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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:
∫_{B}d(ω^{p}) =
∫_{∂B}(ω^{p})
where dim(B) = p+1.
In this statement ω^{p} is a differential pform and the dimension of the manifold B is p+1. The theorem says that the integral of the diffential of ω^{p}, itself a differential (p+1)form, over the manifold B is equal to the integral of ω^{p} over the oriented boundary of B, denoted as ∂B,the dimension of which is p1. The nature of differential forms, pforms, 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:
∫_{∂B}F^{.}dA = ∫_{B}div(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
∫_{∂B}F^{.}ds = ∫_{B}curl(F)^{.}dA.
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