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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 First variation ： ウィキペディア英語版 First variation In applied mathematics and the calculus of variations, the first variation of a functional ''J''(''y'') is defined as the linear functional $\backslash delta\; J(y)$ mapping the function ''h'' to :$\backslash delta\; J(y,h)\; =\; \backslash lim\_\; \backslash frac\; =\; \backslash left.\backslash frac\; J(y\; +\; \backslash varepsilon\; h)\backslash right|\_,$ where ''y'' and ''h'' are functions, and ''ε'' is a scalar. This is recognizable as the Gâteaux derivative of the functional. ==Example== Compute the first variation of :$J(y)=\backslash int\_a^b\; yy\text{'}\; dx.$ From the definition above, :$$ \begin \delta J(y,h)&=\left.\frac J(y + \varepsilon h)\right|_\\ &= \left.\frac \int_a^b (y + \varepsilon h)(y^\prime + \varepsilon h^\prime) \ dx\right|_\\ &= \left.\frac \int_a^b (yy^\prime + y\varepsilon h^\prime + y^\prime\varepsilon h + \varepsilon^2 hh^\prime) \ dx\right|_\\ &= \left.\int_a^b \frac (yy^\prime + y\varepsilon h^\prime + y^\prime\varepsilon h + \varepsilon^2 hh^\prime) \ dx\right|_\\ &= \left.\int_a^b (yh^\prime + y^\prime h + 2\varepsilon hh^\prime) \ dx\right|_\\ &= \int_a^b (yh^\prime + y^\prime h) \ dx \end 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「First variation」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.