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Tensor (intrinsic definition)

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 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 extensively in abstract algebra and homological algebra, where tensors arise naturally. Given a finite set { V1, ..., Vn } of vector spaces over a common field F, one may form their tensor product V1 ⊗ ... ⊗ Vn, 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 type (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 type (0, 2) tensor in An example of such a bilinear form may be defined, termed the associated metric tensor (or sometimes misleadingly the metric or inner product), and is usually denoted g.

[ "Symmetric tensor", "Tensor contraction", "Tensor field", "Geometry", "Algebra", "Matricization", "Tensor product network", "Tensor product of algebras", "Raising and lowering indices", "Penrose graphical notation" ]
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