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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 succinctly:

#### ∫Bd(ωp) = ∫∂B(ωp) 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.

Several important theorems are simply special cases of the Generalized Stokes Theorem.

• One Dimension

The Fundamental Theorem of Calculus:

#### ∫ab(dF/dx)dx = F(b) − F(a)

In terms of the above statement of the GST, the manifold B is the line segment from x=a to x=b. Its boundary is the pair of points x=a and x=b. The ω0 is F(x) and its integral over the boundary is [F(b)−F(a)].

More explicitly the integral over the boundary is h(b)F(b)+h(a)F(a), where h(b) is the orientation of the boundary point x=b and h(a) is that of x=a. The orientation of b is +1 and that a is −1, hence F(b)−F(a).

• Two Dimensions:

Stokes Theorem:

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.

• Two Dimensions:

Green's Theorem: Let B be a region in a plane that is enclosed by a positively oriented smooth simple closed curve ∂B. Then

#### ∫∂B(Pdx + Qdy) = ∫∫B[(∂Q/∂x) − (∂P/∂y)]dxdy

• Three Dimensions:

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: