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In mathematics, the modern component-free approach to the theory of a tensor views a tensor as an abstract object, expressing some definite type of multi-linear concept. Their well-known properties can be derived from their definitions, as linear maps or more generally; and the rules for manipulations of tensors arise as an extension of linear algebra to multilinear algebra. In differential geometry an intrinsic geometric statement may be described by a tensor field on a manifold, and then doesn't need to make reference to coordinates at all. The same is true in general relativity, of tensor fields describing a physical property. The component-free approach is also used heavily in abstract algebra and homological algebra, where tensors arise naturally. :''Note: This article assumes an understanding of the tensor product of vector spaces without chosen bases. An overview of the subject can be found in the main tensor article. ==Definition via tensor products of vector spaces== Given a finite set of vector spaces over a common field ''F'', one may form their tensor product ''V''1 ⊗ ... ⊗ ''V''n, an element of which is termed a tensor. A tensor on the vector space ''V'' is then defined to be an element of (i.e., a vector in) a vector space of the form: : where ''V'' * is the dual space of ''V''. If there are ''m'' copies of ''V'' and ''n'' copies of ''V'' * in our product, the tensor is said to be of type (''m'', ''n'') and contravariant of order ''m'' and covariant order ''n'' and total order ''m''+''n''. The tensors of order zero are just the scalars (elements of the field ''F''), those of contravariant order 1 are the vectors in ''V'', and those of covariant order 1 are the one-forms in ''V'' * (for this reason the last two spaces are often called the contravariant and covariant vectors). The space of all tensors of type (''m'',''n'') is denoted : The (1,1) tensors : are isomorphic in a natural way to the space of linear transformations from ''V'' to ''V''. A bilinear form on a real vector space ''V''; ''V'' × ''V'' → R corresponds in a natural way to a (0,2) tensor in : termed the associated ''metric tensor'' (or sometimes misleadingly the ''metric'' or ''inner product'') and usually denoted ''g''. 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「Tensor (intrinsic definition)」の詳細全文を読む スポンサード リンク
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