Integration on Manifolds

Abstract

In Chapter 11, we introduced line integrals of covector fields, which generalize ordinary integrals to the setting of curves in manifolds. It is also useful to generalize multiple integrals to manifolds. In this chapter, we carry out that generalization. As we show in the beginning of this chapter, there is no way to define the integral of a function in a coordinate-independent way on a smooth manifold. On the other hand, differential forms turn out to have just the right properties for defining integrals intrinsically. After defining integrals of differential forms over oriented smooth manifolds, we prove one of the most important theorems in differential geometry: Stokes’s theorem. It is a generalization of the fundamental theorem of calculus, the fundamental theorem for line integrals, and the classical theorems of vector analysis. Next, we show how these ideas play out on a Riemannian manifold. At the end of the chapter, we introduce densities, which are fields that can be integrated on any manifold, not just oriented ones.