Chapter 6 - Integration of Differential Forms

Abstract

In this chapter we see how to integrate an n-form over an oriented n-dimensional smooth manifold with boundary. We begin by carefully reviewing the ordinary definite integral of a function on a (suitable) subset of Rn, and then see how to generalize it to our context. We give extensive examples to illustrate how to compute integrals of differential forms. In the case of a 1-form we show how exactness of the form is equivalent to path-independence of the integral. We further show how integration of a 1-form on an oriented curve in a smooth manifold with boundary corresponds to the classical notion of a line integral of the corresponding vector field. The particular case of a 1-form has a close relationship to classical notions in physics, and we develop the physical notions of work and energy, and show conservation of energy for conservative vector fields. We conclude by considering the notion of the integral of a k-form over a smooth k-chain in an n-manifold.