\to M$, the $k$th jet prolongation $J^kE\to M$ is again a vector bundle. Its fiber over $x\in M$ consist of all equivalence classes of local sections defined in a neighborhood of $x$ under the equivalence relation that their Taylor series in $x$ (in some local chart for $M$ and some vector bundle chart of $E$) agree up to order $k$. Denoting the class of $s$ as $j^k_xs$, one defines a tautological map from smooth sections of $E$ to smooth sections of $J^kE$. This associates to $s\in\Gamma(E)$ the map $x\mapsto j^k_xs$. More or less by definition, for a $k$th order differential operator $D$, the value of $D(s)$ in $x$ only depends on $j^k_xs$. Then $D$ defines a bundle map $\tilde D:J^kE\to F$ such that $D(s)(x)=\tilde D(j^k_xs)$ and vice versa. For smooth functions, you can similarly use the jet spaces of smooth functions.