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The Geometry of Physics by Theodore Frankel

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CHAPTER 3

Integration of Differential Forms

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EXTERIOR differential forms occur implicitly in all aspects of physics and engineering because they are the natural objects appearing as integrands of line, surface, and volume integrals as well as the n-dimensional generalizations required in, for example, Hamiltonian mechanics, relativity, and string theories. We shall see in this chapter that one does not integrate vectors; one integrates forms. If there is extra structure available, for example, a Riemannian metric, then it is possible to rephrase an integration, say of exterior 1-forms or 2-forms, in terms of a vector integrations involving “arc ...

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