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Encyclopedia > Stokes' theorem

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 (18191903), 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. In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In calculus, the integral of a function is an extension of the concept of a sum. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... A theorem is a proposition that has been or is to be proved on the basis of explicit assumptions. ... Vector calculus (also called vector analysis) is a field of mathematics concerned with multivariate real analysis of vectors in two or more dimensions. ... The silver Anglia knight, commissioned as a trophy in 1850, intended to represent the Black Prince. ... George Gabriel Stokes Sir George Gabriel Stokes, 1st Baronet (13 August 1819â€“1 February 1903) was an Anglo-Irish mathematician and physicist. ... 1819 common year starting on Friday (see link for calendar). ... 1903 (MCMIII) was a common year starting on Thursday (see link for calendar) of the Gregorian calendar or a common year starting on Friday of the 13-day slower Julian calendar. ... William Thomson, 1st Baron Kelvin OM GCVO PC PRS FRSE (26 June 1824 â€“ 17 December 1907) was a mathematical physicist, engineer, and outstanding leader in the physical sciences of the 19th century. ...

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 In mathematics, a function f(x) of a real number variable x is defined piecewise, if f(x) is given by different expressions on various intervals. ... On a sphere, the sum of the angles of a triangle is not equal to 180Â°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... :For other senses of this word, see dimension (disambiguation). ... In mathematics, the support of a numerical function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... On a sphere, the sum of the angles of a triangle is not equal to 180Â°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... In mathematics, an orientation on a real vector space is a choice of which ordered bases are positively oriented (or right-handed) and which are negatively oriented (or left-handed). ...

$int_M domega = oint_{partial M} omega.!,$

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. In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ... The fundamental theorem of calculus is the statement that the two central operations of calculus, differentiation and integration, are inverse functions of one another. ...

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.

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. In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ... In mathematics, a function f(x) of a real number variable x is defined piecewise, if f(x) is given by different expressions on various intervals. ... This article is about algebraic topology. ... In mathematics, both in vector calculus and in differential topology, the concepts of closed form and exact form are defined for differential forms, by the equations d&#945; = 0 for a given form &#945; to be a closed form, and &#945; = d&#946; for an exact form, with &#945; given... The word Boundary has a variety of meanings. ... In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homos = identical) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ... In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. ...

## 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.

### 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. Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ...

The classical Kelvin-Stokes theorem:

$int_{Sigma} nabla times mathbf{F} cdot dmathbf{Sigma} = oint_{partialSigma} mathbf{F} cdot d mathbf{r},$

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. The curve of the line integral ($partialSigma$) must have positive orientation, such that $dmathbf{r}$ points counterclockwise when the surface normal ($dmathbf{Sigma}$) points toward the viewer, following the right-hand rule. In mathematics, a surface integral is a definite integral taken over some surface that may be a curved set in space; it can be thought of as the double integral analog of the path integral. ... To meet Wikipedias quality standards, this article may require cleanup. ... Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ... 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 no other self-intersections) such that when travelling on it one always has the curve interior to the left (and... The left-handed orientation is shown on the left, and the right-handed on the right. ...

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:

$int_{Sigma} left( g left(nabla times mathbf{F}right) + left( nabla g right) times mathbf{F} right) cdot dmathbf{Sigma} = oint_{partialSigma} g mathbf{F} cdot d mathbf{r},$
$int_{Sigma} left( mathbf{F} left(nabla cdot mathbf{G} right) - mathbf{G}left(nabla cdot mathbf{F} right) + left( mathbf{G} cdot nabla right) mathbf{F} - left(mathbf{F} cdot nabla right) mathbf{G} right) cdot dmathbf{Sigma} = oint_{partialSigma} left( mathbf{F} times mathbf{G}right) cdot d mathbf{r}.$

#### 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: Maxwells equations are the set of four equations, attributed to James Clerk Maxwell, that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter. ...

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

In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ... In calculus, the integral of a function is an extension of the concept of a sum. ... Any change in the magnetic environment of a coil of wire will cause a voltage (emf) to be induced in the coil. ... An electric current produces a magnetic field. ...

### Divergence theorem

Likewise the Ostrogradsky-Gauss theorem (also known as the Divergence theorem or Gauss' theorem) In vector calculus, the divergence theorem, also known as Gauss theorem, Ostrogradskys theorem, or Ostrogradskyâ€“Gauss theorem is a result that links the divergence of a vector field to the value of surface integrals of the flow defined by the field. ...

$int_{mathrm{Vol}} nabla cdot mathbf{F} cdot dmathrm{Vol} = oint_{partial mathrm{Vol}} mathbf{F} cdot d mathbf{Sigma}$

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.

### The Fundamental Theorem of Calculus

The fundamental theorem of calculus is the 0+1 dimensional case: the boundary is then the two endpoints, with + on the right and on the left being the orientation. The fundamental theorem of calculus is the statement that the two central operations of calculus, differentiation and integration, are inverse functions of one another. ...

### 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, Greens theorem gives the relationship between a line integral around a simple closed curve C and a double integral over the plane region D bounded by C. Greens theorem was named after British scientist George Green and is a special two-dimensional case of...

## 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

Jerrold E. Marsden. ... Michael David Spivak is a mathematician specializing in differential geometry, an expositor of mathematics, and the founder of Publish-or-Perish Press. ...

Results from FactBites:

 Stokes' theorem - Wikipedia, the free encyclopedia (535 words) Stokes' theorem in differential geometry is a statement about the integration of differential forms which generalizes several theorems from vector calculus. The fundamental theorem of calculus is the 0+1 dimensional case: the boundary is then the two endpoints, with + on the right and − on the left being the orientation. 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.
 George Gabriel Stokes (3233 words) Ray MacSharry, of a memorial at Stokes' birthplace in Skreen on Saturday 10th June 1995 as part of a meeting organised at Sligo RTC by the Institutes of Physics and of Mathematics and its Applications, under the auspices of the Royal Irish Academy, as part of the Sligo 750 celebrations. In 1798, Gabriel Stokes, son of John Stokes and Rector of Skreen, married Elizabeth, the daughter of John Haughton, the Rector of Kilrea. Stokes' manuscript notes still exist in the University Library in Cambridge, although his writing was so bad that he eventually became one of the first people in Britain to make regular use of a typewriter.
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