Abstract
Cartan’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.
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References
Mathematical Tools: Selection of Mathematical Books
Y. Matsushima, Differentiable Manifolds (Marcel Dekker, New York, 1972)
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Straumann, N. (2013). Differential Forms. In: General Relativity. Graduate Texts in Physics. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5410-2_14
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DOI: https://doi.org/10.1007/978-94-007-5410-2_14
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-5409-6
Online ISBN: 978-94-007-5410-2
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