Is there a coordinate-free definition of a differential operator? A differential operator is normally defined to be a sum of partial derivatives with respect to a given basis, i.e. something that looks like $\sum_{\vert {\alpha} \vert \leq n} a_{\alpha}(x) \cdot \partial^{\alpha}$. Differential operators of interest tend to be naturally seen in a coordinate-free way -- for example, the laplacian is the divergence of the gradient. Is there a coordiante-free way to define a general differential operator? 

There is a characterization due to Peetre which ensures that $P:\mathcal{D}(\Omega) \rightarrow \mathcal{D}'(\Omega)$ is a differential operator if, and only if $\hbox{supp }(Pu) \subset \hbox{supp } u$, where $\Omega$ is a open subset of $\mathbb{R}^n$. See: Peetre, J. Une Caractérisation Abstraite des Opérateurs Différentiels and Peetre, J. Réctification a L'article - Une Caractérisation Abstraite des Opérateurs Différentiels Théorème 2.
There are several generalizations of this result for manifolds and fiber bundles.
