We gave the definition of a vector bundle in Section 2.1 and we mentioned that the set Vect(V) of all vector bundles has a sum and a product operation. In the first section of this chapter we examine a set of equivalence classes of bundles over a manifold V which we show to have the structure of a ring. This ring encodes some information on the topology of the manifold V, a result which we state without proof. The vector bundles over a manifold V can be described in terms of modules over the algebra C(V) of functions on V and they have natural noncommutative generalizations. The vector space H of smooth functions ƒ defined on ℝn with values in ℂr can be considered as the space of sections of a trivial vector bundle H over ℝn with fibre ℂr. Within the algebra of all operators on H the differential operators are of special interest. These are polynomials in the partial derivatives of ƒ with respect to the coordinates of ℝn, with smooth functions as coefficients. Different copies of ℂn can be patched together in different ways to form manifolds and at the same time the different copies of Cr can be patched to form non-trivial vector bundles. Consider a differential operator P defined on the corresponding vector space H of smooth sections of one of these bundles.