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Generalized Stokes theorem
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{{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. 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.
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