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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the modern component-free approach to the theory of a tensor views a tensor as an
abstract object In metaphysics, the distinction between abstract and concrete refers to a divide between two types of entities. Many philosophers hold that this difference has fundamental metaphysical significance. Examples of concrete objects include plants, hum ...
, expressing some definite type of multilinear concept. Their properties can be derived from their definitions, as
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
s or more generally; and the rules for manipulations of tensors arise as an extension of
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...
to
multilinear algebra Multilinear algebra is a subfield of mathematics that extends the methods of linear algebra. Just as linear algebra is built on the concept of a vector and develops the theory of vector spaces, multilinear algebra builds on the concepts of ''p' ...
. In
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
an intrinsic geometric statement may be described by a
tensor field In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...
on a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
, and then doesn't need to make reference to coordinates at all. The same is true in
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, of tensor fields describing a
physical property A physical property is any property that is measurable, whose value describes a state of a physical system. The changes in the physical properties of a system can be used to describe its changes between momentary states. Physical properties are ...
. The component-free approach is also used extensively in
abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
and
homological algebra Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...
, where tensors arise naturally. :''Note: This article assumes an understanding of the
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...
of
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
s without chosen bases. An overview of the subject can be found in the main
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
article.''


Definition via tensor products of vector spaces

Given a finite set of
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
s over a common
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
''F'', one may form their
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...
, 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: :V \otimes \cdots \otimes V \otimes V^* \otimes \cdots \otimes V^* where ''V'' is the
dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
of ''V''. If there are ''m'' copies of ''V'' and ''n'' copies of ''V'' in our product, the tensor is said to be of and contravariant of order ''m'' and covariant order ''n'' and total order . 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 is denoted : T^m_n(V) = \underbrace_ \otimes \underbrace_. Example 1. The space of type tensors, T^1_1(V) = V \otimes V^*, is isomorphic in a natural way to the space of
linear transformations In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
from ''V'' to ''V''. Example 2. A
bilinear form In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called ''scalars''). In other words, a bilinear form is a function that is linear i ...
on a real vector space ''V'', V\times V \to F, corresponds in a natural way to a type tensor in T^0_2 (V) = V^* \otimes V^*. An example of such a bilinear form may be defined, termed the associated ''
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
'', and is usually denoted ''g''.


Tensor rank

A simple tensor (also called a tensor of rank one, elementary tensor or decomposable tensor ) is a tensor that can be written as a product of tensors of the form :T=a\otimes b\otimes\cdots\otimes d where ''a'', ''b'', ..., ''d'' are nonzero and in ''V'' or ''V'' – 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 ''T'' is the minimum number of simple tensors that sum to ''T'' . The
zero tensor In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context. Additive identities An additive identi ...
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 ''d'' when each product is of ''n'' vectors from a finite-dimensional vector space of dimension ''d''. The term ''rank of a tensor'' extends the notion of the
rank of a matrix In linear algebra, the rank of a matrix is the dimension of the vector space generated (or spanned) by its columns. p. 48, § 1.16 This corresponds to the maximal number of linearly independent columns of . This, in turn, is identical to the dime ...
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 In linear algebra, the outer product of two coordinate vector In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers (a tuple) that describes the vector in terms of a particular ordered basis. An ea ...
of two nonzero vectors: :A = v w^. The rank of a matrix ''A'' is the smallest number of such outer products that can be summed to produce it: :A = v_1w_1^\mathrm + \cdots + v_k w_k^\mathrm. In indices, a tensor of rank 1 is a tensor of the form :T_^=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 a
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
, and can be determined from
Gaussian elimination In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used ...
for instance. The rank of an order 3 or higher tensor is however often ''very hard'' to determine, and low rank decompositions of tensors are sometimes of great practical interest . Computational tasks such as the efficient multiplication of matrices and the efficient evaluation of
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...
s can be recast as the problem of simultaneously evaluating a set of
bilinear form In mathematics, a bilinear form is a bilinear map on a vector space (the elements of which are called '' vectors'') over a field ''K'' (the elements of which are called ''scalars''). In other words, a bilinear form is a function that is linear i ...
s :z_k = \sum_ T_x_iy_j for given inputs ''xi'' and ''yj''. If a low-rank decomposition of the tensor ''T'' is known, then an efficient
evaluation strategy In a programming language, an evaluation strategy is a set of rules for evaluating expressions. The term is often used to refer to the more specific notion of a ''parameter-passing strategy'' that defines the kind of value that is passed to the f ...
is known .


Universal property

The space T^m_n(V) can be characterized by a
universal property In mathematics, more specifically in category theory, a universal property is a property that characterizes up to an isomorphism the result of some constructions. Thus, universal properties can be used for defining some objects independently fro ...
in terms of
multilinear map In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function :f\colon V_1 \times \cdots \times V_n \to W\text where V_1,\ldots,V_n and W are ...
pings. 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 module In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in t ...
s, and the "universal" approach carries over more easily to more general situations. A scalar-valued function on a
Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is : A\ti ...
(or
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a more ...
) of vector spaces :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 from to ''W'' is denoted ''LN''(''V''1, ..., ''VN''; ''W''). When ''N'' = 1, a multilinear mapping is just an ordinary linear mapping, and the space of all linear mappings from ''V'' to ''W'' is denoted . The universal characterization of the tensor product implies that, for each multilinear function :f\in L^(\underbrace_m,\underbrace_n;W) (where W can represent the field of scalars, a vector space, or a tensor space) there exists a unique linear function :T_f \in L(\underbrace_m \otimes \underbrace_n; W) such that :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) for all v_i \in V and \alpha_i \in V^*. Using the universal property, it follows that the space of (''m'',''n'')-tensors admits a
natural isomorphism In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natural ...
:T^m_n(V) \cong L(\underbrace_m \otimes \underbrace_n; F) \cong L^(\underbrace_m,\underbrace_n; F). Each ''V'' in the definition of the tensor corresponds to a ''V''* inside the argument of the linear maps, and vice versa. (Note that in the former case, there are ''m'' copies of ''V'' and ''n'' copies of ''V''*, and in the latter case vice versa). In particular, one has :\begin 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


Tensor fields

Differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
,
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
and
engineering Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
must often deal with
tensor field In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...
s on
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One ma ...
s. 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.


References

*. *. *. *. * *. *. {{DEFAULTSORT:Tensor (Intrinsic Definition) Tensors