翻訳と辞書 Words near each other ・ Laplace Affair ・ Laplace distribution ・ Laplace equation for irrotational flow ・ Laplace expansion ・ Laplace expansion (potential) ・ Laplace formula ・ Laplace functional ・ Laplace invariant ・ Laplace Island ・ Laplace Island (Antarctica) ・ Laplace Island (Western Australia) ・ Laplace limit ・ Laplace no Ma ・ Laplace number ・ Laplace operator ・ Laplace operators in differential geometry ・ Laplace plane ・ Laplace pressure ・ Laplace principle (large deviations theory) ・ Laplace transform ・ Laplace transform applied to differential equations ・ Laplace's demon ・ Laplace's equation ・ Laplace's law ・ Laplace's method ・ LaPlace, Louisiana ・ Laplace-P ・ LaplacesDemon ・ Laplace–Beltrami operator ・ Laplace–Carson transform Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Laplace operators in differential geometry ： ウィキペディア英語版 Laplace operators in differential geometry In differential geometry there are a number of second-order, linear, elliptic differential operators bearing the name Laplacian. This article provides an overview of some of them. == Connection Laplacian == The connection Laplacian, also known as the rough Laplacian, is a differential operator acting on the various tensor bundles of a manifold, defined in terms of a Riemannian- or pseudo-Riemannian metric. When applied to functions (i.e. tensors of rank 0), the connection Laplacian is often called the Laplace–Beltrami operator. It is defined as the trace of the second covariant derivative: :$\backslash Delta\; T=\; -\backslash text\backslash ;\backslash nabla^2\; T,$ where ''T'' is any tensor, $\backslash nabla$ is the Levi-Civita connection associated to the metric, and the trace is taken with respect to the metric. Recall that the second covariant derivative of ''T'' is defined as :$\backslash nabla^2\_\; T\; =\; -\backslash left(\backslash nabla\_X\; \backslash nabla\_Y\; T\; -\; \backslash nabla\_\; T\backslash right).$ Note that with this definition, the connection Laplacian has negative spectrum. On functions, it agrees with the operator given as the divergence of the gradient. If connection of interest is Levi-Civita connection one can find a convenient formula for Laplacian of scalar function in terms of partial derivatives with respect to chosen coordinates: :$\backslash Delta\; \backslash phi\; =\; |g|^\; \backslash partial\_\backslash mu\backslash left(\; |g|^\; g^\; \backslash partial\_\backslash nu\backslash right)\backslash phi$ where $\backslash phi$ is scalar function, $|g|$ is absolute value of determinant of metric (the use of absolute value is necessary in Pseudo Riemmanian case, for example in General Relativity) and $g^$ denotes inverse of the metric tensor 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「Laplace operators in differential 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.