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Let ''M'' be an oriented piecewise smooth Manifold of Dimension ''n'' and let \omega be an ''n''−1 form that is a Compactly Supported Differential Form on ''M'' of class C1. If ∂''M'' denotes the Boundary of ''M'' with its induced Orientation , then

:\int_M d\omega = \int_{\partial M} \omega.\!\,

Here ''d'' is the Exterior Derivative , which is defined using the manifold structure only. The Stokes theorem can be considered as a generalization of the Fundamental Theorem Of Calculus .

The theorem is often used in situations where ''M'' is an embedded oriented submanifold of some bigger manifold on which the form \omega is defined.

The theorem easily extends to Linear Combination s of Piecewise Smooth submanifolds, so-called chains. The Stokes theorem then shows that closed forms defined up to an Exact Form can be integrated over chains defined only up to a Boundary . This is the basis for the pairing between Homology Groups and De Rham Cohomology .

The classical Kelvin-Stokes theorem:

: \int_{\Sigma}
abla imes \mathbf{F} \cdot d\mathbf{\Sigma} = \int_{\partial\Sigma} \mathbf{F} \cdot d \mathbf{r},

which relates the Surface Integral of the Curl of a Vector Field over a surface \Sigma in Euclidean 3 space to the line integral of the vector field over its boundary, is a special case of the general Stokes theorem (with ''n'' = 2) once we identify a vector field with a 1 form using the metric on Euclidean 3 space. It can be rewritten for the student unacquainted with forms as

:\iint\limits_{\Sigma}\left( rac{\partial R}{\partial y}- rac{\partial Q}{\partial z} ight)\,dydz+\left( rac{\partial P}{\partial z}- rac{\partial R}{\partial x} ight)\,dzdx+\left( rac{\partial Q}{\partial x}- rac{\partial P}{\partial y} ight)\,dxdy=\oint\limits_{\partial\Sigma}P\,dx+Q\,dy+R\,dz

where ''P'', ''Q'' and ''R'' are the components of F.

These variants are frequently used:
: \int_{\Sigma} \left( g \left(
abla imes \mathbf{F} ight) + \left(
abla g ight) imes \mathbf{F} ight) \cdot d\mathbf{\Sigma} = \int_{\partial\Sigma} g \mathbf{F} \cdot d \mathbf{r},

: \int_{\Sigma} d\mathbf{\Sigma}\cdot
abla g = \int_{\partial\Sigma} g d \mathbf{r},

: \int_{\Sigma} \left( \mathbf{F} \left(
abla \cdot \mathbf{G} ight) - \mathbf{G}\left(
abla \cdot \mathbf{F} ight) + \left( \mathbf{G} \cdot
abla ight) \mathbf{F} - \left(\mathbf{F} \cdot
abla ight) \mathbf{G} ight) \cdot d\mathbf{\Sigma} = \int_{\partial\Sigma} \left( \mathbf{F} imes \mathbf{G} ight) \cdot d \mathbf{r}.

Likewise the Ostrogradsky-Gauss Theorem (also known as the Divergence theorem or Gauss' theorem)

:\int_{\mathrm{Vol}}
abla \cdot \mathbf{F} \; d\mathrm{Vol} = \int_{\partial \mathrm{Vol}} \mathbf{F} \cdot d \mathbf{\Sigma}

is a special case if we identify a vector field with the ''n''−1 form obtained by contracting the vector field with the Euclidean volume form.

The Fundamental Theorem Of Calculus and Green's Theorem are also special cases of the general Stokes theorem.

The general form of the Stokes theorem using differential forms is more powerful than the special cases, of course, although the latter are more accessible and are often considered more convenient by practicing scientists and engineers.


REFERENCES

  • Stewart, James. ''Calculus: Concepts and Contexts''. 2nd ed. Pacific Grove, CA: Brooks/Cole, 2001.