Integration on Manifolds

In this chapter, we define the integral of a k-form over a compact oriented k-manifold, and prove the important generalized Stokes’ theorem, which can be regarded as a far-reaching generalization of the fundamental theorem of calculus. We also define the integral of a function over a (not necessarily oriented) manifold, and describe the integral of a form in terms of the integral of a function. The classical theorems of vector analysis (Green’s theorem, divergence theorem, Stokes’ theorem) appear as special cases of the general Stokes’ theorem. Applications are made to topology (the Brouwer fixed point theorem) and to the study of harmonic functions (the mean value property, the maximum principle, Liouville’s theorem, and the Dirichlet principle).