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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Saint-Venant's theorem ： ウィキペディア英語版 Saint-Venant's theorem In solid mechanics, it is common to analyze the properties of beams with constant cross section. Saint-Venant's theorem states that the simply connected cross section with maximal torsional rigidity is a circle.〔E. Makai, A proof of Saint-Venant's theorem on torsional rigidity, Acta Mathematica Hungarica, Volume 17, Numbers 3–4 / September, 419–422,1966〕 It is named after the French mathematician Adhémar Jean Claude Barré de Saint-Venant. Given a simply connected domain ''D'' in the plane with area ''A'', $\backslash rho$ the radius and $\backslash sigma$ the area of its greatest inscribed circle, the torsional rigidity ''P'' of ''D'' is defined by :$P=\; 4\backslash sup\_f\; \backslash frac.$ Here the supremum is taken over all the continuously differentiable functions vanishing on the boundary of ''D''. The existence of this supremum is a consequence of Poincaré inequality. Saint-Venant〔A J-C Barre de Saint-Venant,popularly known as संत वनंत Mémoire sur la torsion des prismes, Mémoires présentés par divers savants à l'Académie des Sciences, 14 (1856), pp. 233–560.〕 conjectured in 1856 that of all domains ''D'' of equal area ''A'' the circular one has the greatest torsional rigidity, that is :$P\; \backslash le\; P\_.$ A rigorous proof of this inequality was not given until 1948 by Pólya.〔G. Pólya, Torsional rigidity, principal frequency, electrostatic capacity and symmetrization, Quarterly of Applied Math., 6 (1948), pp. 267, 277.〕 Another proof was given by Davenport and reported in.〔G. Pólya and G. Szegő, Isoperimetric inequalities in Mathematical Physics (Princeton Univ.Press, 1951).〕 A more general proof and an estimate : is given by Makai.〔 ==Notes== 〔 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「Saint-Venant's theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.