Differential Forms

Abstract

AbstractCartan’s calculus of differential forms is particularly useful in general relativity (and also in other fields of physics). We begin our discussion by repeating some algebraic preliminaries on exterior algebras. Then exterior differential forms and the associated exterior algebra are introduced. On this we study general properties of derivations and antiderivations. The most important one is Cartan’s exterior derivative. Poincaré’s Lemma is also an important tool in physics. A proof of it will be given in Chap. 15. Useful formulas for the exterior derivative will be derived, as well as relations with the Lie derivative and the interior product. After that we introduce the ∗-operation and the codifferential, for which we establish various properties. An important subsection is devoted to the integral theorems of Stokes and Gauss.KeywordsDifferential FormVolume FormLocal Coordinate SystemExterior DerivativeExterior AlgebraThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.