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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 Calculus of moving surfaces ： ウィキペディア英語版 Calculus of moving surfaces The calculus of moving surfaces (CMS) 〔Grinfeld, P. (2010). "Hamiltonian Dynamic Equations for Fluid Films". Studies in Applied Mathematics. . ISSN 00222526.〕 is an extension of the classical tensor calculus to deforming manifolds. Central to the CMS is the $\backslash delta\; /\backslash delta\; t$-derivative whose original definition 〔J. Hadamard, Lecons Sur La Propagation Des Ondes et Les Equations De l’Hydrodynamique. Paris: Hermann, 1903.〕 was put forth by Jacques Hadamard. It plays the role analogous to that of the covariant derivative $\backslash nabla\; \_$ on differential manifolds. In particular, it has the property that it produces a tensor when applied to a tensor. Suppose that $S\_t$ is the evolution of the surface $S$ indexed by a time-like parameter $t$. The definitions of the surface velocity $C$ and the operator $\backslash delta\; /\backslash delta\; t$ are the geometric foundations of the CMS. The velocity C is the rate of deformation of the surface $S$ in the instantaneous normal direction. The value of $C$ at a point $P$ is defined as the limit : $C=\backslash lim\_\; \backslash frac$ where $P^$ is the point on $S\_$ that lies on the straight line perpendicular to $S\_$ at point P. This definition is illustrated in the first geometric figure below. The velocity $C$ is a signed quantity: it is positive when $\backslash overline$ and $C$ is analogous to the relationship between location and velocity in elementary calculus: knowing either quantity allows one to construct the other by differentiation or integration. The $\backslash delta\; /\backslash delta\; t$-derivative for a scalar field F defined on $S\_$ is the rate of change in $F$ in the instantaneously normal direction: : $\backslash frac=\backslash lim\_\; \backslash frac$ This definition is also illustrated in second geometric figure. The above definitions are ''geometric''. In analytical settings, direct application of these definitions may not be possible. The CMS gives ''analytical'' definitions of C and $\backslash delta\; /\backslash delta\; t$ in terms of elementary operations from calculus and differential geometry. ==Analytical definitions== For analytical definitions of $C$ and $\backslash delta\; /\backslash delta\; t$, consider the evolution of $S$ given by : $Z^i\; =\; Z^i\; \backslash left(\; t\; ,S\; \backslash right)\; \backslash ,$ where $Z^$ are general curvilinear space coordinates and $S^$ are the surface coordinates. By convention, tensor indices of function arguments are dropped. Thus the above equations contains $S$ rather than $S^\backslash alpha$.The velocity object $v^$ is defined as the partial derivative : $v^i\; =\backslash frac$ The velocity $C$ can be computed most directly by the formula : $C=v^i\; N\_i\; \backslash ,$ where $N\_i$ are the covariant components of the normal vector $\backslash vec$. The definition of the $\backslash delta\; /\backslash delta\; t$-derivative for an invariant ''F'' reads : $\backslash frac=\backslash frac-v^Z^\_\backslash nabla\; \_F$ where $Z^\backslash alpha\_i$ is the shift tensor and $\backslash nabla\_\backslash alpha$ is the covariant derivative on S. For ''tensors'', an appropriate generalization is needed. The proper definition for a representative tensor $T^\_$ reads : $\backslash frac\}=\backslash frac\}-v^\backslash nabla\; \_T^\_+v^\backslash Gamma\; ^\_T^\_-v^\backslash Gamma\; ^\_T^\_+\backslash nabla\; \_v^T^\_-\backslash nabla\; \_v^T^\_$ where $\backslash Gamma^k\_$ are Christoffel symbols. 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「Calculus of moving surfaces」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.