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In mathematics, Green's theorem gives the relationship between a line integral around a simple closed curve ''C'' and a double integral over the plane region ''D'' bounded by ''C''. It is named after George Green 〔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 (pages 10-12 ) of his ''Essay''. In 1846, the form of "Green's theorem" which appears in this article was first published, without proof, in an article by Augustin Cauchy: A. Cauchy (1846) ("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.) A proof of the theorem was finally provided in 1851 by Bernhard Riemann in his inaugural dissertation: Bernhard Riemann (1851) (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.〕 and is the two-dimensional special case of the more general Kelvin–Stokes theorem. ==Theorem== Let ''C'' be a positively oriented, piecewise smooth, simple closed curve in a plane, and let ''D'' be the region bounded by ''C''. If ''L'' and ''M'' are functions of (''x'', ''y'') defined on an open region containing ''D'' and have continuous partial derivatives there, then〔Mathematical methods for physics and engineering, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press, 2010, ISBN 978-0-521-86153-3〕〔Vector Analysis (2nd Edition), M.R. Spiegel, S. Lipschutz, D. Spellman, Schaum’s Outlines, McGraw Hill (USA), 2009, ISBN 978-0-07-161545-7〕 : where the path of integration along ''C'' is counterclockwise. In physics, Green's theorem is mostly used to solve two-dimensional flow integrals, stating that the sum of fluid outflows from a volume is equal to the total outflow summed about an enclosing area. In plane geometry, and in particular, area surveying, Green's theorem can be used to determine the area and centroid of plane figures solely by integrating over the perimeter. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Green's theorem」の詳細全文を読む スポンサード リンク
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