Tensor (intrinsic definition)
Template:Short description {{#invoke:Hatnote|hatnote}} Template:More footnotes needed
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 multilinear concept. Their 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 intrinsicTemplate:Definition needed 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.
Definition via tensor products of vector spacesEdit
Given a finite set Template:Math of vector spaces over a common field Template:Mvar, one may form their tensor product Template:Math, an element of which is termed a tensor.
A tensor on the vector space Template:Mvar is then defined to be an element of (i.e., a vector in) a vector space of the form: <math display="block">V \otimes \cdots \otimes V \otimes V^* \otimes \cdots \otimes V^*</math> where Template:Mvar is the dual space of Template:Mvar.
If there are Template:Mvar copies of Template:Mvar and Template:Mvar copies of Template:Mvar in our product, the tensor is said to be of Template:Nowrap and contravariant of order Template:Mvar and covariant of order Template:Mvar and of total order Template:Math. The tensors of order zero are just the scalars (elements of the field Template:Mvar), those of contravariant order 1 are the vectors in Template:Mvar, and those of covariant order 1 are the one-forms in Template:Mvar (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 Template:Math is denoted <math display="block"> T^m_n(V) = \underbrace{ V\otimes \dots \otimes V}_{m} \otimes \underbrace{ V^*\otimes \dots \otimes V^*}_{n}.</math>
Example 1. The space of type Template:Math tensors, <math>T^1_1(V) = V \otimes V^*,</math> is isomorphic in a natural way to the space of linear transformations from Template:Mvar to Template:Mvar.
Example 2. A bilinear form on a real vector space Template:Mvar, <math>V\times V \to F,</math> corresponds in a natural way to a type Template:Math tensor in <math>T^0_2 (V) = V^* \otimes V^*.</math> An example of such a bilinear form may be defined,Template:Clarify termed the associated metric tensor, and is usually denoted Template:Mvar.
Tensor rankEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}
A simple tensor (also called a tensor of rank one, elementary tensor or decomposable tensorTemplate:Sfnp) is a tensor that can be written as a product of tensors of the form <math display="block">T=a\otimes b\otimes\cdots\otimes d</math> where Template:Math are nonzero and in Template:Mvar or Template:Mvar – that is, if the tensor is nonzero and completely factorizable. Every tensor can be expressed as a sum of simple tensors. The rank of a tensor Template:Mvar is the minimum number of simple tensors that sum to Template:Mvar.Template:Sfnp
The zero 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 is Template:Math when each product is of Template:Mvar vectors from a finite-dimensional vector space of dimension Template:Mvar.
The term rank of a tensor extends the notion of the rank 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 the range of the matrix. A matrix thus has rank one if it can be written as an outer product of two nonzero vectors: <math display="block">A = v w^{\mathrm{T}}.</math>
The rank of a matrix Template:Mvar is the smallest number of such outer products that can be summed to produce it: <math display="block">A = v_1w_1^\mathrm{T} + \cdots + v_k w_k^\mathrm{T}.</math>
In indices, a tensor of rank 1 is a tensor of the form <math display="block">T_{ij\dots}^{k\ell\dots}=a_i b_j \cdots c^k d^\ell\cdots.</math>
The rank of a tensor of order 2 agrees with the rank when the tensor is regarded as a matrix,Template:Sfnp and can be determined from Gaussian elimination for instance. The rank of an order 3 or higher tensor is however often Template:Em to determine, and low rank decompositions of tensors are sometimes of great practical interest.Template:Sfnp In fact, the problem of finding the rank of an order 3 tensor over any finite field is NP-Complete, and over the rationals, is NP-Hard.Template:Sfnp Computational tasks such as the efficient multiplication of matrices and the efficient evaluation of polynomials can be recast as the problem of simultaneously evaluating a set of bilinear forms <math display="block">z_k = \sum_{ij} T_{ijk}x_iy_j</math> for given inputs Template:Mvar and Template:Mvar. If a low-rank decomposition of the tensor Template:Mvar is known, then an efficient evaluation strategy is known.Template:Sfnp
Universal propertyEdit
The space <math>T^m_n(V)</math> can be characterized by a universal property in terms of multilinear 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 for free modules, and the "universal" approach carries over more easily to more general situations.
A scalar-valued function on a Cartesian product (or direct sum) of vector spaces <math display="block">f : V_1\times\cdots\times V_N \to F</math> is multilinear if it is linear in each argument. The space of all multilinear mappings from Template:Math to Template:Mvar is denoted Template:Math. When Template:Math, a multilinear mapping is just an ordinary linear mapping, and the space of all linear mappings from Template:Mvar to Template:Mvar is denoted Template:Math.
The universal characterization of the tensor product implies that, for each multilinear function <math display="block">f\in L^{m+n}(\underbrace{V^*,\ldots,V^*}_m,\underbrace{V,\ldots,V}_n;W)</math> (where Template:Mvar can represent the field of scalars, a vector space, or a tensor space) there exists a unique linear function <math display="block">T_f \in L(\underbrace{V^*\otimes\cdots\otimes V^*}_m \otimes \underbrace{V\otimes\cdots\otimes V}_n; W)</math> such that <math display="block">f(\alpha_1,\ldots,\alpha_m, v_1,\ldots,v_n) = T_f(\alpha_1\otimes\cdots\otimes\alpha_m \otimes v_1\otimes\cdots\otimes v_n)</math> for all Template:Mvar in Template:Mvar and Template:Mvar in Template:Mvar.
Using the universal property, it follows, when Template:Mvar is finite dimensional, that the space of Template:Math-tensors admits a natural isomorphism <math display="block">T^m_n(V) \cong L(\underbrace{V^* \otimes \cdots \otimes V^*}_m \otimes \underbrace{V \otimes \cdots \otimes V}_n; F) \cong L^{m+n}(\underbrace{V^*, \ldots,V^*}_m,\underbrace{V,\ldots,V}_n; F).</math>
Each Template:Mvar in the definition of the tensor corresponds to a Template:Mvar inside the argument of the linear maps, and vice versa. (Note that in the former case, there are Template:Mvar copies of Template:Mvar and Template:Mvar copies of Template:Mvar, and in the latter case vice versa). In particular, one has <math display="block">\begin{align} T^1_0(V) &\cong L(V^*;F) \cong V,\\ T^0_1(V) &\cong L(V;F) = V^*,\\ T^1_1(V) &\cong L(V;V). \end{align}</math>
Tensor fieldsEdit
{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}
Differential geometry, physics and engineering must often deal with tensor fields on smooth manifolds. The term tensor is sometimes used as a shorthand for tensor field. A tensor field expresses the concept of a tensor that varies from point to point on the manifold.