Stokes\' theorem - Citizendia

Stokes' theorem (or Stokes's theorem) in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is In Mathematics, a theorem is a statement proven on the basis of previously accepted or established statements Vector calculus (also called vector analysis) is a field of Mathematics concerned with multivariable Real analysis of vectors in an Inner 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. Knight is the English term for a social position originating in the Middle Ages. Sir George Gabriel Stokes 1st Baronet FRS ( 13 August 1819 &ndash 1 February 1903) was a mathematician and physicist Year 1819 ( MDCCCXIX) was a Common year starting on Friday (link will display the full calendar in the Gregorian Calendar (or a Common year Year 1903 ( MCMIII) was a Common year starting on Thursday (link will display calendar of the Gregorian calendar or a Common year starting William Thomson 1st Baron Kelvin (or Lord Kelvin) OM, GCVO, PC, PRS, FRSE, (26 June 1824 &ndash 17 December 1907 ^[1] The theorem acquired its name from Stokes' habit of including it in the Cambridge prize examinations. The University of Cambridge (often Cambridge University) located in Cambridge, England, is the second-oldest university in the In 1854, he asked his students to prove the theorem on an examination; it is unknown if anyone was able to do so.

1 Introduction
2 General formulation
3 Topological reading
4 Special cases
5 References
6 Further Reading
7 External links

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:

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

Stokes's theorem is a vast generalization of this theorem in the following sense. The fundamental theorem of calculus specifies the relationship between the two central operations of Calculus, differentiation and integration. The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set In Calculus, an antiderivative, primitive or indefinite integral of a function f is a function F whose Derivative

By the choice of F, $\frac{dF}{dx}=f$ . In the parlance of differential forms, this is saying that f(x) dx is the exterior derivative of the 0-form (i. In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is In Differential geometry, the exterior derivative extends the concept of the differential of a function which is a form of degree zero to Differential forms 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. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be 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. A surface S in the Euclidean space R 3 is orientable if a two-dimensional figure (for example) cannot be moved around the surface and back In Mathematics, the support of a function is the set of points where the function is not zero or the closure of that set
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

Let M be an oriented smooth manifold of dimension n and let $α$ be an n-form that is a compactly supported on M. A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is In Mathematics, the support of a function is the set of points where the function is not zero or the closure of that set The integral of $α$ over M is defined as follows: Let {f_i} be a partition of unity associated with a locally finite cover {U_i} 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. In Mathematics, a partition of unity of a Topological space X is a set of continuous functions \{\rho_i\}_{i\in I} from X This is well-defined.

Stokes' theorem reads: If $ω$ 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. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be See also Orientation (geometry. In Mathematics, an orientation on a real Vector space is a choice of which In Differential geometry, the exterior derivative extends the concept of the differential of a function which is a form of degree zero to Differential forms

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.

Topological reading

Let M be a smooth manifold. A (C^∞-)singular k-simplex of M is a smooth map from the standard simplex in R^k to M. The free abelian group S_k generated by singular k-simplices is said to consist of singular k-chains of M. In Algebraic topology, a simplicial k - chain is a formal linear combination of k - simplices. These groups, together with boundary map ∂, defines a chain complex. For a different notion of boundary related to Manifolds see that article In Mathematics, a chain complex is a construct originally used in the field of Algebraic topology. The corresponding homology (resp. cohomology) is called the (C^∞-)singular homology (resp. In Algebraic topology, a branch of Mathematics, singular homology refers to the study of a certain set of Topological invariants of a Topological space 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. In Mathematics, de Rham cohomology (after Georges de Rham) is a tool belonging both to Algebraic topology and to Differential topology, capable

Differential k-forms can be integrated over a k-simplex in a natural way, by pulling back to R^k. 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 S_k*, the linear functionals on S_k. In other words, a k-form $ω$ 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

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

de Rham's theorem shows that this homomorphism is in fact an isomorphism. In Mathematics, de Rham cohomology (after Georges de Rham) is a tool belonging both to Algebraic topology and to Differential topology, capable So the converse to 1 and 2 above hold true. In other words, if {c_i} are cycles generating the k-th homology group, then for any corresponding real numbers {a_i}, there exist a closed form $ω$ such that

$\int_{c_i} \omega = a_i ,$

and this form is unique up to exact forms.

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 geometry they are more accessible, older and have familiar names. In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane 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

An illustration of Kelvin-Stokes theorem with surface

Σ

, 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). In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space. 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 $Σ$ 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. In Mathematics, a surface integral is a Definite integral taken over a Surface (which may be a curved set in Space) it can be thought cURL is a Command line tool for transferring files with URL syntax. In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean 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. In Mathematics, a positively oriented curve is a planar Simple closed curve (that is a curve in the plane whose starting point is also the end point and which has For the related yet different principle relating to electromagnetic coils see Right hand grip 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

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. In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric 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

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. In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space Faraday's law of induction describes an important basic law of electromagnetism which is involved in the working of Transformers Inductors and many forms of In Vector calculus, the divergence theorem, also known as Gauss&rsquos theorem ( Carl Friedrich Gauss) Ostrogradsky&rsquos theorem ( Mikhail

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. In Physics and Mathematics, Green's theorem gives the relationship between a Line integral around a simple closed curve C and a Double integral

References

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

External links

Proof of general Stokes theorem on PlanetMath
Differential Forms and Stokes’ Theorem Jerrold E. Marsden Control and Dynamical Systems, Caltech
Calculus 3 - Stokes Theorem from lamar.edu - a human-readable explanation

PlanetMath is a free, collaborative online Mathematics Encyclopedia.

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