Stokes' theorem

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Stokes' theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. It is named after Sir George Gabriel Stokes (1819-1903), although the first known statement of the theorem is by William Thomson (Lord Kelvin) and appears in a letter of his to Stokes. The theorem acquired its name from Stokes' habit of including it in the Cambridge prize examinations.

Let M be an oriented piecewise smooth manifold of dimension n and let <math>\omega</math> 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

<math>\int_M d\omega = \int_{\partial M} \omega.\!\,</math>

Here d is the exterior derivative, which is defined using the manifold structure only. The Stokes theorem can be considered as a generalisation of the fundamental theorem of calculus; and the latter indeed follows easily from the former.

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

The theorem easily extends to linear combinations 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:

<math> \int_{\Sigma} \nabla \times \mathbf{F} \cdot d\mathbf{\Sigma} = \int_{\partial\Sigma} \mathbf{F} \cdot d \mathbf{r}, </math>

which relates the surface integral of the curl of a vector field over a surface <math>\Sigma</math> 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

<math>\iint\limits_{\Sigma}\left(\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z}\right)\,dydz+\left(\frac{\partial P}{\partial z}-\frac{\partial R}{\partial x}\right)\,dzdx+\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)\,dxdy=\oint\limits_{\partial\Sigma}P\,dx+Q\,dy+R\,dz</math>

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

These variants are frequently used:

<math> \int_{\Sigma} \left( g \left(\nabla \times \mathbf{F}\right) + \left( \nabla g \right) \times \mathbf{F} \right) \cdot d\mathbf{\Sigma} = \int_{\partial\Sigma} g \mathbf{F} \cdot d \mathbf{r}, </math>

<math> \int_{\Sigma} d\mathbf{\Sigma}\cdot \nabla g = \int_{\partial\Sigma} g d \mathbf{r}, </math>

<math> \int_{\Sigma} \left( \mathbf{F} \left(\nabla \cdot \mathbf{G} \right) - \mathbf{G}\left(\nabla \cdot \mathbf{F} \right) + \left( \mathbf{G} \cdot \nabla \right) \mathbf{F} - \left(\mathbf{F} \cdot \nabla \right) \mathbf{G} \right) \cdot d\mathbf{\Sigma} = \int_{\partial\Sigma} \left( \mathbf{F} \times \mathbf{G}\right) \cdot d \mathbf{r}.</math>

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

<math>\int_{\mathrm{Vol}} \nabla \cdot \mathbf{F} \; d\mathrm{Vol} = \int_{\partial \mathrm{Vol}} \mathbf{F} \cdot d \mathbf{\Sigma}</math>

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.

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