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The Generalized Stokes Theorem

The Generalized Stokes Theorem
and Differential Forms

Mathematics is a very practical subject but it also has its aesthetic elements. One of the most beautiful topics is the Generalized Stokes Theorem. This beauty comes from bringing together a variety of topics: integration, differentiation, manifolds and boundaries. The Generalized Stokes Theorem can be stated quite succintly:


Bd(ωp) = ∫∂Bp)
where dim(B) = p+1.
 

In this statement ωp is a differential p-form and the dimension of the manifold B is p+1. The theorem says that the integral of the diffential of ωp, itself a differential (p+1)-form, over the manifold B is equal to the integral of ωp over the oriented boundary of B, denoted as ∂B,the dimension of which is p-1. The nature of differential forms, p-forms, is explained in more detail elsewhere. Here the nature of the Generalized Stokes Theorem will be illustrated.

The Generalized Stokes Theorem incorporates two theorems important in physics:

 

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