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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:
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:
Gauss' Theorem: The integral of a vector function F(x,y,z) over the surface of a closed three dimensional volume B is equal to the integral of the divergence of F(x,y,z) over the volume B. Restated in symbols this is:
∫∂BF.dA = ∫Bdiv(F)dV.
Stokes' Theorem: The integral of a vector function F(x,y,z) around a directed closed curve ∂B, which is the oriented boundary of an oriented surface B is equal to the integral of the curl of F over the surface B. Restated this is
∫∂BF.ds = ∫Bcurl(F).dA.
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