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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 List of formulas in Riemannian geometry ： ウィキペディア英語版 List of formulas in Riemannian geometry This is a list of formulas encountered in Riemannian geometry. ==Christoffel symbols, covariant derivative== In a smooth coordinate chart, the Christoffel symbols of the first kind are given by :$\backslash Gamma\_=\backslash frac12\; \backslash left($ \frac g_ +\frac g_ -\frac g_ \right) =\frac12 \left( g_ + g_ - g_ \right) \,, and the Christoffel symbols of the second kind by :$\backslash begin$ \Gamma^m\Gamma_\\ &=\frac\, g^ \left( \frac g_ +\frac g_ -\frac g_ \right) =\frac\, g^ \left( g_ + g_ - g_ \right) \,. \end Here $g^$ is the inverse matrix to the metric tensor $g\_$. In other words, :$$ \delta^i and thus :$$ n = \delta^ig_ is the dimension of the manifold. Christoffel symbols satisfy the symmetry relations :$\backslash Gamma\_\; =\; \backslash Gamma\_$ or, respectively, $\backslash Gamma^i\_$, the second of which is equivalent to the torsion-freeness of the Levi-Civita connection. The contracting relations on the Christoffel symbols are given by :$\backslash Gamma^i\; g^\backslash frac=\backslash frac\; \backslash frac\; =\; \backslash frac\; \backslash$ and :$g^\backslash Gamma^i\backslash ,g^\backslash right)\}$ where |''g''| is the absolute value of the determinant of the metric tensor $g\_\backslash$. These are useful when dealing with divergences and Laplacians (see below). The covariant derivative of a vector field with components $v^i$ is given by: :$$ v^i +\Gamma^i=(\nabla_j v)_i=\frac-\Gamma^k this becomes :$$ v^=\frac +\Gamma^i+\Gamma^j and likewise for tensors with more indices. The covariant derivative of a function (scalar) $\backslash phi$ is just its usual differential: :$$ \nabla_i \phi=\phi_=\phi_=\frac Because the Levi-Civita connection is metric-compatible, the covariant derivatives of metrics vanish, :$$ (\nabla_k g)_ = (\nabla_k g)^ = 0 as well as the covariant derivatives of the metric's determinant (and volume element) :$$ \nabla_k \sqrt=0 The geodesic $X(t)$ starting at the origin with initial speed $v^i$ has Taylor expansion in the chart: :$$ X(t)^i=tv^i-\frac\Gamma^i{}_{jk}v^jv^k+O(t^3) 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「List of formulas in Riemannian geometry」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.