Chapter 2 - Differential Forms in [formula omitted], II

Abstract

In this chapter we consider the meaning of differential forms in Rn: k-forms are alternating multilinear functions on k-tuples of tangent vectors all based at the same point. We provide extensive examples, showing how to effectively compute with differential forms. We further see how to interpret exterior differentiation from this point of view. We develop the notion of an orientation for Rn and we see how to interpret the values of certain differential forms in terms of signed volume. We prove the converse of Poincaré’s Lemma in the case of forms defined on a star-shaped region in Rn.