Differential forms constitutes an approach to multivariable calculus that simplifies the study of integration over surfaces of any dimension in Rp. This topic introduces algebraic techniques into the study of higher dimensional geometry and allows us to recapture with rigor the results obtained in the preceding chapter.

There are many approaches to defining and exploring differential forms, all leading to the same objective. One is to just define a form as a symbol with certain properties. That was the approach taken when I first encountered the subject. Though this appeals to many and ultimately leads to the same objective, it strikes me as inconsistent with the approach we have taken. So in this chapter I have decided to follow the approach in [7] where a form is defined in terms that are more in line with what I think is the background of readers of this book. The reader might also want to look at [1], [9], and [10] where there are different approaches.

Introduction

Here we will define differential forms and explore their algebraic properties and the process of differentiating them. The first definition extends that of a surface as given in the preceding chapter.

Definition. Let q ≥ 1. A q-surface domain or just a surface domain is a compact Jordan subset R of Rq such that int R is connected and R = cl (int R). A q-surface in Rp is a pair where R is a q-surface domain and is a function from R into Rp that is smooth on some neighborhood of R. The trace of is the set. If G is an open subset of Rp and, we say is a q-surface in G; let Sq(G) be the collection of all q-surfaces contained in G.