Editing Generalized Stokes theorem

{{Short description|Statement about integration on manifolds}}
{{about|the generalized theorem|the classical theorem|Stokes' theorem|the equation governing viscous drag in fluids|Stokes' law}}{{Calculus|Vector}}

In [[vector calculus]] and [[differential geometry]] the '''generalized Stokes theorem''' (sometimes with apostrophe as '''Stokes' theorem''' or '''Stokes's theorem'''), also called the '''Stokes–Cartan theorem''',<ref>{{Cite book| url=https://www.springer.com/gp/book/9789400745575|title=Physics of Collisional Plasmas – Introduction to |author1=Michel Moisan |author2= Jacques Pelletier |publisher=Springer|language=en}}</ref> is a statement about the [[integral|integration]] of [[differential form]]s on [[manifolds]], which both simplifies and generalizes several [[theorem]]s from [[vector calculus]].  In particular, the [[fundamental theorem of calculus]] is the special case where the manifold is a [[line segment]], [[Green’s theorem]] and [[Stokes' theorem]] are the cases of a [[surface (mathematics)|surface]] in <math>\R^2</math> or <math>\R^3,</math> and the [[divergence theorem]] is the case of a volume in <math>\R^3.</math><ref>"The Man Who Solved the Market", Gregory Zuckerman, Portfolio November 2019, ASIN: B07P1NNTSD</ref> Hence, the theorem is sometimes referred to as the '''fundamental theorem of multivariate calculus'''.<ref>{{Cite book |last=Spivak |first=Michael |title=Calculus on manifolds : a modern approach to classical theorems of advanced calculus |date=1965 |isbn=0-8053-9021-9 |location=New York |oclc=187146 |publisher=Avalon Publishing}}</ref>

Stokes' theorem says that the integral of a differential form <math>\omega</math> over the [[boundary of a manifold|boundary]] <math>\partial\Omega</math> of some [[orientation (vector space)#Orientation on manifolds|orientable]] manifold <math>\Omega</math> is equal to the integral of its [[exterior derivative]] <math>d\omega</math> over the whole of <math>\Omega</math>, i.e.,
<math display="block">\int_{\partial \Omega} \omega = \int_\Omega \operatorname{d}\omega\,.</math>

Stokes' theorem was formulated in its modern form by [[Élie Cartan]] in 1945,<ref>{{Cite book|title=Les Systèmes Différentiels Extérieurs et leurs Applications Géométriques| last=Cartan| first=Élie| publisher=Hermann|year=1945| location=Paris}}</ref> following earlier work on the generalization of the theorems of vector calculus by [[Vito Volterra]], [[Édouard Goursat]], and [[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 [[Stokes' theorem|classical result]] that [[William Thomson, 1st Baron Kelvin|Lord Kelvin]] communicated to [[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| 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| author1-link=William Thomson, 1st Baron Kelvin|author2-link=George Gabriel Stokes}}
* 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|author-link=Hermann Hankel}} 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 | 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 |author-link=George Gabriel Stokes|editor1-first=Joseph|editor1-last=Larmor|editor2-first=John William |editor2-last=Strutt}}</ref><ref>{{cite book| first=Olivier|last=Darrigol |title=Electrodynamics from Ampère to Einstein| page=146|isbn=0198505930 |location=Oxford, England |publisher=OUP Oxford |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 [[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|author-link=James 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|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|author-link=James 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|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 case relates the [[surface integral]] of the [[Curl (mathematics)|curl]] of a [[vector field]] <math>\textbf{F}</math> over a surface (that is, the [[flux]] of <math>\text{curl}\,\textbf{F}</math>) in Euclidean three-space to the [[line integral]] of the vector field over the surface boundary.

== Introduction ==
The [[fundamental theorem of calculus|second fundamental theorem of calculus]] states that the [[integral]] of a function <math>f</math> over the [[interval (mathematics)|interval]] <math>[a,b]</math> can be calculated by finding an [[antiderivative]] <math>F</math> of <math>f</math>:
<math display="block">\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 <math>F</math>, <math>\frac{dF}{dx}=f(x)</math>. In the parlance of [[differential form]]s, this is saying that <math>f(x)\,dx</math> is the [[exterior derivative]] of the 0-form, i.e. function, <math>F</math>: in other words, that <math>dF=f\,dx</math>. The general Stokes theorem applies to higher [[Degree_of_a_polynomial|degree]] differential forms <math>\omega</math> instead of just 0-forms such as <math>F</math>.
* A closed interval <math>[a,b]</math> is a simple example of a one-dimensional [[manifold with boundary]]. Its boundary is the set consisting of the two points <math>a</math> and <math>b</math>. Integrating <math>f</math> 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 [[compact support|compactly supported]] in order to give a well-defined integral.
* The two points <math>a</math> and <math>b</math> form the boundary of the closed interval. More generally, Stokes' theorem applies to oriented manifolds <math>M</math> with boundary. The boundary <math>\partial M</math> of <math>M</math> is itself a manifold and inherits a natural orientation from that of <math>M</math>. For example, the natural orientation of the interval gives an orientation of the two boundary points. Intuitively, <math>a</math> inherits the opposite orientation as <math>b</math>, as they are at opposite ends of the interval. So, "integrating" <math>F</math> over two boundary points <math>a</math>, <math>b</math> is taking the difference <math>F(b)-F(a)</math>.
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>f\,dx=dF</math>) over a 1-dimensional manifold (<math>[a,b]</math>) by considering the anti-derivative (<math>F</math>) at the 0-dimensional boundaries (<math>\{a,b\}</math>), one can generalize the fundamental theorem of calculus, with a few additional caveats, to deal with the value of integrals (<math>d\omega</math>) over <math>n</math>-dimensional manifolds (<math>\Omega</math>) by considering the antiderivative (<math>\omega</math>) at the <math>(n-1)</math>-dimensional boundaries (<math>\partial\Omega</math>) of the manifold.

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

==Formulation for smooth manifolds with boundary==
Let <math>\Omega</math> be an [[oriented manifold|oriented]] [[smooth manifold]] of [[dimension]] <math>n</math> with boundary and let <math>\alpha</math> be a [[smooth function|smooth]] <math>n</math>-[[differential form]] that is [[Support (mathematics)#Compact support|compactly supported]] on <math>\Omega</math>. First, suppose that <math>\alpha</math> is compactly supported in the domain of a single, oriented [[coordinate chart]] <math>\{U,\varphi\}</math>. In this case, we define the integral of <math>\alpha</math> over <math>\Omega</math> as
<math display="block">\int_\Omega \alpha = \int_{\varphi(U)} (\varphi^{-1})^* \alpha\,,</math>
i.e., via the [[Pullback (differential geometry)|pullback]] of <math>\alpha</math> to <math>\R^n</math>.

More generally, the integral of <math>\alpha</math> over <math>\Omega</math> is defined as follows: Let <math>\{\psi_i\}</math> be a [[partition of unity]] associated with a [[locally finite collection|locally finite]] [[cover (topology)|cover]] <math>\{U_i,\varphi_i\}</math> of (consistently oriented) coordinate charts, then define the integral
<math display="block">\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^n</math> 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:
{{math theorem
| note = ''Stokes–Cartan''
| math_statement = Let <math>\omega</math> be a [[infinitely differentiable|smooth]] <math>(n-1)</math>-[[differential form|form]] with [[compact support]] on an [[oriented]], <math>n</math>-dimensional [[manifold|manifold-with-boundary]] <math>\Omega</math>, where <math>\partial \Omega</math> is given the induced orientation. Then <math display="block">\int_{\Omega} d\omega = \int_{\partial\Omega} \omega.</math>
}}

Here <math>d</math> is the [[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 [[thermodynamics]], where frequently expressions as <math>\oint_W\{\text{d}_\text{total}U\}</math> appear (wherein the total derivative, see below, should not be confused with the exterior one), the integration path <math>W</math> is a one-dimensional closed line on a much higher-dimensional manifold. That is, in a thermodynamic application, where <math>U</math> is a function of the temperature <math>\alpha_1=T</math>, the volume <math>\alpha_2=V</math>, and the electrical polarization <math>\alpha_3=P</math> of the sample, one has
<math display="block">\{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 display="block">\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>\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]] on the set of singular {{mvar|k}}-simplices in {{mvar|M}}. These groups, together with the boundary map, {{math|∂}}, define a [[chain complex]]. The corresponding homology (resp. cohomology) group is isomorphic to the usual [[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]] groups <math>H_{dR}^k(M, \mathbf{R})</math>.

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 display="block">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|1=''dω'' = 0}}, have zero integral over ''boundaries'', i.e. over manifolds that can be written as {{math|∂Σ<sub>''c''</sub> ''M<sub>c</sub>''}}, and
#exact forms, i.e., {{math|1=''ω'' = ''dσ''}}, have zero integral over ''cycles'', i.e. if the boundaries sum up to the empty set: {{math|1=∂Σ<sub>''c''</sub> ''M<sub>c</sub>'' = ∅}}.

[[De Rham's theorem]] shows that this homomorphism is in fact an [[isomorphism]]. So the converse to 1 and 2 above hold true. In other words, if {{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 display="block">\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>

{{math theorem
| note = ''Stokes' theorem for chains''
| math_statement = If {{mvar|c}} is a smooth {{mvar|k}}-chain in a smooth manifold {{mvar|M}}, and {{mvar|ω}} is a smooth {{math|(''k'' − 1)}}-form on {{mvar|M}}, then
<math display="block">\int_{\partial c}\omega = \int_c d\omega.</math>
}}

== 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|1=''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}}

== Classical vector analysis example ==
Let <math>\gamma:[a,b]\to\R^2</math> be a [[piecewise]] smooth [[Jordan curve|Jordan plane curve]]. The [[Jordan curve theorem]] implies that <math>\gamma</math> divides <math>\R^2</math> into two components, a [[compact space|compact]] one and another that is non-compact. Let <math>D</math> denote the compact part that is bounded by <math>\gamma</math> and suppose <math>\psi:D\to\R^3</math> is smooth, with <math>S=\psi(D)</math>. If <math>\Gamma</math> is the [[space curve]] defined by <math>\Gamma(t)=\psi(\gamma(t))</math><ref name="cgamma" group="note"><math>\gamma</math> and <math>\Gamma</math> are both loops, however, <math>\Gamma</math> is not necessarily a [[Jordan curve]]</ref> and <math>\textbf{F}</math> is a smooth vector field on <math>\R^3</math>, then:<ref name="Jame">{{cite book |last=Stewart |first=James |url={{Google books |plainurl=yes |id=btIhvKZCkTsC |page=786 }} |title=Essential Calculus: Early Transcendentals |publisher=Cole |year=2010}}</ref><ref name="bath">This proof is based on the Lecture Notes given by Prof. Robert Scheichl ([[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">{{cite web |title=This proof is also same to the proof shown in |url=http://www.proofwiki.org/wiki/Classical_Stokes'_Theorem}}</ref>
<math display="block">\oint_\Gamma \mathbf{F}\, \cdot\, d{\mathbf{\Gamma}}  = \iint_S \left( \nabla \times \mathbf{F} \right) \cdot\, d\mathbf{S} </math>

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 
<math display="block">\begin{pmatrix}
F_x \\ 
F_y \\ 
F_z \\
\end{pmatrix}\cdot d\Gamma \to F_x \,dx + F_y \,dy + F_z \,dz</math>
<math display="block">\begin{align}
&\nabla \times \begin{pmatrix} F_x \\ F_y \\ F_z \end{pmatrix} \cdot d\mathbf{S} = 
\begin{pmatrix}
\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{pmatrix} \cdot d\mathbf{S} \to \\[1.4ex]
&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{align}</math>

==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>\Omega</math> 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 <math>x</math>-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>\Omega</math> 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>\Omega</math> 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 <math>D</math> is a connected bounded open subset of <math>\R^n</math>.  Call <math>D</math> a ''standard domain'' if it satisfies the following property: there exists a subset <math>P</math> of <math>\partial D</math>, open in <math>\partial D</math>, whose complement in <math>\partial D</math> has [[Hausdorff measure|Hausdorff <math>(n-1)</math>-measure]] zero; and such that every point of <math>P</math> has a ''generalized normal vector''. This is a vector <math>\textbf{v}(x)</math> such that, if a coordinate system is chosen so that <math>\textbf{v}(x)</math> is the first basis vector, then, in an open neighborhood around <math>x</math>, there exists a smooth function <math>f(x_2,\dots,x_n)</math> such that <math>P</math> is the graph <math>\{x_1=f(x_2,\dots,x_n)\}</math> and <math>D</math> is the region <math>\{x_1:x_1<f(x_2,\dots,x_n)\}</math>. Whitney remarks that the boundary of a standard domain is the union of a set of zero Hausdorff <math>(n-1)</math>-measure and a finite or countable union of smooth <math>(n-1)</math>-manifolds, each of which has the domain on only one side. He then proves that if <math>D</math> is a standard domain in <math>\R^n</math>, <math>\omega</math> is an <math>(n-1)</math>-form which is defined, continuous, and bounded on <math>D\cup P</math>, smooth on <math>D</math>, integrable on <math>P</math>, and such that <math>d\omega</math> is integrable on <math>D</math>, then Stokes' theorem holds, that is,
<math display="block">\int_P \omega = \int_D d\omega\,.</math>

The study of measure-theoretic properties of rough sets leads to [[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|bibcode=1993math.....10231H|s2cid=17436511}}</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.

===Classical (vector calculus) case===
{{Main|Stokes' theorem}}

[[Image:Stokes' Theorem.svg|thumb|right|An illustration of the vector-calculus Stokes theorem, with surface <math>\Sigma</math>, its boundary <math>\partial\Sigma</math> 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]]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 Stokes' theorem relates the [[surface integral]] of the [[Curl (mathematics)|curl]] of a [[vector field]] over a surface <math>\Sigma</math> in Euclidean three-space to the [[line integral]] of the vector field over its boundary. It is a special case of the general Stokes theorem (with <math>n=2</math>) once we identify a vector field with a 1-form using the metric on Euclidean 3-space. The curve of the line integral, <math>\partial\Sigma</math>, must have positive [[curve orientation|orientation]], meaning that <math>\partial\Sigma</math> points counterclockwise when the [[normal (geometry)|surface normal]], <math>n</math>, points toward the viewer.

One consequence of this theorem is that the [[field line]]s of a vector field with zero curl cannot be closed contours. The formula can be rewritten as:{{clear}}
{{math theorem
| math_statement = Suppose <math>\textbf{F}=\big(P(x,y,z),Q(x,y,z),R(x,y,z)\big)</math> is defined in a region with smooth surface <math>\Sigma</math> and has continuous first-order [[partial derivatives]]. Then
<math display="block">
 \iint_\Sigma \Biggl(\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\Biggr)
  = \oint_{\partial\Sigma} \Big(P\,dx + Q\,dy + R\,dz\Big)\,,
</math>
where <math>P,Q</math> and <math>R</math> are the components of <math>\textbf{F}</math>, and <math>\partial\Sigma</math> is the boundary of the region <math>\Sigma</math>.
}}

===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 [[Maxwell equations]] 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]]. 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 three-dimensional Stokes theorem plus relativistic invariance, <math>\textstyle\int\tfrac{\partial}{\partial t}\dots\to\tfrac{d}{dt}\textstyle\int\cdots</math>)
|- valign="center"
| Maxwell–Faraday equation<br> [[Faraday's law of induction]]:
| style="text-align: center;" | <math>\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}</math>
| style="text-align: center;" | <math>\begin{align}
\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} </math>
(with {{mvar|C}} and {{mvar|S}} not necessarily stationary)
|- valign="center"
| [[Ampère's law]]<br /> (with Maxwell's extension):
| style="text-align: center;" | <math>\nabla \times \mathbf{H} = \mathbf{J} + \frac{\partial \mathbf{D}} {\partial t}</math>
| style="text-align: center;" | <math>\begin{align}
\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} </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 [[Maxwell's equations#CGS units|CGS]] or [[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| isbn=9780471431329 }}</ref><ref>{{cite book|first1=M.|last1=Born|first2=E.|last2=Wolf| title=[[Principles of Optics]]|edition=6th|publisher=Cambridge University Press|location=Cambridge, England|date=1980}}</ref>
<math display="block">\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 [[speed of light]] in vacuum.

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

===Volume integral of gradient of scalar field===
Let <math>f : \Omega \to \mathbb{R}</math> be a [[scalar field]]. Then
<math display="block">\int_\Omega \vec{\nabla} f = \int_{\partial \Omega} \vec{n} f</math>
where <math>\vec{n}</math> is the [[normal vector]] to the surface <math>\partial \Omega</math> at a given point.

Proof:
Let <math>\vec{c}</math> be a vector. Then
<math display="block">
\begin{align}
0
&= \int_\Omega \vec{\nabla} \cdot \vec{c} f - \int_{\partial \Omega} \vec{n} \cdot \vec{c} f & \text{by the divergence theorem} \\
&= \int_\Omega \vec{c} \cdot \vec{\nabla} f - \int_{\partial \Omega} \vec{c} \cdot \vec{n} f \\
&= \vec{c} \cdot \int_\Omega \vec{\nabla} f - \vec{c} \cdot \int_{\partial \Omega} \vec{n} f \\
&= \vec{c} \cdot \left( \int_\Omega \vec{\nabla} f - \int_{\partial \Omega} \vec{n} f \right)
\end{align}
</math>
Since this holds for any <math>\vec{c}</math> (in particular, for every [[basis vector]]), the result follows.

==See also==
{{Portal|Mathematics}}
*[[Chandrasekhar–Wentzel lemma]]

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

==References==
{{reflist}}

==Further reading==
* {{cite book |last=Grunsky |first=Helmut |author-link=Helmut Grunsky |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|author-link2=Shlomo Sternberg|last1=Loomis|first1=Lynn Harold|author1-link=Lynn Harold Loomis}}
* {{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|last1=Madsen|first1=Ib|author1-link=Ib Madsen}}
* {{cite book|author-link=Jerrold E. Marsden|last1=Marsden|first1=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}}
* {{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|author-link=Walter Rudin}}
* {{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 |author1-link=Michael Spivak }}
* {{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|author1-link=Loring W. Tu}}

==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

{{Calculus topics}}

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