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, which was given to students and asked to be proven.
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[edit] Introduction
The fundamental theorem of calculus states that the integral of a function f over the interval [a, b] can be calculated by finding an antiderivative F of f:
.
Stoke's theorem is a vast generalization of this theorem in the following sense.
- By the choice of F,
. In the parlance of differential forms, this is saying that f(x) dx is the exterior derivative of the 0-form (i.e. function) F: dF = f dx. The general Stokes theorem applies to higher differential forms ω instead of F. - In a fancy language, the open interval (a, b) is a one-dimensional manifold. Its boundary is the set consisting of the two points a and b. Integrating f over the interval may be generalized to integrating forms on a higher-dimensional manifold. Two technical conditions are needed: the manifold has to be orientable, and the form has to be compactly supported in order to give a well-defined integral.
- The two points a and b form the boundary of the open interval. More generally, Stokes' theorem applies to (oriented) manifolds M with boundary. The boundary ∂M of M is itself a manifold and inherits a natural orientation from that of the boundary. For example, the natural orientation of the interval gives an orientation of the two boundary points. Intuitively, a inherits the opposite orientation as b, as they are at opposite ends of the interval. So, "integrating" F over two boundary points a, b is taking the difference F(b) - F(a).
So the fundamental theorem reads:
.
[edit] General formulation
Let M be an oriented piecewise smooth manifold of dimension n and let ω 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
Here d is the exterior derivative, which is defined using the manifold structure only.
The theorem is often used in situations where M is an embedded oriented submanifold of some bigger manifold on which the form ω 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 and easier to use than the special cases . Because in Cartesian coordinates the traditional versions can be formulated without the machinery of differential they are more accessible, older and have familiar names. The traditional forms are often considered more convenient by practicing scientists and engineers but the non naturalness of the traditional formulation becomes apparent when using other coordinate systems, even familiar ones like spherical or cylindrical coordinates. 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:
which relates the surface integral of the curl of a vector field over a surface Σ 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 (
) must have positive orientation, such that 
points counterclockwise when the surface normal (
) points toward the viewer, following the right-hand rule.
It can be rewritten for the student acquainted with forms as
where P, Q and R are the components of F.
These variants are frequently used:
[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: |  |
 |
| Ampère's law (with Maxwell's extension): |
 |
 |
[edit] Divergence theorem
Likewise the Ostrogradsky-Gauss theorem (also known as the Divergence theorem or Gauss' theorem)
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] 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
