翻訳と辞書 Words near each other ・ Jacobel Glacier ・ Jacobellis ・ Jacobellis v. Ohio ・ Jacobello da Messina ・ Jacobello del Fiore ・ Jacobello di Bonomo ・ Jacobena Angliss ・ Jacobethan ・ Jacobetty Rosa ・ Jacobi ・ Jacobi (crater) ・ Jacobi (surname) ・ Jacobi coordinates ・ Jacobi eigenvalue algorithm ・ Jacobi elliptic functions ・ Jacobi field ・ Jacobi form ・ Jacobi group ・ Jacobi identity ・ Jacobi integral ・ Jacobi matrix ・ Jacobi Medical Center ・ Jacobi method ・ Jacobi method for complex Hermitian matrices ・ Jacobi operator ・ Jacobi polynomials ・ Jacobi Robinson ・ Jacobi rotation ・ Jacobi set ・ Jacobi sum 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 Jacobi field ： ウィキペディア英語版 Jacobi field In Riemannian geometry, a Jacobi field is a vector field along a geodesic $\backslash gamma$ in a Riemannian manifold describing the difference between the geodesic and an "infinitesimally close" geodesic. In other words, the Jacobi fields along a geodesic form the tangent space to the geodesic in the space of all geodesics. They are named after Carl Jacobi. ==Definitions and properties== Jacobi fields can be obtained in the following way: Take a smooth one parameter family of geodesics $\backslash gamma\_\backslash tau$ with $\backslash gamma\_0=\backslash gamma$, then :$J(t)=\backslash left.\backslash frac\backslash right|\_$ is a Jacobi field, and describes the behavior of the geodesics in an infinitesimal neighborhood of a given geodesic $\backslash gamma$. A vector field ''J'' along a geodesic $\backslash gamma$ is said to be a Jacobi field if it satisfies the Jacobi equation: :$\backslash fracJ(t)+R(J(t),\backslash dot\backslash gamma(t))\backslash dot\backslash gamma(t)=0,$ where ''D'' denotes the covariant derivative with respect to the Levi-Civita connection, ''R'' the Riemann curvature tensor, $\backslash dot\backslash gamma(t)=d\backslash gamma(t)/dt$ the tangent vector field, and ''t'' is the parameter of the geodesic. On a complete Riemannian manifold, for any Jacobi field there is a family of geodesics $\backslash gamma\_\backslash tau$ describing the field (as in the preceding paragraph). The Jacobi equation is a linear, second order ordinary differential equation; in particular, values of $J$ and $\backslash fracJ$ at one point of $\backslash gamma$ uniquely determine the Jacobi field. Furthermore, the set of Jacobi fields along a given geodesic forms a real vector space of dimension twice the dimension of the manifold. As trivial examples of Jacobi fields one can consider $\backslash dot\backslash gamma(t)$ and $t\backslash dot\backslash gamma(t)$. These correspond respectively to the following families of reparametrisations: $\backslash gamma\_\backslash tau(t)=\backslash gamma(\backslash tau+t)$ and $\backslash gamma\_\backslash tau(t)=\backslash gamma((1+\backslash tau)t)$. Any Jacobi field $J$ can be represented in a unique way as a sum $T+I$, where $T=a\backslash dot\backslash gamma(t)+bt\backslash dot\backslash gamma(t)$ is a linear combination of trivial Jacobi fields and $I(t)$ is orthogonal to $\backslash dot\backslash gamma(t)$, for all $t$. The field $I$ then corresponds to the same variation of geodesics as $J$, only with changed parameterizations. 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「Jacobi field」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.