ABSTRACT
In this chapter, we give meaning to the various notions of the previous chapter. We show first how to integrate a differential p-form over a so-called singular p-chain. The result is an integral which includes both line and surface integrals of vector fields as special cases. We then derive a general Stokes’ theorem for this integral and show that both Green’s theorem and Stokes’ theorem are special cases of it. As another special case, we prove Gauss’ theorem, which relates certain surface integrals in R 3 to volume integrals, and give an application to hydrodynamics.
