Exterior algebra and differential calculus

Abstract

In this chapter we introduce the algebra and calculus of differential forms, which also goes by the name “exterior differential calculus.” In Section 6.3 we defined the concept of differential form of degree 1, and in Section 6.4, the integral ∫ _γ ω of such a differential form ω along a curve γ. This integral changes sign if the direction in which γ is traversed is reversed. In Chapter 8 we define the integral of a differential form ω of higher degree r over a portion A of an r-dimensional manifold. For example, for r = 2 a differential form of degree 2 can be integrated over a piece of surface A. The integral of a differential form depends on the orientation assigned to A and changes sign if the orientation is reversed.