Differential Forms

Abstract

Distributions can be viewed as linear functionals acting on a suitable space of test functions; correspondingly currents, as an extension of the concept of distributions, are continuous linear functionals acting on a suitable space of differential forms. To this aim the space of differential forms is endowed with a suitable topology.

This chapter is devoted to a presentation of differential forms together with their relevant properties which will be required in later chapters, more specifically dedicated to currents. In particular it includes operations on differential forms, the definition of pullback of a form, and the definitions of line integrals and surface integrals of forms. The close link between differential forms and vector fields is taken into account.