Stokes' theorem

From Wikipedia, the free encyclopedia

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, which was given to students and asked to be proven.

1 General formulation
2 Topological reading
3 Special cases
4 References
5 External links

[edit] General formulation

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 = \oint_{\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 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 <math>\omega</math> is defined.

[edit] Topological reading

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.

[edit] Special cases

The general form of the Stokes theorem using differential forms is more powerful than the special cases. The latter are more accessible and have familiar names. They are often cited, and considered more convenient, by practicing scientists and engineers. There is potential for confusion in the way names are applied, and the use of dual formulations.

[edit] Kelvin-Stokes theorem

This is the (dualized) 1+1 dimensional case, for a 1-form (dualized because it is a statement about vector fields). This special case is often just referred to as the Stokes' theorem in many introductory university vector calculus courses.

The classical Kelvin-Stokes theorem:

<math> \int_{\Sigma} \nabla \times \mathbf{F} \cdot d\mathbf{\Sigma} = \oint_{\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 three-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 three-space. The curve of the line integral (<math>\partial\Sigma</math>) must have positive orientation, such that <math>d\mathbf{r}</math> points counterclockwise when the surface normal (<math>d\mathbf{\Sigma}</math>) points toward the viewer, following the right-hand rule.

It can be rewritten for the student acquainted 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} = \oint_{\partial\Sigma} g \mathbf{F} \cdot 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} = \oint_{\partial\Sigma} \left( \mathbf{F} \times \mathbf{G}\right) \cdot d \mathbf{r}.</math>

[edit] In Electromagnetism

Two of the four Maxwell equations involve curls of 3-D vector fields and their differential and integral forms are related by the Kelvin-Stokes theorem:

Name	Differential form	Integral form (using Kelvin-Stokes theorem)
Faraday's law of induction:	<math>\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}</math>	<math>\oint_C \mathbf{E} \cdot d\mathbf{l} = \int_S \nabla \times \mathbf{E} \cdot d\mathbf{A} = - \ { d \over dt } \int_S \mathbf{B} \cdot d\mathbf{A}</math>
Ampère's law (with Maxwell's extension):	<math>\nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}} {\partial t}</math>	<math>\oint_C \mathbf{H} \cdot d\mathbf{l} = \int_S \nabla \times \mathbf{H} \cdot d \mathbf{A} = \int_S \mathbf{J} \cdot d \mathbf{A} + {d \over dt} \int_S \mathbf{D} \cdot d \mathbf{A}</math>

[edit] Divergence theorem

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

<math>\int_{\mathrm{Vol}} \nabla \cdot \mathbf{F} \cdot d\mathrm{Vol} = \oint_{\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.

[edit] The Fundamental Theorem of Calculus

The fundamental theorem of calculus is the 0+1 dimensional case: the boundary is then the two endpoints, with <math>+</math> on the right and <math>-</math> on the left being the orientation.

[edit] Green's theorem

Green's theorem is immediately recognizable as the third integrand of both sides in the integral in terms of P, Q, and R cited above.

[edit] References

Stewart, James. Calculus: Concepts and Contexts. 2nd ed. Pacific Grove, CA: Brooks/Cole, 2001.
Marsden, Jerrold E., Anthony Tromba. Vector Calculus. 5th edition W. H. Freeman: 2003.
Spivak, Michael. Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. HarperCollins Publishers (June 1965). ISBN 0-8053-9021-9.
Joos, Georg. Theoretische Physik. 13th ed. Akademische Verlagsgesellschaft Wiesbaden 1980. ISBN 3-400-00013-2