  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 intÃ©grales qui s'Ã©tendent Ã  tous les points d'une courbe fermÃ©e" (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) fÃ¼r eine allgemeine theorie der functionen einer verÃ¤nderlichen complexen grÃ¶sse (basis for a general theory of functions of a variable complex quantity), (gÃ¶ttingen, (germany): adalbert rente, 1867); see pages 8 - 9. and is the two-dimensional special case of the more general kelvinâstokes 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, thenmathematical methods for physics and engineering, k.f. riley, m.p. hobson, s.j. bence, cambridge university press, 2010, isbn 978-0-521-86153-3vector analysis (2nd edition), m.r. spiegel, s. lipschutz, d. spellman, schaumâs outlines, mcgraw hill (usa), 2009, isbn 978-0-07-161545-7 :\oint_{c} (l\, dx + m\, dy) = \iint_{d} \left(\frac{\partial m}{\partial x} - \frac{\partial l}{\partial y}\right)\, dx\, dy where the path of integration along c is counterclockwise. in physics
Theorems_in_calculus