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In mathematics and physics, the Christoffel symbols are an array of numbers describing an affine connection.〔See, for instance, and 〕 In other words, when a surface or other manifold is endowed with a sense of differential geometry — parallel transport, covariant derivatives, geodesics, etc. — the Christoffel symbols are a concrete representation of that geometry in coordinates on that surface or manifold. Frequently, but not always, the connection in question is the Levi-Civita connection of a metric tensor. At each point of the underlying ''n''-dimensional manifold, for any local coordinate system around that point, the Christoffel symbols are denoted Γ''i''''jk'' for ''i'', ''j'', ''k'' = 1, 2, ..., ''n''. Each entry of this ''n'' × ''n'' × ''n'' array is a real number. Under ''linear'' coordinate transformations on the manifold, the Christoffel symbols transform like the components of a tensor, but under general coordinate transformations they do not. Christoffel symbols are used for performing practical calculations. For example, the Riemann curvature tensor can be expressed entirely in terms of the Christoffel symbols and their first partial derivatives. In general relativity, the connection plays the role of the gravitational force field with the corresponding gravitational potential being the metric tensor. When the coordinate system and the metric tensor share some symmetry, many of the Γ''i''''jk'' are zero. The Christoffel symbols are named for Elwin Bruno Christoffel (1829–1900). ==Preliminaries== The definitions given below are valid for both Riemannian manifolds and pseudo-Riemannian manifolds, such as those of general relativity, with careful distinction being made between upper and lower indices (contra-variant and co-variant indices). The formulas hold for either sign convention, unless otherwise noted. Einstein summation convention is used in this article. The connection coefficients of the Levi-Civita connection (or pseudo-Riemannian connection) expressed in a coordinate basis are called the 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「Christoffel symbols」の詳細全文を読む スポンサード リンク
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