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{{Calculus|Vector}}

In [[Vector calculus|vector calculus]], and more generally [[Differential geometry|differential geometry]], '''Stokes' theorem''' (sometimes spelled '''Stokes's theorem''', and also called the '''generalized Stokes theorem''' or the '''Stokes–Cartan theorem'''<ref>{{Cite book|url=https://www.springer.com/gp/book/9789400745575|title=Physics of Collisional Plasmas – Introduction to {{!}} Michel Moisan {{!}} Springer|language=en}}</ref>) is a statement about the [[Integral|integration]] of [[Differential form|differential form]]s on manifolds, which both simplifies and generalizes several [[Theorem|theorem]]s from [[Vector calculus|vector calculus]].  Stokes' theorem says that the integral of a differential form {{mvar|ω}} over the boundary of some [[Orientation (vector space)#Orientation on manifolds|orientable]] manifold {{math|Ω}} is equal to the integral of its [[Exterior derivative|exterior derivative]] {{mvar|dω}} over the whole of {{math|Ω}}, i.e.,

:<math>\int_{\partial \Omega}\omega=\int_\Omega d\omega\,.</math>

Stokes' theorem was formulated in its modern form by [[Biography:Élie Cartan|Élie Cartan]] in 1945,<ref>{{Cite book|title=Les Systèmes Différentiels Extérieurs et leurs Applications Géométriques|last=Cartan|first=Élie|publisher=Hermann|year=1945|isbn=|location=Paris|pages=}}</ref> following earlier work on the generalization of the theorems of vector calculus by [[Biography:Vito Volterra|Vito Volterra]], [[Biography:Édouard Goursat|Édouard Goursat]], and [[Biography:Henri Poincaré|Henri Poincaré]].<ref>{{Cite journal|last=Katz|first=Victor J.|date=1979-01-01|title=The History of Stokes' Theorem|jstor=2690275|journal=Mathematics Magazine|volume=52|issue=3|pages=146–156|doi=10.2307/2690275}}</ref><ref>{{Cite book|title=History of Topology|last=Katz|first=Victor J.|publisher=Elsevier|year=1999|isbn=9780444823755|editor-last=James|editor-first=I. M.|location=Amsterdam|pages=111–122|chapter=5. Differential Forms}}</ref>

This modern form of Stokes' theorem is a vast generalization of a classical result that [[Biography:William Thomson, 1st Baron Kelvin|Lord Kelvin]] communicated to [[Biography:Sir George Stokes, 1st Baronet|George Stokes]] in a letter dated July 2, 1850.<ref>See:
* {{cite journal|first=Victor J.|last=Katz|date=May 1979|title=The history of Stokes' theorem|journal=Mathematics Magazine|volume=52|issue=3|pages=146–156|doi=10.1080/0025570x.1979.11976770}}
* The letter from Thomson to Stokes appears in: {{cite book|url=https://books.google.com/books?id=YrjkOEdC83gC&pg=PA97#v=onepage&q&f=false|title=The Correspondence between Sir George Gabriel Stokes and Sir William Thomson, Baron Kelvin of Largs, Volume 1: 1846–1869|last2=Stokes|first2=George Gabriel|date=1990|publisher=Cambridge University Press|editor-last=Wilson|editor-first=David B.|location=Cambridge, England|pages=96–97|first1=William|last1=Thomson|isbn=9780521328319|}}
* Neither Thomson nor Stokes published a proof of the theorem. The first published proof appeared in 1861 in: {{cite book|url=http://babel.hathitrust.org/cgi/pt?id=mdp.39015035826760#page/34/mode/1up|title=Zur allgemeinen Theorie der Bewegung der Flüssigkeiten|last=Hankel|first=Hermann|date=1861|publisher=Dieterische University Buchdruckerei|location=Göttingen, Germany|pages=34–37|trans-title=On the general theory of the movement of fluids|}} Hankel doesn't mention the author of the theorem.
* In a footnote, Larmor mentions earlier researchers who had integrated, over a surface, the curl of a vector field. See: {{cite book|url=https://books.google.com/books?id=O28ssiqLT9AC&pg=PA320#v=onepage&q&f=false|title=Mathematical and Physical Papers by the late Sir George Gabriel Stokes|last=Stokes|first=George Gabriel|date=1905|publisher=University of Cambridge Press|volume=5|location=Cambridge, England|pages=320–321|editor1-first=Joseph|editor1-last=Larmor|editor2-first=John William, Baron Rayleigh|editor2-last=Strutt}}</ref><ref>{{cite book|first=Olivier|last=Darrigol|title=Electrodynamics from Ampère to Einstein|page=146|isbn=0198505930|location=Oxford, England|date=2000}}</ref><ref name=spivak65>Spivak (1965), p. vii, Preface.</ref> Stokes set the theorem as a question on the 1854 Smith's Prize exam, which led to the result bearing his name.  It was first published by [[Biography:Hermann Hankel|Hermann Hankel]] in 1861.<ref name=spivak65 /><ref>See:
* The 1854 Smith's Prize Examination is available online at: [http://www.clerkmaxwellfoundation.org/SmithsPrizeExam_Stokes.pdf Clerk Maxwell Foundation]. Maxwell took this examination and tied for first place with Edward John Routh. See: {{cite book|first1=James|last1=Clerk Maxwell|editor-first=P.&nbsp;M.|editor-last=Harman|title=The Scientific Letters and Papers of James Clerk Maxwell, Volume I: 1846–1862|location=Cambridge, England|publisher=Cambridge University Press|date=1990|url=https://books.google.com/books?id=zfM8AAAAIAAJ&pg=PA237#v=onepage&q&f=false|page=237, footnote 2|isbn=9780521256254}} See also Smith's prize or the [http://www.clerkmaxwellfoundation.org/SmithsPrizeSolutions2008_2_14.pdf Clerk Maxwell Foundation].
* {{cite book|first=James|last=Clerk Maxwell|title=A Treatise on Electricity and Magnetism|location=Oxford, England|publisher=Clarendon Press|date=1873|volume=1|url=https://books.google.com/books?id=92QSAAAAIAAJ&pg=PA27#v=onepage&q&f=false|pages=25–27}} In a footnote on page 27, Maxwell mentions that Stokes used the theorem as question 8 in the Smith's Prize Examination of 1854.  This footnote appears to have been the cause of the theorem's being known as "Stokes' theorem".</ref> This classical [[Physics:Kelvin–Stokes theorem|Kelvin–Stokes theorem]] relates the [[Surface integral|surface integral]] of the [[Curl (mathematics)|curl]] of a [[Vector field|vector field]] {{math|'''F'''}} over a surface (that is, the [[Physics:Flux|flux]] of {{math|curl '''F'''}}) in Euclidean three-space to the [[Line integral|line integral]] of the vector field over its boundary (also known as the loop integral).

'''Mathematical Interpretation'''

Let {{math|''γ'': [''a'', ''b''] → '''R'''<sup>2</sup>}} be a [[Piecewise|piecewise]] smooth Jordan plane curve. The [[Jordan curve theorem]] implies that {{mvar|γ}} divides {{math|'''R'''<sup>2</sup>}} into two components, a [[Compact space|compact]] one and another that is non-compact. Let {{mvar|D}} denote the compact part that is bounded by {{mvar|γ}} and suppose {{math|''ψ'': ''D'' → '''R'''<sup>3</sup>}} is smooth, with {{math|''S'' :{{=}} ''ψ''(''D'')}}. If {{math|Γ}} is the space curve defined by {{math|Γ(''t'') {{=}} ''ψ''(''γ''(''t''))}}<ref group="note" name=cgamma>{{mvar|γ}} and {{math|Γ}} are both loops, however, {{math|Γ}} is not necessarily a Jordan curve</ref> and {{math|'''F'''}} is a smooth vector field on {{math|'''R'''<sup>3</sup>}}, then:<ref name="Jame">{{cite book |first=James |last=Stewart |title=Essential Calculus: Early Transcendentals |publisher=Cole |year=2010 |url={{Google books |plainurl=yes |id=btIhvKZCkTsC |page=786 }} }}</ref><ref name="bath">This proof is based on the Lecture Notes given by Prof. Robert Scheichl ([[Organization:University of Bath|University of Bath]], U.K) [http://www.maths.bath.ac.uk/~masrs/ma20010/], please refer the [http://www.maths.bath.ac.uk/~masrs/ma20010/stokesproofs.pdf]</ref><ref name="proofwik">[http://www.proofwiki.org/wiki/Classical_Stokes'_Theorem This proof is also same to the proof shown in]</ref>

:<math>\oint_\Gamma \mathbf{F}\, \cdot\, d{\mathbf{\Gamma}}  = \iint_S \nabla\times\mathbf{F}\, \cdot\, d\mathbf{S} </math>

This classical statement, along with the classical [[Divergence theorem|divergence theorem]], the [[Fundamental theorem of calculus|fundamental theorem of calculus]], and [[Green's theorem]] are simply special cases of the general formulation stated above.

== Introduction ==
The [[Fundamental theorem of calculus|fundamental theorem of calculus]] states that the [[Integral|integral]] of a function {{mvar|f}} over the [[Interval (mathematics)|interval]] {{math|[''a'', ''b'']}} can be calculated by finding an [[Antiderivative|antiderivative]] {{mvar|F}} of&nbsp;{{mvar|f}}:

:<math>\int_a^b f(x)\,dx = F(b) - F(a)\,.</math>

Stokes' theorem is a vast generalization of this theorem in the following sense. 
* By the choice of {{mvar|F}}, {{math|1={{sfrac|''dF''|''dx''}} = ''f''(''x'')}}. In the parlance of [[Differential form|differential form]]s, this is saying that {{math|''f''(''x'')&thinsp;''dx''}} is the [[Exterior derivative|exterior derivative]] of the 0-form, i.e. function, {{mvar|F}}: in other words, that {{math|1=''dF'' = ''f''&thinsp;''dx''}}. The general Stokes theorem applies to higher differential forms {{mvar|ω}} instead of just 0-forms such as {{mvar|F}}.
* A closed interval {{math|[''a'', ''b'']}} is a simple example of a one-dimensional manifold with boundary.  Its boundary is the set consisting of the two points {{mvar|a}} and {{mvar|b}}. Integrating {{mvar|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 {{mvar|a}} and {{mvar|b}} form the boundary of the closed interval. More generally, Stokes' theorem applies to oriented manifolds {{mvar|M}} with boundary. The boundary {{math|∂''M''}} of {{mvar|M}} is itself a manifold and inherits a natural orientation from that of {{mvar|M}}. For example, the natural orientation of the interval gives an orientation of the two boundary points. Intuitively, {{mvar|a}} inherits the opposite orientation as {{mvar|b}}, as they are at opposite ends of the interval. So, "integrating" {{mvar|F}} over two boundary points {{mvar|a}}, {{mvar|b}} is taking the difference {{math|''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 ({{math|1=''f''&thinsp;''dx'' = ''dF''}}) over a 1-dimensional manifold ({{math|[''a'', ''b'']}}) by considering the anti-derivative ({{mvar|F}}) at the 0-dimensional boundaries ({{math|{''a'', ''b''}|}}), one can generalize the fundamental theorem of calculus, with a few additional caveats, to deal with the value of integrals ({{mvar|dω}}) over {{mvar|n}}-dimensional manifolds ({{math|Ω}}) by considering the antiderivative ({{mvar|ω}}) at the {{math|(''n'' − 1)}}-dimensional boundaries ({{math|∂Ω}}) of the manifold.

So the fundamental theorem reads:
:<math>\int_{[a, b]} f(x)\,dx = \int_{[a, b]} dF = \int_{\{a\}^- \cup \{b\}^+} F = F(b) - F(a)\,.</math>

==Formulation for smooth manifolds with boundary==
Let {{math|Ω}} be an oriented smooth manifold with boundary of [[Dimension|dimension]] {{mvar|n}} and let {{mvar|α}} be a smooth {{mvar|n}}-[[Differential form|differential form]] that is [[Support (mathematics)#Compact support|compactly supported]] on {{math|Ω}}. First, suppose that {{mvar|α}} is compactly supported in the domain of a single, oriented coordinate chart {{math|{''U'', ''φ''}|}}. In this case, we define the integral of {{mvar|α}} over {{math|Ω}} as

:<math>\int_\Omega \alpha = \int_{\varphi(U)} \left(\varphi^{-1}\right)^* \alpha\,,</math>

i.e., via the [[Pullback (differential geometry)|pullback]] of {{mvar|α}} to {{math|'''R'''<sup>''n''</sup>}}.

More generally, the integral of {{mvar|α}} over {{math|Ω}} is defined as follows: Let {{math|{''ψ<sub>i</sub>''}|}} be a [[Partition of unity|partition of unity]] associated with a [[Locally finite collection|locally finite]] [[Cover (topology)|cover]] {{math|{''U<sub>i</sub>'', ''φ<sub>i</sub>''}|}} of (consistently oriented) coordinate charts, then define the integral

:<math>\int_\Omega \alpha \equiv \sum_i \int_{U_i} \psi_i \alpha\,,</math>

where each term in the sum is evaluated by pulling back to {{math|'''R'''<sup>''n''</sup>}} 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:

{{quotebox|align=center|quote='''Theorem.''' ''(Stokes–Cartan) If <math>\omega</math> is a smooth <math>(n-1)</math>-[[Differential form|form]] with compact support on smooth <math>n</math>-dimensional [[Manifold|manifold-with-boundary]] <math>\Omega</math>, <math>\partial\Omega</math> denotes the boundary of <math>\Omega</math> given the induced orientation, and <math>i:\partial \Omega\hookrightarrow \Omega</math> is the [[Inclusion map|inclusion map]], then

:<math>\int_\Omega d\omega = \int_{\partial \Omega} i^*\omega</math>.''}}

Conventionally, <math display="inline">\int_{\partial\Omega} i^*\omega</math> is abbreviated as <math display="inline">\int_{\partial\Omega} \omega</math>, since the pullback of a differential form by the inclusion map is simply its restriction to its domain: <math>i^*\omega=\omega|_{\partial\Omega}</math>.  Here <math>d</math> is the [[Exterior derivative|exterior derivative]], which is defined using the manifold structure only.  The right-hand side is sometimes written as <math display="inline">\oint_{\partial\Omega} \omega</math> to stress the fact that the <math>(n-1)</math>-manifold <math>\partial\Omega</math> has no boundary.<ref group="note">For  mathematicians this fact is known, therefore the circle is redundant and often omitted.  However, one should keep in mind here that in [[Physics:Thermodynamics|thermodynamics]], where frequently expressions as {{math|∮<sub>''W''</sub> {''d''<sub>total</sub>''U''}|}} appear (wherein the total derivative, see below, should not be confused with the exterior one), the integration path {{mvar|W}} is a one-dimensional closed line on a much higher-dimensional manifold. That is, in a thermodynamic application, where {{mvar|U}} is a function of the temperature {{math|''α''<sub>1</sub> :{{=}} ''T''}}, the volume {{math|''α''<sub>2</sub> :{{=}} ''V''}}, and the electrical polarization {{math|''α''<sub>3</sub> :{{=}} ''P''}} of the sample, one has

:<math>\{d_\text{total}U\}=\sum_{i=1}^3\frac{\partial U}{\partial\alpha_i}\,d\alpha_i\,,</math>
and the circle is really necessary, e.g. if one considers the ''differential'' consequences of the ''integral'' postulate

:<math>\oint_W\,\{d_\text{total}U\}\, \stackrel{!}{=}\,0\,.</math></ref> (This fact is also an implication of Stokes' theorem, since for a given smooth <math>n</math>-dimensional manifold <math>\Omega</math>, application of the theorem twice gives <math display="inline">\int_{\partial(\partial \Omega)}\omega=\int_\Omega d(d\omega)=0</math> for any <math>(n-2)</math>-form <math>\omega</math>, which implies that <math>\partial(\partial\Omega)=\emptyset</math>.) 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 <math>\Omega</math> is an embedded oriented submanifold of some bigger manifold, often <math>\mathbf{R}^k</math>, on which the form <math>\omega</math> is defined.

==Topological preliminaries; integration over chains==

Let {{mvar|M}} be a smooth manifold. A (smooth) singular [[Simplex|{{mvar|k}}-simplex]] in {{mvar|M}} is defined as a smooth map from the standard simplex in {{math|'''R'''<sup>''k''</sup>}} to {{mvar|M}}. The group {{math|''C''<sub>''k''</sub>(''M'', '''Z''')}} of singular {{mvar|k}}-[[Chain (algebraic topology)|chains]] on {{mvar|M}} is defined to be the [[Free abelian group|free abelian group]] on the set of singular {{mvar|k}}-simplices in {{mvar|M}}. These groups, together with the boundary map, {{math|∂}}, define a [[Chain complex|chain complex]]. The corresponding homology (resp. cohomology) group is isomorphic to the usual [[Singular homology|singular homology]] group {{math|''H''<sub>''k''</sub>(''M'', '''Z''')}} (resp. the singular cohomology group {{math|''H''<sup>''k''</sup>(''M'', '''Z''')}}), defined using continuous rather than smooth simplices in {{mvar|M}}.

On the other hand, the differential forms, with exterior derivative, {{mvar|d}}, as the connecting map, form a cochain complex, which defines the [[De Rham cohomology|de Rham cohomology]] groups {{math|''H''{{su|p=''k''|b=''dR''}}(''M'', '''R''')}}.

Differential {{mvar|k}}-forms can be integrated over a {{mvar|k}}-simplex in a natural way, by pulling back to {{math|'''R'''<sup>''k''</sup>}}. Extending by linearity allows one to integrate over chains. This gives a linear map from the space of {{mvar|k}}-forms to the {{mvar|k}}th group of singular cochains, {{math|''C<sup>k</sup>''(''M'', '''Z''')}}, the linear functionals on {{math|''C<sub>k</sub>''(''M'', '''Z''')}}. In other words, a {{mvar|k}}-form {{mvar|ω}} defines a functional

:<math>I(\omega)(c) = \oint_c \omega</math>.

on the {{mvar|k}}-chains. Stokes' theorem says that this is a chain map from de Rham cohomology to singular cohomology with real coefficients; the exterior derivative, {{mvar|d}}, behaves like the ''dual'' of {{math|∂}} on forms. This gives a homomorphism from de Rham cohomology to singular cohomology. On the level of forms, this means:

#closed forms, i.e., {{math|''dω'' {{=}} 0}}, have zero integral over ''boundaries'', i.e. over manifolds that can be written as {{math|∂∑<sub>''c''</sub> ''M<sub>c</sub>''}}, and
#exact forms, i.e., {{math|''ω'' {{=}} ''dσ''}}, have zero integral over ''cycles'', i.e. if the boundaries sum up to the empty set: {{math|∑<sub>''c''</sub> ''M<sub>c</sub>'' {{=}} ∅}}.

[[De Rham's theorem]] shows that this homomorphism is in fact an [[Isomorphism|isomorphism]]. So the converse to 1 and 2 above hold true. In other words, if {{math|{''c<sub>i</sub>''}|}} are cycles generating the {{mvar|k}}th homology group, then for any corresponding real numbers, {{math|{''a<sub>i</sub>''}|}}, there exist a closed form, {{mvar|ω}}, such that

:<math>\oint_{c_i} \omega = a_i\,,</math>

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.<ref>{{Cite book|title=Manifolds, Tensors, and Forms|last=Renteln|first=Paul|publisher=Cambridge University Press|year=2014|isbn=9781107324893|location=Cambridge, UK|pages=158–175}}</ref>  Formally stated, the latter reads:<ref>{{Cite book|title=Introduction to Smooth Manifolds|last=Lee|first=John M.|publisher=Springer|year=2013|isbn=9781441999818|location=New York|pages=481}}</ref>

{{quotebox|quote='''Theorem.''' ''(Stokes' theorem for chains) If <math>c</math> is a smooth <math>k</math>-chain in a smooth manifold <math>M</math>, and <math>\omega</math> is a smooth <math>(k-1)</math>-form on <math>M</math>, then

:<math>\int_{\partial c}\omega = \int_c d\omega</math>.''|align=center}}

== Underlying principle ==
[[Image:Stokes patch.svg|200px|left]]
To simplify these topological arguments, it is worthwhile to examine the underlying principle by considering an example for {{math|''d'' {{=}} 2}} 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, [[Simplex|simplices]]), which usually is not difficult.

{{Clear}}

==Generalization to rough sets==
[[Image:Green's-theorem-simple-region.svg|thumb|upright|180px|A region (here called {{mvar|D}} instead of {{math|Ω}}) with piecewise smooth boundary.  This is a manifold with corners, so its boundary is not a smooth manifold.]]

The formulation above, in which {{math|Ω}} 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 {{mvar|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 {{math|Ω}} 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 {{math|Ω}} and its boundary are well-behaved away from a small set of points (a measure zero set).

A version of Stokes' theorem that allows for roughness was proved by Whitney.<ref>Whitney, ''Geometric Integration Theory,'' III.14.</ref>  Assume that {{mvar|D}} is a connected bounded open subset of {{math|'''R'''<sup>''n''</sup>}}.  Call {{mvar|D}} a ''standard domain'' if it satisfies the following property: There exists a subset {{mvar|P}} of {{math|∂''D''}}, open in {{math|∂''D''}}, whose complement in {{math|∂''D''}} has [[Hausdorff measure|Hausdorff {{math|(''n'' − 1)}}-measure]] zero; and such that every point of {{mvar|P}} has a ''generalized normal vector''.  This is a vector {{math|'''v'''(''x'')}} such that, if a coordinate system is chosen so that {{math|'''v'''(''x'')}} is the first basis vector, then, in an open neighborhood around {{mvar|x}}, there exists a smooth function {{math|''f''(''x''<sub>2</sub>, …, ''x''<sub>''n''</sub>)}} such that {{mvar|P}} is the graph {{math|{ ''x''<sub>1</sub> {{=}} ''f''(''x''<sub>2</sub>, …, ''x''<sub>''n''</sub>) }|}} and {{mvar|D}} is the region {{math|{''x''<sub>1</sub> :  ''x''<sub>1</sub> < ''f''(''x''<sub>2</sub>, …, ''x''<sub>''n''</sub>) }|}}.  Whitney remarks that the boundary of a standard domain is the union of a set of zero Hausdorff {{math|(''n'' − 1)}}-measure and a finite or countable union of smooth {{math|(''n'' − 1)}}-manifolds, each of which has the domain on only one side.  He then proves that if {{mvar|D}} is a standard domain in {{math|'''R'''<sup>''n''</sup>}}, {{mvar|ω}} is an {{math|(''n'' − 1)}}-form which is defined, continuous, and bounded on {{math|''D'' ∪ ''P''}}, smooth on {{mvar|D}}, integrable on {{mvar|P}}, and such that {{mvar|dω}} is integrable on {{mvar|D}}, then Stokes' theorem holds, that is,
:<math>\int_P \omega = \int_D d\omega\,.</math>

The study of measure-theoretic properties of rough sets leads to [[Geometric measure theory|geometric measure theory]].  Even more general versions of Stokes' theorem have been proved by Federer and by Harrison.<ref>{{cite journal|last=Harrison|first=J.|title=Stokes' theorem for nonsmooth chains|journal=Bulletin of the American Mathematical Society |series=New Series|volume=29|issue=2|date=October 1993|pages=235–243|doi=10.1090/S0273-0979-1993-00429-4|arxiv=math/9310231}}</ref>

==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 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.

===Kelvin–Stokes theorem===
{{Main|Physics:Kelvin–Stokes theorem}}

[[Image:Stokes' Theorem.svg|thumb|right|An illustration of the Kelvin–Stokes theorem, with surface {{math|Σ}}, its boundary {{math|∂Σ}} and the "normal" vector {{mvar|n}}.]]

This is a (dualized) (1&nbsp;+&nbsp;1)-dimensional case, for a 1-form (dualized because it is a statement about [[Vector field|vector field]]s). 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 (mathematics)|curl]]''' theorem.

The classical Kelvin–Stokes theorem relates the [[Surface integral|surface integral]] of the [[Curl (mathematics)|curl]] of a [[Vector field|vector field]] over a surface {{math|Σ}} in Euclidean three-space to the [[Line integral|line integral]] of the vector field over its boundary. It is a special case of the general Stokes theorem (with {{math|''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, {{math|∂Σ}}, must have positive [[Curve orientation|orientation]], meaning that {{math|∂Σ}} points counterclockwise when the [[Normal (geometry)|surface normal]], {{math|'''n'''}}, points toward the viewer.

One consequence of the Kelvin–Stokes theorem is that the [[Field line|field line]]s of a vector field with zero curl cannot be closed contours. The formula can be rewritten as:

<div align=left>
<div class="messagebox standard-talk NavFrame"><div align=left>
Suppose {{math|1='''F'''=(''P''(''x'',''y'',''z''),''Q''(''x'',''y'',''z''),''R''(''x'',''y'',''z''))}} is defined in region with smooth surface {{math|Σ}} and has first-order continuous partial derivatives. Then
: <math>\begin{align}
& \iint_\Sigma \Bigg(\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\Bigg) \\[4pt]
= {} & \oint_{\partial\Sigma} \Big(P\,dx+Q\,dy+R\,dz\Big)\,,\end{align}</math>
<br>where {{mvar|P}}, {{mvar|Q}} and {{mvar|R}} are the components of {{math|'''F'''}}, and {{math|∂Σ}} is boundary of region with smooth surface {{math|Σ}}.
</div>
</div>
</div>

===Green's theorem===
[[Green's theorem]] is immediately recognizable as the third integrand of both sides in the integral in terms of {{mvar|P}}, {{mvar|Q}}, and {{mvar|R}} cited above.

====In electromagnetism====
Two of the four [[Physics:Maxwell equations|Maxwell equations]] involve curls of 3-D vector fields, and their differential and integral forms are related by the [[Physics:Kelvin–Stokes theorem|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 (see Differentiation under the integral sign):

{| class="wikitable" border="1"
! Name
! [[Partial differential equation|Differential]] form
! [[Integral]] form (using Kelvin–Stokes theorem plus relativistic invariance, {{math|∫ {{sfrac|∂|∂''t''}} … → {{sfrac|''d''|''dt''}} ∫ …}})
|- valign="center"
| Maxwell–Faraday equation<br> [[Physics:Faraday's law of induction|Faraday's law of induction]]:
| <math>\quad \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t} \quad</math>
| <math>\begin{align}\quad \oint_C \mathbf{E} \cdot d\mathbf{l} &= \iint_S  \nabla \times \mathbf{E} \cdot d\mathbf{A} \\ &= -\iint_S \frac{\partial \mathbf{B}}{\partial t} \cdot d\mathbf{A}\end{align}\quad </math> (with {{mvar|C}} and {{mvar|S}} not necessarily stationary)
|- valign="center"
| Ampère's law<br /> (with Maxwell's extension):
| <math>\quad \nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}} {\partial t} \quad</math>
| <math>\begin{align}\quad \oint_C \mathbf{H} \cdot d\mathbf{l} &= \iint_S \nabla \times \mathbf{H} \cdot d\mathbf{A}\\ &= \iint_S \mathbf{J} \cdot d\mathbf{A} + \iint_S \frac{\partial \mathbf{D}}{\partial t} \cdot d\mathbf{A}\end{align}\quad </math> (with {{mvar|C}} and {{mvar|S}} not necessarily stationary)
|}
The above listed subset of Maxwell's equations are valid for electromagnetic fields expressed in SI units. In other systems of units, such as [[Physics:Maxwell's equations#CGS units|CGS]] or [[Physics:Gaussian units|Gaussian units]], the scaling factors for the terms differ. For example, in Gaussian units, Faraday's law of induction and Ampère's law take the forms:<ref>{{cite book|first=J.&nbsp;D.|last=Jackson|title=Classical Electrodynamics|url=https://archive.org/details/classicalelectro00jack_0|url-access=registration|edition=2nd|publisher=Wiley|location=New York, NY|date=1975}}</ref><ref>{{cite book|first=M.|last=Born|first2=E.|last2=Wolf|title=Principles of Optics|edition=6th|publisher=Cambridge University Press|location=Cambridge, England|date=1980}}</ref>

:<math>\begin{align} \nabla \times \mathbf{E} &= -\frac{1}{c} \frac{\partial \mathbf{B}} {\partial t}\,, \\
\nabla \times \mathbf{H} &= \frac{1}{c} \frac{\partial \mathbf{D}} {\partial t} + \frac{4\pi}{c} \mathbf{J}\,, \end{align}</math>

respectively, where {{mvar|c}} is the [[Physics:Speed of light|speed of light]] in vacuum.

===Divergence theorem===
Likewise, the [[Divergence theorem|Ostrogradsky–Gauss theorem]] (also known as the divergence theorem or Gauss's theorem)

:<math>\int_\mathrm{Vol} \nabla \cdot \mathbf{F} \, d_\mathrm{Vol} = \oint_{\partial \mathrm{Vol}} \mathbf{F} \cdot d\boldsymbol{\Sigma}</math>

is a special case if we identify a vector field with the {{math|(''n'' − 1)}}-form obtained by contracting the vector field with the Euclidean volume form. An application of this is the case {{math|'''F''' {{=}} ''f'''''c'''}} where {{math|'''c'''}} is an arbitrary constant vector. Working out the divergence of the product gives
:<math>\mathbf c \cdot \int_\mathrm{Vol} \nabla f \, d_\mathrm{Vol} = \mathbf c \cdot \oint_{\partial \mathrm{Vol}} f\, d\boldsymbol{\Sigma}\,.</math>
Since this holds for all {{math|'''c'''}} we find
:<math>\int_\mathrm{Vol} \nabla f \, d_\mathrm{Vol} = \oint_{\partial \mathrm{Vol}} f\, d\boldsymbol{\Sigma}\,.</math>

==See also==
*[[Chandrasekhar–Wentzel lemma]]

==Footnotes==
{{reflist|group=note}}

==References==
{{reflist}}

==Further reading==
* {{cite book |last=Grunsky |first=Helmut |title=The General Stokes' Theorem |location=Boston |publisher=Pitman |year=1983 |isbn=0-273-08510-7 |url-access=registration |url=https://archive.org/details/generalstokesthe0000grun }}
* {{cite journal| last=Katz | first=Victor J. | title=The History of Stokes' Theorem | journal=Mathematics Magazine |date=May 1979 | volume=52 | number=3 | pages=146–156 | jstor=2690275 | doi=10.2307/2690275}}
* {{cite book|url=https://archive.org/details/LoomisL.H.SternbergS.AdvancedCalculusRevisedEditionJonesAndBartlett|title=Advanced Calculus|last2=Sternberg|first2=Shlomo|publisher=World Scientific|year=2014|isbn=978-981-4583-93-0|location=Hackensack, New Jersey|pages=|last1=Loomis|first1=Lynn Harold|}}
* {{cite book|url=https://archive.org/details/MadsenI.H.TornehaveJ.FromCalculusToCohomologyDeRhamCohomologyAndCharacteristicClasses1996|title=From Calculus to Cohomology: De Rham cohomology and characteristic classes|last2=Tornehave|first2=Jørgen|publisher=Cambridge University Press|year=1997|isbn=0-521-58956-8|location=Cambridge, UK|pages=|last1=Madsen|first1=Ib|}}
* {{cite book|last=Marsden|first=Jerrold E.|last2=Anthony|first2=Tromba|title=Vector Calculus|edition=5th|publisher=W.&nbsp;H. Freeman|date=2003}}
* {{cite book|url=https://archive.org/details/GraduateTextsInMathematics218LeeJ.M.IntroductionToSmoothManifoldsSpringer2012|title=Introduction to Smooth Manifolds|last=Lee|first=John|date=2003|publisher=Springer-Verlag|isbn=978-0-387-95448-6|location=|pages=}}
* {{cite book|url=https://archive.org/details/1979RudinW|title=Principles of Mathematical Analysis|last=Rudin|first=Walter|publisher=McGraw–Hill|year=1976|isbn = 0-07-054235-X|location=New York, NY|pages=|}}
* {{cite book |title=Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus |last1=Spivak|first1=Michael|title-link=Calculus on Manifolds (book)|publisher=	Benjamin Cummings |year=1965 |isbn=0-8053-9021-9 |location=San Francisco |pages= |}}
* {{cite book |first=James |last=Stewart |title=Calculus: Concepts and Contexts |url=https://books.google.com/books?id=Vou3MZu_7tcC&pg=PA960 |date=2009 |publisher=Cengage Learning |pages=960–967 |isbn=978-0-495-55742-5}}
* {{cite book|last=Stewart|first=James|title=Calculus: Early Transcendental Functions|edition=5th|publisher=Brooks/Cole|date=2003}}
* {{cite book|title=An Introduction to Manifolds|last1=Tu|first1=Loring W.|publisher=Springer|year=2011|isbn=978-1-4419-7399-3|edition=2nd|location=New York|pages=|}}

==External links==
* {{springer|title=Stokes formula|id=p/s090310}}
* [http://higheredbcs.wiley.com/legacy/college/hugheshallett/0471484822/theory/hh_focusontheory_sectionm.pdf Proof of the Divergence Theorem and Stokes' Theorem]
* [http://tutorial.math.lamar.edu/classes/calcIII/stokestheorem.aspx Calculus 3 – Stokes Theorem from lamar.edu] – an expository explanation

[[Category:Differential topology]]
[[Category:Differential forms]]
[[Category:Duality theories]]
[[Category:Integration on manifolds]]
[[Category:Theorems in calculus]]
[[Category:Theorems in differential geometry]]

{{Sourceattribution|Stokes' theorem}}