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The Geometry of Physics by Theodore Frankel

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CHAPTER 5

The Poincaré Lemma and Potentials

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5.1. A More General Stokes’s Theorem

We shall accept the following technical generalizations of results already proven.

Let Vp be a compact oriented submanifold (perhaps with boundary) of Mn and let F : MnWm be a smooth map into a manifold Wm. The image F(V) in W need not be a submanifold. It might have self-intersections and all sorts of pathologies. Still, if βp is a form on W, it makes sense to talk of the integral of β over F(V) and in fact

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(5.1)

which generalizes (3.17). In a sense, ...

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