Inmathematics, the moderncomponent-free approach to the theory of atensor views a tensor as anabstract object, expressing some definite type ofmultilinear concept. Their properties can be derived from their definitions, aslinear maps or more generally; and the rules for manipulations of tensors arise as an extension oflinear algebra tomultilinear algebra.

Indifferential geometry, an intrinsic^{[definition needed]} geometric statement may be described by atensor field on amanifold, and then doesn't need to make reference to coordinates at all. The same is true ingeneral relativity, of tensor fields describing aphysical property. The component-free approach is also used extensively inabstract algebra andhomological algebra, where tensors arise naturally.

Given a finite set{V₁, ...,V_n} ofvector spaces over a commonfieldF, one may form theirtensor productV₁ ⊗ ... ⊗V_n, an element of which is termed atensor.

Atensor on the vector spaceV is then defined to be an element of (i.e., a vector in) a vector space of the form:$${\displaystyle V\otimes \cdots \otimes V\otimes V^{\ast }\otimes \cdots \otimes V^{\ast }}$$ whereV^∗ is thedual space ofV.

If there arem copies ofV andn copies ofV^∗ in our product, the tensor is said to be oftype (m,n) andcontravariant of orderm and covariant of ordern and of totalorderm +n. The tensors of order zero are just the scalars (elements of the fieldF), those of contravariant order 1 are the vectors inV, and those of covariant order 1 are theone-forms inV^∗ (for this reason, the elements of the last two spaces are often called the contravariant and covariant vectors). The space of all tensors of type(m,n) is denoted$${\displaystyle T_n^m(V)=\underset{m}{\underbrace{V\otimes \cdots \otimes V}}\otimes \underset{n}{\underbrace{V^{\ast }\otimes \cdots \otimes V^{\ast }}}.}$$ 

Example 1. The space of type(1, 1) tensors,${\displaystyle T_1^1(V)=V\otimes V^{\ast },}$  isisomorphic in a natural way to the space oflinear transformations fromV toV.

Example 2. Abilinear form on a real vector spaceV,${\displaystyle V\times V\to F,}$  corresponds in a natural way to a type(0, 2) tensor in${\displaystyle T_2^0(V)=V^{\ast }\otimes V^{\ast }.}$  An example of such a bilinear form may be defined,^{[clarification needed]} termed the associatedmetric tensor, and is usually denotedg.

Asimple tensor (also called a tensor of rank one, elementary tensor or decomposable tensor^[1]) is a tensor that can be written as a product of tensors of the form$${\displaystyle T=a\otimes b\otimes \cdots \otimes d}$$ wherea,b, ...,d are nonzero and inV orV^∗ – that is, if the tensor is nonzero and completelyfactorizable. Every tensor can be expressed as a sum of simple tensors. Therank of a tensorT is the minimum number of simple tensors that sum toT.^[2]

Thezero tensor has rank zero. A nonzero order 0 or 1 tensor always has rank 1. The rank of a non-zero order 2 or higher tensor is less than or equal to the product of the dimensions of all but the highest-dimensioned vectors in (a sum of products of) which the tensor can be expressed, which isdⁿ⁻¹ when each product is ofn vectors from a finite-dimensional vector space of dimensiond.

The termrank of a tensor extends the notion of therank of a matrix in linear algebra, although the term is also often used to mean the order (or degree) of a tensor. The rank of a matrix is the minimum number of column vectors needed to span therange of the matrix. A matrix thus has rank one if it can be written as anouter product of two nonzero vectors:$${\displaystyle A=v{w}^{\mathrm{T}}.}$$ 

The rank of a matrixA is the smallest number of such outer products that can be summed to produce it:$${\displaystyle A=v_1{w}_1^{\mathrm{T}}+\cdots +v_k{w}_k^{\mathrm{T}}.}$$ 

In indices, a tensor of rank 1 is a tensor of the form$${\displaystyle T_{ij\dots }^{k\ell \dots }=a_i{b}_j\cdots c^k{d}^{\ell }\cdots .}$$ 

The rank of a tensor of order 2 agrees with the rank when the tensor is regarded as amatrix,^[3] and can be determined fromGaussian elimination for instance. The rank of an order 3 or higher tensor is however oftenvery difficult to determine, and low rank decompositions of tensors are sometimes of great practical interest.^[4] In fact, the problem of finding the rank of an order 3 tensor over anyfinite field isNP-Complete, and over the rationals, isNP-Hard.^[5] Computational tasks such as the efficient multiplication of matrices and the efficient evaluation ofpolynomials can be recast as the problem of simultaneously evaluating a set ofbilinear forms$${\displaystyle z_k=\sum _{ij}{T}_{ijk}{x}_i{y}_j}$$ for given inputsx_i andy_j. If a low-rank decomposition of the tensorT is known, then an efficientevaluation strategy is known.^[6]

The space${\displaystyle T_n^m(V)}$  can be characterized by auniversal property in terms ofmultilinear mappings. Amongst the advantages of this approach are that it gives a way to show that many linear mappings are "natural" or "geometric" (in other words are independent of any choice of basis). Explicit computational information can then be written down using bases, and this order of priorities can be more convenient than proving a formula gives rise to a natural mapping. Another aspect is that tensor products are not used only forfree modules, and the "universal" approach carries over more easily to more general situations.

A scalar-valued function on aCartesian product (ordirect sum) of vector spaces$${\displaystyle f:V_1\times \cdots \times V_N\to F}$$ is multilinear if it is linear in each argument. The space of all multilinear mappings fromV₁ × ... ×V_N toW is denotedL^N(V₁, ...,V_N; W). WhenN = 1, a multilinear mapping is just an ordinary linear mapping, and the space of all linear mappings fromV toW is denotedL(V;W).

Theuniversal characterization of the tensor product implies that, for each multilinear function$${\displaystyle f\in L^{m+n}(\underset{m}{\underbrace{V^{\ast },\dots ,V^{\ast }}},\underset{n}{\underbrace{V,\dots ,V}};W)}$$ (whereW can represent the field of scalars, a vector space, or a tensor space) there exists a unique linear function$${\displaystyle T_f\in L(\underset{m}{\underbrace{V^{\ast }\otimes \cdots \otimes V^{\ast }}}\otimes \underset{n}{\underbrace{V\otimes \cdots \otimes V}};W)}$$ such that$${\displaystyle f({\alpha }_1,\dots ,{\alpha }_m,v_1,\dots ,v_n)=T_f({\alpha }_1\otimes \cdots \otimes {\alpha }_m\otimes v_1\otimes \cdots \otimes v_n)}$$ for allv_i inV andα_i inV^∗.

Using the universal property, it follows, whenV isfinite dimensional, that the space of(m,n)-tensors admits anatural isomorphism$${\displaystyle T_n^m(V)\cong L(\underset{m}{\underbrace{V^{\ast }\otimes \cdots \otimes V^{\ast }}}\otimes \underset{n}{\underbrace{V\otimes \cdots \otimes V}};F)\cong L^{m+n}(\underset{m}{\underbrace{V^{\ast },\dots ,V^{\ast }}},\underset{n}{\underbrace{V,\dots ,V}};F).}$$ 

EachV in the definition of the tensor corresponds to aV^∗ inside the argument of the linear maps, and vice versa. (Note that in the former case, there arem copies ofV andn copies ofV^∗, and in the latter case vice versa). In particular, one has 

Differential geometry,physics andengineering must often deal withtensor fields onsmooth manifolds. The termtensor is sometimes used as a shorthand fortensor field. A tensor field expresses the concept of a tensor that varies from point to point on the manifold.

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