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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 Covariant transformation ： ウィキペディア英語版 Covariant transformation : ''See also Covariance and contravariance of vectors'' In physics, a covariant transformation is a rule (specified below) that specifies how certain entities change under a change of basis. In particular, the term is used for vectors and tensors. The transformation that describes the new basis vectors as a linear combination of the old basis vectors is ''defined'' as a covariant transformation. Conventionally, indices identifying the basis vectors are placed as lower indices and so are all entities that transform in the same way. The inverse of a covariant transformation is a contravariant transformation. Since a vector should be ''invariant'' under a change of basis, its ''components'' must transform according to the contravariant rule. Conventionally, indices identifying the components of a vector are placed as upper indices and so are all indices of entities that transform in the same way. The sum over pairwise matching indices of a product with the same lower and upper indices are invariant under a transformation. A vector itself is a geometrical quantity, in principle, independent (invariant) of the chosen basis. A vector v is given, say, in components ''v''''i'' on a chosen basis e''i''. On another basis, say e′''j'', the same vector v has different components ''v''′''j'' and :$\backslash mathbf\; =\; \backslash sum\_i\; v^i\; \_i\; =\; \backslash sum\_j\; ^j\; \backslash mathbf\text{'}\_j.$ With v being invariant and the e''i'' transforming covariantly, it must be that the ''v''''i'' (the set of numbers identifying the components) transform in a different way, being the inverse called the contravariant transformation rule. If, for example in a 2-dimensional Euclidean space, the new basis vectors are rotated anti-clockwise with respect to the old basis vectors, then it will appear in terms of the new system that the ''componentwise representation'' of the vector was rotated in the opposite direction, i.e. clockwise (see figure). ---- Image:Transformation2polar_basis_vectors.svg|A vector v, and local tangent basis vectors and . Image:Transformation2polar contravariant vector.svg|Coordinate representations of v. A vector v is described in a given coordinate grid (black lines) on a basis which are the tangent vectors to the (here rectangular) coordinate grid. The basis vectors are ex and ey. In another coordinate system (dashed and red), the new basis vectors are tangent vectors in the radial direction and perpendicular to it. These basis vectors are indicated in red as er and eφ. They appear rotated anticlockwise with respect to the first basis. The covariant transformation here is thus an anticlockwise rotation. If we view the vector v with eφ pointed upwards, its representation in this frame appears rotated to the right. The contravariant transformation is a clockwise rotation. . ---- ==Examples of covariant transformation== 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「Covariant transformation」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.