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In
vector calculus Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subject ...
and
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilin ...
the generalized Stokes theorem (sometimes with apostrophe as Stokes' theorem or Stokes's theorem), also called the Stokes–Cartan theorem, is a statement about the
integration Integration may refer to: Biology *Multisensory integration *Path integration * Pre-integration complex, viral genetic material used to insert a viral genome into a host genome *DNA integration, by means of site-specific recombinase technology, ...
of
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
s on
manifolds In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
, which both simplifies and generalizes several
theorem In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of th ...
s from
vector calculus Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subject ...
. In particular, the
fundamental theorem of calculus The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, o ...
is the special case where the manifold is a
line segment In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. The length of a line segment is given by the Euclidean distance between i ...
, and
Stokes' theorem Stokes's theorem, also known as the Kelvin–Stokes theoremNagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Bai-Fu-Kan(j ...
is the case of a
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is t ...
in \R^3. Hence, the theorem is sometimes referred to as the Fundamental Theorem of Multivariate Calculus. Stokes' theorem says that the integral of a differential form \omega over the boundary \partial\Omega of some
orientable In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
manifold \Omega is equal to the integral of its
exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The res ...
d\omega over the whole of \Omega, i.e., \int_ \omega = \int_\Omega d\omega\,. Stokes' theorem was formulated in its modern form by
Élie Cartan Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry ...
in 1945, following earlier work on the generalization of the theorems of vector calculus by
Vito Volterra Vito Volterra (, ; 3 May 1860 – 11 October 1940) was an Italian mathematician and physicist, known for his contributions to mathematical biology and integral equations, being one of the founders of functional analysis. Biography Born in An ...
,
Édouard Goursat Édouard Jean-Baptiste Goursat (21 May 1858 – 25 November 1936) was a French mathematician, now remembered principally as an expositor for his ''Cours d'analyse mathématique'', which appeared in the first decade of the twentieth century. It se ...
, and
Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
. This modern form of Stokes' theorem is a vast generalization of a classical result that
Lord Kelvin William Thomson, 1st Baron Kelvin, (26 June 182417 December 1907) was a British mathematician, mathematical physicist and engineer born in Belfast. Professor of Natural Philosophy at the University of Glasgow for 53 years, he did important ...
communicated to George Stokes in a letter dated July 2, 1850.Spivak (1965), p. vii, Preface. Stokes set the theorem as a question on the 1854
Smith's Prize The Smith's Prize was the name of each of two prizes awarded annually to two research students in mathematics and theoretical physics at the University of Cambridge from 1769. Following the reorganization in 1998, they are now awarded under the ...
exam, which led to the result bearing his name. It was first published by
Hermann Hankel Hermann Hankel (14 February 1839 – 29 August 1873) was a German mathematician. Having worked on mathematical analysis during his career, he is best known for introducing the Hankel transform and the Hankel matrix. Biography Hankel was born on 1 ...
in 1861. This classical case relates the
surface integral In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one m ...
of the
curl cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL". History cURL was fir ...
of a vector field \textbf over a surface (that is, the
flux Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ph ...
of \text\,\textbf) in Euclidean three-space to the
line integral In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integral'' is used as well, a ...
of the vector field over the surface boundary (also known as the loop integral). Classical generalizations of the
fundamental theorem of calculus The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, o ...
like the
divergence theorem In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface to the ''divergence'' of the field in the vol ...
, and
Green's theorem In vector calculus, Green's theorem relates a line integral around a simple closed curve to a double integral over the plane region bounded by . It is the two-dimensional special case of Stokes' theorem. Theorem Let be a positively oriente ...
from vector calculus are special cases of the general formulation stated above after making a standard identification of vector fields with differential forms (different for each of the classical theorems).


Introduction

The
second fundamental theorem of calculus The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each time) with the concept of integrating a function (calculating the area under its graph, or ...
states that the
integral In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
of a function f over the interval ,b/math> can be calculated by finding an
antiderivative In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolical ...
F of f: \int_a^b f(x)\,dx = F(b) - F(a)\,. Stokes' theorem is a vast generalization of this theorem in the following sense. * By the choice of \textbf, \frac=f(x). In the parlance of
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
s, this is saying that f(x)\,dx is the
exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The res ...
of the 0-form, i.e. function, F: in other words, that dF=f\,dx. The general Stokes theorem applies to higher differential forms \omega instead of just 0-forms such as F. * A closed interval ,b/math> is a simple example of a one-dimensional
manifold with boundary In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
. 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 In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
, and the form has to be
compactly supported In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smalle ...
in order to give a well-defined integral. * The two points a and b form the boundary of the closed interval. More generally, Stokes' theorem applies to oriented manifolds M with boundary. The boundary \partial M of M is itself a manifold and inherits a natural orientation from that of M. 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). In even simpler terms, one can consider the points as boundaries of curves, that is as 0-dimensional boundaries of 1-dimensional manifolds. So, just as one can find the value of an integral (f\,dx=dF) over a 1-dimensional manifold ( ,b/math>) by considering the anti-derivative (F) at the 0-dimensional boundaries (\), one can generalize the fundamental theorem of calculus, with a few additional caveats, to deal with the value of integrals (d\omega) over n-dimensional manifolds (\Omega) by considering the antiderivative (\omega) at the (n-1)-dimensional boundaries (\partial\Omega) of the manifold. So the fundamental theorem reads: \int_ f(x)\,dx = \int_ \,dF = \int_ \,F = \int_ F = F(b) - F(a)\,.


Formulation for smooth manifolds with boundary

Let \Omega be an
oriented In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
with boundary of
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordin ...
n and let \alpha be a
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebraic ...
n-
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
that is
compactly supported In mathematics, the support of a real-valued function f is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smalle ...
on \Omega. First, suppose that \alpha is compactly supported in the domain of a single, oriented
coordinate chart In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout math ...
\. In this case, we define the integral of \alpha over \Omega as \int_\Omega \alpha = \int_ (\varphi^)^* \alpha\,, i.e., via the
pullback In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: i ...
of \alpha to \R^n. More generally, the integral of \alpha over \Omega is defined as follows: Let \ be a
partition of unity In mathematics, a partition of unity of a topological space is a set of continuous functions from to the unit interval ,1such that for every point x\in X: * there is a neighbourhood of where all but a finite number of the functions of are ...
associated with a locally finite cover \ of (consistently oriented) coordinate charts, then define the integral \int_\Omega \alpha \equiv \sum_i \int_ \psi_i \alpha\,, where each term in the sum is evaluated by pulling back to \R^n as described above. This quantity is well-defined; that is, it does not depend on the choice of the coordinate charts, nor the partition of unity. The generalized Stokes theorem reads: Here d is the
exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The res ...
, which is defined using the manifold structure only. The right-hand side is sometimes written as \oint_ \omega to stress the fact that the (n-1)-manifold \partial\Omega has no boundary.For mathematicians this fact is known, therefore the circle is redundant and often omitted. However, one should keep in mind here that in
thermodynamics Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of ther ...
, where frequently expressions as \oint_W\ appear (wherein the total derivative, see below, should not be confused with the exterior one), the integration path W is a one-dimensional closed line on a much higher-dimensional manifold. That is, in a thermodynamic application, where U is a function of the temperature \alpha_1=T, the volume \alpha_2=V, and the electrical polarization \alpha_3=P of the sample, one has \ = \sum_^3\frac\,d\alpha_i\,, and the circle is really necessary, e.g. if one considers the ''differential'' consequences of the ''integral'' postulate \oint_W\,\\, \stackrel\,0\,.
(This fact is also an implication of Stokes' theorem, since for a given smooth n-dimensional manifold \Omega, application of the theorem twice gives \int_\omega=\int_\Omega d(d\omega)=0 for any (n-2)-form \omega, which implies that \partial(\partial\Omega)=\emptyset.) The right-hand side of the equation is often used to formulate ''integral'' laws; the left-hand side then leads to equivalent ''differential'' formulations (see below). The theorem is often used in situations where \Omega is an embedded oriented submanifold of some bigger manifold, often \R^k, on which the form \omega is defined.


Topological preliminaries; integration over chains

Let be a
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
. A (smooth) singular -simplex in is defined as a
smooth map In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
from the standard simplex in to . The group of singular -
chains A chain is a serial assembly of connected pieces, called links, typically made of metal, with an overall character similar to that of a rope in that it is flexible and curved in compression but linear, rigid, and load-bearing in tension. A ...
on is defined to be the
free abelian group In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a sub ...
on the set of singular -simplices in . These groups, together with the boundary map, , define a
chain complex In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is included in the kernel of ...
. The corresponding homology (resp. cohomology) group is isomorphic to the usual
singular homology In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space ''X'', the so-called homology groups H_n(X). Intuitively, singular homology counts, for each dimension ''n'', the ''n''- ...
group (resp. the
singular cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
group ), defined using continuous rather than smooth simplices in . On the other hand, the differential forms, with exterior derivative, , as the connecting map, form a cochain complex, which defines the
de Rham cohomology In mathematics, de Rham cohomology (named after Georges de Rham) 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 adapt ...
groups H_^k(M, \mathbf). Differential -forms can be integrated over a -simplex in a natural way, by pulling back to . Extending by linearity allows one to integrate over chains. This gives a linear map from the space of -forms to the th group of singular cochains, , the linear functionals on . In other words, a -form defines a functional I(\omega)(c) = \oint_c \omega. on the -chains. Stokes' theorem says that this is a chain map from de Rham cohomology to singular cohomology with real coefficients; the exterior derivative, , 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, i.e., , have zero integral over ''boundaries'', i.e. over manifolds that can be written as , and #exact forms, i.e., , have zero integral over ''cycles'', i.e. if the boundaries sum up to the empty set: . De Rham's theorem shows that this homomorphism is in fact an
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
. So the converse to 1 and 2 above hold true. In other words, if are cycles generating the th homology group, then for any corresponding real numbers, , there exist a closed form, , such that \oint_ \omega = a_i\,, and this form is unique up to exact forms. Stokes' theorem on smooth manifolds can be derived from Stokes' theorem for chains in smooth manifolds, and vice versa. Formally stated, the latter reads:


Underlying principle

To simplify these topological arguments, it is worthwhile to examine the underlying principle by considering an example for dimensions. The essential idea can be understood by the diagram on the left, which shows that, in an oriented tiling of a manifold, the interior paths are traversed in opposite directions; their contributions to the path integral thus cancel each other pairwise. As a consequence, only the contribution from the boundary remains. It thus suffices to prove Stokes' theorem for sufficiently fine tilings (or, equivalently,
simplices In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
), which usually is not difficult.


Classical vector analysis example

Let \gamma: ,bto\R^2 be a
piecewise In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. ...
smooth Jordan plane curve. The
Jordan curve theorem In topology, the Jordan curve theorem asserts that every ''Jordan curve'' (a plane simple closed curve) divides the plane into an "interior" region bounded by the curve and an " exterior" region containing all of the nearby and far away exterior ...
implies that \gamma divides \R^2 into two components, a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Briti ...
one and another that is non-compact. Let D denote the compact part that is bounded by \gamma and suppose \psi:D\to\R^3 is smooth, with S=\psi(D). If \Gamma is the
space curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
defined by \Gamma(t)=\psi(\gamma(t))\gamma and \Gamma are both loops, however, \Gamma is not necessarily a
Jordan curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
and \textbf is a smooth vector field on \R^3, then:This proof is based on the Lecture Notes given by Prof. Robert Scheichl (
University of Bath (Virgil, Georgics II) , mottoeng = Learn the culture proper to each after its kind , established = 1886 (Merchant Venturers Technical College) 1960 (Bristol College of Science and Technology) 1966 (Bath University of Technology) 1971 (univ ...
, U.K

please refer th

/ref> \oint_\Gamma \mathbf\, \cdot\, d = \iint_S \left( \nabla \times \mathbf \right) \cdot\, d\mathbf This classical statement, is a special case of the general formulation after making an identification of vector field with a 1-form and its curl with a two form through \begin F_x \\ F_y \\ F_z \\ \end\cdot d\Gamma \to F_x \,dx + F_y \,dy + F_z \,dz \begin &\nabla \times \begin F_x \\ F_y \\ F_z \end \cdot d\mathbf = \begin \partial_y F_z - \partial_z F_y \\ \partial_z F_x - \partial_x F_z \\ \partial_x F_y - \partial_y F_x \\ \end \cdot d\mathbf \to \\ .4ex&d(F_x \,dx + F_y \,dy + F_z \,dz) = \left(\partial_y F_z - \partial_z F_y\right) dy \wedge dz + \left(\partial_z F_x -\partial_x F_z\right) dz \wedge dx + \left(\partial_x F_y - \partial_y F_x\right) dx \wedge dy. \end


Generalization to rough sets

The formulation above, in which \Omega is a smooth manifold with boundary, does not suffice in many applications. For example, if the domain of integration is defined as the plane region between two x-coordinates and the graphs of two functions, it will often happen that the domain has corners. In such a case, the corner points mean that \Omega is not a smooth manifold with boundary, and so the statement of Stokes' theorem given above does not apply. Nevertheless, it is possible to check that the conclusion of Stokes' theorem is still true. This is because \Omega and its boundary are well-behaved away from a small set of points (a
measure zero In mathematical analysis, a null set N \subset \mathbb is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length. The notion of null ...
set). A version of Stokes' theorem that allows for roughness was proved by Whitney. Assume that D is a connected bounded open subset of \R^n. Call D a ''standard domain'' if it satisfies the following property: there exists a subset P of \partial D, open in \partial D, whose complement in \partial D has Hausdorff (n-1)-measure zero; and such that every point of P has a ''generalized normal vector''. This is a vector \textbf(x) such that, if a coordinate system is chosen so that \textbf(x) is the first basis vector, then, in an open neighborhood around x, there exists a smooth function f(x_2,\dots,x_n) such that P is the graph \ and D is the region \. Whitney remarks that the boundary of a standard domain is the union of a set of zero Hausdorff (n-1)-measure and a finite or countable union of smooth (n-1)-manifolds, each of which has the domain on only one side. He then proves that if D is a standard domain in \R^n, \omega is an (n-1)-form which is defined, continuous, and bounded on D\cup P, smooth on D, integrable on P, and such that d\omega is integrable on D, then Stokes' theorem holds, that is, \int_P \omega = \int_D d\omega\,. The study of measure-theoretic properties of rough sets leads to
geometric measure theory In mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows mathematicians to extend tools from differential geometry to a much larger class of surfac ...
. Even more general versions of Stokes' theorem have been proved by Federer and by Harrison.


Special cases

The general form of the Stokes theorem using differential forms is more powerful and easier to use than the special cases. The traditional versions can be formulated using
Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
without the machinery of differential geometry, and thus are more accessible. Further, they are older and their names are more familiar as a result. 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.


Classical (vector calculus) case

This is a (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 ''Stokes' theorem'' in many introductory university vector calculus courses and is used in physics and engineering. It is also sometimes known as the
curl cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL". History cURL was fir ...
theorem. The classical Stokes' theorem relates the
surface integral In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, one m ...
of the
curl cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL". History cURL was fir ...
of a vector field over a surface \Sigma in Euclidean three-space to the
line integral In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integral'' is used as well, a ...
of the vector field over its boundary. It 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. The curve of the line integral, \partial\Sigma, must have positive
orientation Orientation may refer to: Positioning in physical space * Map orientation, the relationship between directions on a map and compass directions * Orientation (housing), the position of a building with respect to the sun, a concept in building des ...
, meaning that \partial\Sigma points counterclockwise when the
surface normal In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at ...
, n, points toward the viewer. One consequence of this theorem is that the
field line A field line is a graphical visual aid for visualizing vector fields. It consists of an imaginary directed line which is tangent to the field vector at each point along its length. A diagram showing a representative set of neighboring field l ...
s of a vector field with zero curl cannot be closed contours. The formula can be rewritten as:


Green's theorem

Green's theorem In vector calculus, Green's theorem relates a line integral around a simple closed curve to a double integral over the plane region bounded by . It is the two-dimensional special case of Stokes' theorem. Theorem Let be a positively oriente ...
is immediately recognizable as the third integrand of both sides in the integral in terms of , , and cited above.


In electromagnetism

Two of the four
Maxwell equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. ...
involve curls of 3-D vector fields, and their differential and integral forms are related by the special 3-dimensional (vector calculus) case of
Stokes' theorem Stokes's theorem, also known as the Kelvin–Stokes theoremNagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Bai-Fu-Kan(j ...
. 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 (see
Differentiation under the integral sign In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form \int_^ f(x,t)\,dt, where -\infty < a(x), b(x) < \infty and the integral are
): The above listed subset of Maxwell's equations are valid for electromagnetic fields expressed in
SI units The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms and initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wid ...
. In other systems of units, such as CGS or
Gaussian units Gaussian units constitute a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs (centimetre–gram–second) units. It is also called the Gaussian unit system, Gaussian-cgs unit ...
, the scaling factors for the terms differ. For example, in Gaussian units, Faraday's law of induction and Ampère's law take the forms: \begin \nabla \times \mathbf &= -\frac \frac \,, \\ \nabla \times \mathbf &= \frac \frac + \frac \mathbf\,, \end respectively, where is the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit for ...
in vacuum.


Divergence theorem

Likewise, the
divergence theorem In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface to the ''divergence'' of the field in the vol ...
\int_\mathrm \nabla \cdot \mathbf \, d_\mathrm = \oint_ \mathbf \cdot d\boldsymbol 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. An application of this is the case \textbf=f\vec where \vec is an arbitrary constant vector. Working out the divergence of the product gives \vec \cdot \int_\mathrm \nabla f \, d_\mathrm = \vec \cdot \oint_ f\, d\boldsymbol\,. Since this holds for all \vec we find \int_\mathrm \nabla f \, d_\mathrm = \oint_ f\, d\boldsymbol\,.


Volume integral of gradient of scalar field

Let f : \Omega \to \mathbb be a
scalar field In mathematics and physics, a scalar field is a function associating a single number to every point in a space – possibly physical space. The scalar may either be a pure mathematical number (dimensionless) or a scalar physical quantity ...
. Then \int_\Omega \vec f = \int_ \vec f where \vec is the
normal vector In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at ...
to the surface \partial \Omega at a given point. Proof: Let \vec be a vector. Then \begin 0 &= \int_\Omega \vec \cdot \vec f - \int_ \vec \cdot \vec f & \text \\ &= \int_\Omega \vec \cdot \vec f - \int_ \vec \cdot \vec f \\ &= \vec \cdot \int_\Omega \vec f - \vec \cdot \int_ \vec f \\ &= \vec \cdot \left( \int_\Omega \vec f - \int_ \vec f \right) \end Since this holds for any \vec (in particular, for every
basis vector In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as componen ...
), the result follows.


See also

* Chandrasekhar–Wentzel lemma


Footnotes


References


Further reading

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External links

* *
Proof of the Divergence Theorem and Stokes' Theorem

Calculus 3 – Stokes Theorem from lamar.edu
– an expository explanation {{Calculus topics Differential topology Differential forms Duality theories Integration on manifolds Theorems in calculus Theorems in differential geometry