Abstract
In this chapter, we begin to develop the theory of differential forms, which are alternating tensor fields on manifolds. It might come as a surprise, but these innocent-sounding objects turn out to be considerably more important in smooth manifold theory than symmetric tensor fields, because they provide a framework for generalizing a variety of concepts from multivariable calculus. The purpose of this chapter is to describe the main tools for manipulating differential forms. The most important such tool is the exterior derivative, which generalizes the differential of a smooth function that we introduced in Chapter 11, as well as the gradient, divergence, and curl operators of multivariable calculus. At the end of the chapter, we will see how the exterior derivative can be used to simplify the computation of Lie derivatives of differential forms.
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References
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Lee, J.M. (2013). Differential Forms. In: Introduction to Smooth Manifolds. Graduate Texts in Mathematics, vol 218. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-9982-5_14
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DOI: https://doi.org/10.1007/978-1-4419-9982-5_14
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