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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 Symmetric tensor ： ウィキペディア英語版 Symmetric tensor In mathematics, a symmetric tensor is a tensor that is invariant under a permutation of its vector arguments: :$T(v\_1,v\_2,\backslash dots,v\_r)\; =\; T(v\_,v\_,\backslash dots,v\_)$ for every permutation σ of the symbols . Alternatively, a symmetric tensor of order ''r'' represented in coordinates as a quantity with ''r'' indices satisfies :$T\_\; =\; T\_\backslash dots\; i\_\}.$ The space of symmetric tensors of order ''r'' on a finite-dimensional vector space is naturally isomorphic to the dual of the space of homogeneous polynomials of degree ''r'' on ''V''. Over fields of characteristic zero, the graded vector space of all symmetric tensors can be naturally identified with the symmetric algebra on ''V''. A related concept is that of the antisymmetric tensor or alternating form. Symmetric tensors occur widely in engineering, physics and mathematics. ==Definition== Let ''V'' be a vector space and :$T\backslash in\; V^$ a tensor of order ''k''. Then ''T'' is a symmetric tensor if :$\backslash tau\_\backslash sigma\; T\; =\; T\backslash ,$ for the braiding maps associated to every permutation σ on the symbols (or equivalently for every transposition on these symbols). Given a basis of ''V'', any symmetric tensor ''T'' of rank ''k'' can be written as :$T\; =\; \backslash sum\_^N\; T\_\; e^\; \backslash otimes\; e^\backslash otimes\backslash cdots\; \backslash otimes\; e^$ for some unique list of coefficients $T\_$ (the ''components'' of the tensor in the basis) that are symmetric on the indices. That is to say :$T\_\backslash dots\; i\_\}\; =\; T\_$ for every permutation σ. The space of all symmetric tensors of order ''k'' defined on ''V'' is often denoted by ''S''''k''(''V'') or Sym''k''(''V''). It is itself a vector space, and if ''V'' has dimension ''N'' then the dimension of Sym''k''(''V'') is the binomial coefficient :$\backslash dim\backslash ,\; \backslash operatorname^k(V)\; =\; .$ We then construct Sym(''V'') as the direct sum of Sym''k''(''V'') for ''k'' = 0,1,2,… :$\backslash operatorname(V)=\; \backslash bigoplus\_^\backslash infty\; \backslash operatorname^k(V).$ 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「Symmetric tensor」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.