Differential operator between vector bundles

Summary

A differential operator between vector bundles over a manifold is a linear map from sections of one bundle to sections of the other. In local coordinates and trivialisations, it is a matrix of partial derivatives with smooth coefficients. It is the coordinate-free form of a linear system of partial differential equations on a manifold. The exterior derivative, the Laplacian on forms, Dirac operators and connections are all examples. Global analysis studies these operators on compact manifolds, where their properties are tied to the topology of the manifold.