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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 Second covariant derivative ： ウィキペディア英語版 Second covariant derivative In the math branches of differential geometry and vector calculus, the second covariant derivative, or the second order covariant derivative, of a vector field is the derivative of its derivative with respect to another two tangent vector fields. Formally, given a (pseudo)-Riemannian manifold (''M'', ''g'') associated with a vector bundle ''E'' → ''M'', let ∇ denote the Levi-Civita connection given by the metric ''g'', and denote by Γ(''E'') the space of the smooth sections of the total space ''E''. Denote by ''T *M'' the cotangent bundle of ''M''. Then the second covariant derivative can be defined as the composition of the two ∇s as follows: 〔, pp. 7〕 :$\backslash Gamma(E)\; \backslash stackrel\; \backslash Gamma(T^$ *M \otimes E) \stackrel \Gamma(T^ *M \otimes T^ *M \otimes E). For example, given vector fields ''u'', ''v'', ''w'', a second covariant derivative can be written as :$(\backslash nabla^2\_\; w)^a\; =\; u^c\; v^b\; \backslash nabla\_c\; \backslash nabla\_b\; w^a$ by using abstract index notation. It is also straightforward to verify that :$(\backslash nabla\_u\; \backslash nabla\_v\; w)^a\; =\; u^c\; \backslash nabla\_c\; v^b\; \backslash nabla\_b\; w^a\; =\; u^c\; v^b\; \backslash nabla\_c\; \backslash nabla\_b\; w^a\; +\; (u^c\; \backslash nabla\_c\; v^b)\; \backslash nabla\_b\; w^a\; =\; (\backslash nabla^2\_\; w)^a\; +\; (\backslash nabla\_\; w)^a.$ Thus :$\backslash nabla^2\_\; w\; =\; \backslash nabla\_u\; \backslash nabla\_v\; w\; -\; \backslash nabla\_\; w.$ One may use this fact to write Riemann curvature tensor as follows: 〔(【引用サイトリンク】title=Chapter 13: Curvature in Riemannian Manifolds )〕 :$R(u,v)\; w=\backslash nabla^2\_\; w\; -\; \backslash nabla^2\_\; w.$ Similarly, one may also obtain the second covariant derivative of a function ''f'' as :$\backslash nabla^2\_\; f\; =\; u^c\; v^b\; \backslash nabla\_c\; \backslash nabla\_b\; f\; =\; \backslash nabla\_u\; \backslash nabla\_v\; f\; -\; \backslash nabla\_\; f.$ Since Levi-Civita connection is torsion-free, for any vector fields ''u'' and ''v'', we have :$\backslash nabla\_u\; v\; -\; \backslash nabla\_v\; u\; =\; (v).$ By feeding the function ''f'' on both sides of the above equation, we have :$(\backslash nabla\_u\; v\; -\; \backslash nabla\_v\; u)(f)\; =\; (v)(f)\; =\; u(v(f))\; -\; v(u(f)).$ :$\backslash nabla\_\; f\; -\; \backslash nabla\_\; f\; =\; \backslash nabla\_u\; \backslash nabla\_v\; f\; -\; \backslash nabla\_v\; \backslash nabla\_u\; f.$ Thus :$\backslash nabla^2\_\; f\; =\; \backslash nabla^2\_\; f.$ That is, the value of the second covariant derivative of a function is independent on the order of taking derivatives. ==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「Second covariant derivative」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.