Vector Bundles and the Global Point of View
In this chapter we develop a broader context in which to view the notion of a differential form: that of a vector bundle. We begin by defining vector bundles in general, and by considering related constructions, in particular that of the dual bundle. We then see how to the tangent bundle and cotangent bundle of a smooth manifold with boundary are (important) special cases of vector bundles. We can then view vector fields as sections of the tangent bundle, and -forms as sections of the cotangent bundle, with an analogous result holding for forms of higher degree.