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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 Reynolds transport theorem ： ウィキペディア英語版 Reynolds transport theorem In differential calculus, the Reynolds transport theorem (also known as the Leibniz–Reynolds transport theorem), or in short Reynolds' theorem, is a three-dimensional generalization of the Leibniz integral rule which is also known as differentiation under the integral sign. The theorem is named after Osborne Reynolds (1842–1912). It is used to recast derivatives of integrated quantities and is useful in formulating the basic equations of continuum mechanics. Consider integrating $\backslash mathbf\; =\; \backslash mathbf(\backslash mathbf,t)$ over the time-dependent region $\backslash Omega(t)$ that has boundary $\backslash partial\; \backslash Omega(t)$, then taking the derivative with respect to time: :$$ \cfract}\int_ \mathbf~\text ~. If we wish to move the derivative within the integral, there are two issues: the time dependence of $\backslash mathbf$, and the introduction of and removal of space from $\backslash Omega$ due to its dynamic boundary. Reynolds' transport theorem provides the necessary framework. == General form == Reynolds' transport theorem, derived in,〔L. G. Leal, 2007, p. 23.〕〔O. Reynolds, 1903, Vol. 3, p. 12–13〕〔J.E. Marsden and A. Tromba, 5th ed. 2003〕 is: :$$ \cfract}\int_ \mathbf~\text = \int_ \frac~\text + \int_ (\mathbf^\cdot\mathbf)\mathbf~\text ~ in which $\backslash mathbf(\backslash mathbf,t)$ is the outward-pointing unit-normal, $\backslash mathbf$ is a point in the region and is the variable of integration, $\backslash text$ and $\backslash text$ are volume and surface elements at $\backslash mathbf$, and $\backslash mathbf^(\backslash mathbf,t)$ is the velocity of the area element – so not necessarily the flow velocity.〔Only for a material element there is $\backslash mathbf^b=\backslash mathbf.$〕 The function $\backslash mathbf$ may be tensor, vector or scalar valued.〔H. Yamaguchi, ''Engineering Fluid Mechanics, ''Springer c2008 p23〕 Note that the integral on the left hand side is a function solely of time, and so the total derivative has been used. 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「Reynolds transport 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.