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 [\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_ \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 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:

[ \int_ \nabla \times \mathbf \cdot d\mathbf = \int_ \mathbf \cdot d \mathbf, ]

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 acquainted with forms as

[\iint\limits_\left(\frac-\frac\right)\,dydz+\left(\frac-\frac\right)\,dzdx+\left(\frac-\frac\right)\,dxdy=\oint\limits_P\,dx+Q\,dy+R\,dz]

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

These variants are frequently used:

[ \int_ \left( g \left(\nabla \times \mathbf\right) + \left( \nabla g \right) \times \mathbf \right) \cdot d\mathbf = \int_ g \mathbf \cdot d \mathbf, ]

[ \int_ d\mathbf\cdot \nabla g = \int_ g d \mathbf, ]

[ \int_ \left( \mathbf \left(\nabla \cdot \mathbf \right) - \mathbf\left(\nabla \cdot \mathbf \right) + \left( \mathbf \cdot \nabla \right) \mathbf - \left(\mathbf \cdot \nabla \right) \mathbf \right) \cdot d\mathbf = \int_ \left( \mathbf \times \mathbf\right) \cdot d \mathbf.]

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

[\int_} \nabla \cdot \mathbf \; d\mathrm = \int_} \mathbf \cdot d \mathbf]

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.

Stokes' theorem in physics

Altough Stokes' theorem is valid as it stands for arbitrary dimensions [n], it is the case [n=3] that is the most interesting to physicists. The way in which the theorem is conventionally stated by them is

[ \int_ \nabla \times \mathbf\, \mathbf = \oint_ \mathbf\, \mathbf. ]

Here, A is a vector field. On the left-hand-side the integration is performed on a surface denoted by [S]. [S] might for instance stand for a flat disc. On the right-hand-side the integral is to be calculated over the boundary of [S]. In the case of the flat disc, [\partial S] would be the perimeter. The vector da denotes the outward surface normal vector of magnitude [da] and dl is a line element along the boundary of [S], of magnitude [dl].

One of the reasons for the importance of Stokes' Theorem for [n=3] is its use in the theory of electricity and magnetism. Here the closed loop integral of the electric field E is equal to the work done on a unit charge by moving it along the closed loop. This amount is experimentally found to be proportional to the rate of change of the magnetic flux through any surface whose boundary is the closed loop. This results in the equation

[\oint_\mathbf\cdot \mathbf=-\frac\frac\int_\mathbf\cdot\mathbf]

where B is the magnetic field vector. If Stokes' theorem is now applied to the left-hand-side, the integrands may be set equal and one arrives at

[\nabla\times\mathbf=-\frac\frac\mathbf].

This is one of the four fundamental Maxwell's equations of electromagnetism.

References

Stewart, James. Calculus: Concepts and Contexts. 2nd ed. Pacific Grove, CA: Brooks/Cole, 2001.
Jerrold E. Marsden, 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 0805390219.
Joos, Georg. Theoretische Physik. 13th ed. Akademische Verlagsgesellschaft Wiesbaden 1980. ISBN 3-400-00013-2

External links

[Proof of general Stokes theorem] on PlanetMath

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