Differential Forms

This chapter shows how to generalize line and surface integrals to more general sorts of integrals on Rⁿ. It provides a far-reaching generalization of Green's theorem and Stokes' theorem. The chapter describes a good deal of material, much of it formal and not obviously connected with integration. It is concerned with the more formal aspects of the theory. The chapter presents algebraic operations on differential forms. It explores differential forms and shows how to add, multiply, and differentiate them, and discusses the relationships among these operations. The chapter introduces the notion of a differential form on an open subset of Euclidean space. It discusses the relationship between the exterior differential and certain important operations on vector fields and functions. The reason for introducing differential forms is, in fact, to allow people to generalize line integrals to higher dimensional surfaces.