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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 Abstract index notation ： ウィキペディア英語版 Abstract index notation Abstract index notation is a mathematical notation for tensors and spinors that uses indices to indicate their types, rather than their components in a particular basis. The indices are mere placeholders, not related to any fixed basis and, in particular, are non-numerical. Thus it should not be confused with the Ricci calculus. The notation was introduced by Roger Penrose as a way to use the formal aspects of the Einstein summation convention in order to compensate for the difficulty in describing contractions and covariant differentiation in modern abstract tensor notation, while preserving the explicit covariance of the expressions involved. Let ''V'' be a vector space, and ''V'' * its dual. Consider, for example, a rank-2 covariant tensor $\backslash scriptstyle\; h\backslash in\; V^$ *\otimes V^ *. Then ''h'' can be identified with a bilinear form on ''V''. In other words, it is a function of two arguments in ''V'' which can be represented as a pair of ''slots'': :$h\; =\; h(-,-).\backslash ,$ Abstract index notation is merely a ''labelling'' of the slots by Latin letters, which have no significance apart from their designation as labels of the slots (i.e., they are non-numerical): : $h\; =\; h\_.\backslash ,$ A contraction between two tensors is represented by the repetition of an index label, where one label is contravariant (an ''upper index'' corresponding to a tensor in ''V'') and one label is covariant (a ''lower index'' corresponding to a tensor in ''V'' *). Thus, for instance, : $\{t\_\{ab\}\}^b$ is the trace of a tensor ''t'' = ''t''abc over its last two slots. This manner of representing tensor contractions by repeated indices is formally similar to the Einstein summation convention. However, as the indices are non-numerical, it does not imply summation: rather it corresponds to the abstract basis-independent trace operation (or duality pairing) between tensor factors of type ''V'' and those of type ''V'' *. == Abstract indices and tensor spaces == A general homogeneous tensor is an element of a tensor product of copies of ''V'' and ''V''∗, such as :$V\backslash otimes\; V^$ *\otimes V^ * \otimes V\otimes V^ *. Label each factor in this tensor product with a Latin letter in a raised position for each contravariant ''V'' factor, and in a lowered position for each covariant ''V''∗ position. In this way, write the product as :$V^a\; V\_b\; V\_c\; V^d\; V\_e\backslash ,$ or, simply :$^d\}\_e.$ The last two expressions denote the same object as the first. Tensors of this type are denoted using similar notation, for example: :$^d\}\_e\; \backslash in\; ^d\}\_e\; =\; V\backslash otimes\; V^$ *\otimes V^ * \otimes V\otimes 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「Abstract index notation」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.