Chapter 1 - Differential Forms in [formula omitted], I

Abstract

In this chapter we consider differential forms as formal objects in Rn. We show how to do algebra with differential forms, this algebra being known as exterior algebra, and we introduce the operation of exterior differentiation. We state and prove Poincaré’s Lemma. Focusing on R3, we show how, in R3, differential forms “correspond” to functions and vector fields and how, under this correspondence, the classical operations of gradient, curl, and divergence on vector fields are all special cases of exterior differentiation. We prove the converse of Poincaré’s Lemma in the case of forms defined on all of Rn.