Stokes' theorem

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In differential geometry, Stokes' theorem 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 in July 1850.[1] The theorem acquired its name from Stokes's habit of including it in the Cambridge prize examinations. In 1854, he asked his students to prove the theorem on an examination.

Introduction[edit]

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:

\int_a^b f(x)\,\mathrm dx = F(b) - F(a).

Stokes' theorem is a vast generalization of this theorem in the following sense.

  • By the choice of F, \frac{dF}{dx}=f. In the parlance of differential forms, this is saying that f(xdx is the exterior derivative of the 0-form (i.e. function) F: dF = f dx. The general Stokes theorem applies to higher differential forms \omega instead of F.
  • In fancy language, the closed interval [a, b] is a one-dimensional manifold with boundary. 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 manifold. 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:

\int_{[a, b]} f(x)\,dx = \int_{[a, b]} dF = \int_{\{a\}^- \cup \{b\}^+} F = F(b) - F(a).

General formulation[edit]

Let M be an oriented smooth manifold of dimension n and let \alpha be an n-differential form that is compactly supported on M. The integral of \alpha over M is defined as follows: Let {fi} be a partition of unity associated with a locally finite cover {Ui} of (consistently oriented) coordinate neighborhoods, then the integral

\int_M \alpha \,

is defined to be

\sum_i \int_{U_i} f_i \, \alpha ,\,

where each term in the sum is evaluated by pulling back to Rn. This is well-defined.

Stokes' theorem reads: If \omega is an (n − 1)-form with compact support on M and ∂M denotes the boundary of M with its induced orientation, then

\int_M \mathrm {d}\omega = \oint_{\partial M} \omega.\!\,

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 \omega is defined.

Topological reading[edit]

Let M be a smooth manifold. A (C-)singular k-simplex of M is a smooth map from the standard simplex in Rk to M. The free abelian group Sk generated by singular k-simplices is said to consist of singular k-chains of M. These groups, together with boundary map ∂, defines a chain complex. The corresponding homology (resp. cohomology) is called the (C-)singular homology (resp. cohomology) of M.

On the other hand, the differential forms, with exterior derivative d as the connecting map, form a cochain complex, which defines de Rham cohomology.

Differential k-forms can be integrated over a k-simplex in a natural way, by pulling back to Rk. Extending by linearity allows one to integrate over chains. This gives a linear map from the space of k-forms to the k-th group in the singular cochain Sk*, the linear functionals on Sk. In other words, a k-form \omega defines a functional

I(\omega)(c) = \int_c \omega \,

on the k-chains. Stokes' theorem says that this is a chain map from de Rham cohomology to singular cohomology; the exterior derivative d behaves like the "dual" of ∂ on forms. This gives a homomorphism from de Rham cohomology to singular cohomology. On the level of forms, this means

  1. closed forms have zero integral over boundaries and,
  2. exact forms have zero integral over cycles.

de Rham's theorem shows that this homomorphism is in fact an isomorphism. So the converse to 1 and 2 above hold true. In other words, if {ci} are cycles generating the k-th homology group, then for any corresponding real numbers {ai}, there exist a closed form \omega such that

\int_{c_i} \omega = a_i ,

and this form is unique up to exact forms.

Special cases[edit]

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 geometry 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.

Kelvin-Stokes theorem[edit]

An illustration of Kelvin-Stokes theorem with surface \Sigma, its boundary \scriptstyle{\partial \Sigma,} and orientation n.

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. It is also sometimes known as the curl theorem.

The classical Kelvin-Stokes theorem:

 \int_{\Sigma} \nabla \times \mathbf{F} \cdot d\mathbf{\Sigma} = \oint_{\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 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 ( ∂Σ ) must have positive orientation, meaning that d r points counterclockwise when the surface normal ( d Σ ) points toward the viewer, following the right-hand rule.

It can be rewritten for the student acquainted with forms as

\iint\limits_{\Sigma}\left(\frac{\partial R}{\partial y}-\frac{\partial Q}{\partial z}\right)\,dy\,dz  +\left(\frac{\partial P}{\partial z}-\frac{\partial R}{\partial x}\right)\,dz\,dx  +\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)\,dx\,dy    =\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(\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},
 \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}.

In electromagnetism[edit]

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. Caution must be taken to avoid cases with moving boundaries: the partial time derivatives are intended to exclude such cases. If moving boundaries are included, interchange of integration and differentiation introduces terms related to boundary motion not included in the results below:

Name Differential form Integral form (using Kelvin-Stokes theorem plus relativistic invariance, \int \frac{\partial }{\partial t } ...\to \frac{\mathrm d}{\mathrm dt} \int ...    )
Maxwell-Faraday equation
Faraday's law of induction:
  \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t} \oint_C \mathbf{E} \cdot d\mathbf{l} = \int_S  \nabla \times \mathbf{E} \cdot d\mathbf{A} = - \ { \mathrm d \over {\mathrm d t} }   \int_S   \mathbf{B} \cdot d\mathbf{A}   C and S stationary
Ampère's law
(with Maxwell's extension):
  \nabla \times \mathbf{H} = \mathbf{j} + \frac{\partial  \mathbf{D}} {\partial t}     \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} +
{\mathrm d \over {\mathrm dt}} \int_S \mathbf{D} \cdot d \mathbf{A}    C and S stationary

Divergence theorem[edit]

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

\int_{\mathrm{Vol}} \nabla \cdot \mathbf{F} \ d_\mathrm{Vol} = \oint_{\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.

Green's theorem[edit]

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.

References[edit]

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Further reading[edit]

External links[edit]

  • Olivier Darrigol,Electrodynamics from Ampere to Einstein, p. 146,ISBN 0198505930 Oxford (2000)