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Green's theorem
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== History == It is named after [[George Green (mathematician)|George Green]], who stated a similar result in an 1828 paper titled ''[[An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism]]''. In 1846, [[Augustin-Louis Cauchy]] published a paper stating Green's theorem as the penultimate sentence. This is in fact the first printed version of Green's theorem in the form appearing in modern textbooks. George Green, ''An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism'' (Nottingham, England: T. Wheelhouse, 1828). Green did not actually derive the form of "Green's theorem" which appears in this article; rather, he derived a form of the "divergence theorem", which appears on [https://books.google.com/books?id=GwYXAAAAYAAJ&pg=PA10 pages 10–12] of his ''Essay''.<br /> In 1846, the form of "Green's theorem" which appears in this article was first published, without proof, in an article by [[Augustin-Louis Cauchy|Augustin Cauchy]]: A. Cauchy (1846) [https://archive.org/stream/ComptesRendusAcademieDesSciences0023/ComptesRendusAcadmieDesSciences-Tome023-Juillet-dcembre1846#page/n254/mode/1up "Sur les intégrales qui s'étendent à tous les points d'une courbe fermée"] (On integrals that extend over all of the points of a closed curve), ''Comptes rendus'', '''23''': 251–255. (The equation appears at the bottom of page 254, where (''S'') denotes the line integral of a function ''k'' along the curve ''s'' that encloses the area ''S''.)<br /> A proof of the theorem was finally provided in 1851 by [[Bernhard Riemann]] in his inaugural dissertation: Bernhard Riemann (1851) [https://books.google.com/books?id=PpALAAAAYAAJ&pg=PP5 ''Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse''] (Basis for a general theory of functions of a variable complex quantity), (Göttingen, (Germany): Adalbert Rente, 1867); see pages 8–9.<ref>{{Cite book |last=Katz |first=Victor J. |author-link=Victor J. Katz |url=https://edisciplinas.usp.br/pluginfile.php/6075667/mod_resource/content/1/Victor%20J.%20Katz%20-%20A%20History%20of%20Mathematics-Pearson%20%282008%29.pdf |title=A history of mathematics: an introduction |publisher=[[Addison-Wesley]] |year=2009 |isbn=978-0-321-38700-4 |edition=3. |location=Boston, Mass. Munich |pages=801–5 |chapter=22.3.3: Complex Functions and Line Integrals}}</ref>
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