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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the modern component-free approach to the theory of a tensor views a tensor as an
abstract object In metaphysics Metaphysics is the branch of philosophy that studies the first principles of being, identity and change, space and time, causality, necessity and possibility. It includes questions about the nature of consciousness and the rela ...
, expressing some definite type of multilinear concept. Their properties can be derived from their definitions, as
linear map In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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 matrix (mat ...
to
multilinear algebra In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
. In
differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ...
an intrinsic geometric statement may be described by a
tensor field In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
on a
manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

, 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, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
, of tensor fields describing a
physical property A physical property is any property Property is a system of rights that gives people legal control of valuable things, and also refers to the valuable things themselves. Depending on the nature of the property, an owner of property may have ...
. The component-free approach is also used extensively in
abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
and
homological algebra Homological algebra is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and thei ...
, where tensors arise naturally. :''Note: This article assumes an understanding of the
tensor product In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

of
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
s without chosen bases. An overview of the subject can be found in the main
tensor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

article.''

# Definition via tensor products of vector spaces

Given a finite set of
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
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 grassl ...
''F'', one may form their
tensor product In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, 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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
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 Order, ORDER or Orders may refer to: * Orderliness Orderliness is a quality that is characterized by a person’s interest in keeping their surroundings and themselves well organized, and is associated with other qualities such as cleanliness a ...
. 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 :$T^m_n\left(V\right) = \underbrace_ \otimes \underbrace_.$ Example 1. The space of type tensors, $T^1_1\left(V\right) = V \otimes V^*,$ is isomorphic in a natural way to the space of
linear transformations In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
from ''V'' to ''V''. Example 2. A
bilinear form In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
on a real vector space ''V'', $V\times V \to F,$ corresponds in a natural way to a type tensor in $T^0_2 \left(V\right) = V^* \otimes V^*.$ An example of such a bilinear form may be defined, termed the associated ''
metric tensor In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population ...
'', 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 . 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 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 d ...
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 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 t ...
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 or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols, or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the material in between a eukaryoti ...
, and can be determined from Gaussian elimination 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s can be recast as the problem of simultaneously evaluating a set of
bilinear form In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
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 whether to evaluate the Parameter (computer ...
is known .

# Universal property

The space $T^m_n\left(V\right)$ can be characterized by a
universal property In category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), ...
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 ar ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

s, and the "universal" approach carries over more easily to more general situations. A scalar-valued function on a
Cartesian product In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
(or
direct sum The direct sum is an operation from abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathema ...
) 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 :$f\in L^\left(\underbrace_m,\underbrace_n;W\right)$ (where $W$ can represent the field of scalars, a vector space, or a tensor space) there exists a unique linear function :$T_f \in L\left(\underbrace_m \otimes \underbrace_n; W\right)$ such that :$f\left(\alpha_1,\ldots,\alpha_m, v_1,\ldots,v_n\right) = T_f\left(\alpha_1\otimes\cdots\otimes\alpha_m \otimes v_1\otimes\cdots\otimes v_n\right)$ 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 Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled dir ...
:$T^m_n\left(V\right) \cong L\left(\underbrace_m \otimes \underbrace_n; F\right) \cong L^\left(\underbrace_m,\underbrace_n; F\right).$ 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 :$\begin T^1_0\left(V\right) &\cong L\left(V^*;F\right) \cong V\\ T^0_1\left(V\right) &\cong L\left(V;F\right) = V^* \\ T^1_1\left(V\right) &\cong L\left(V;V\right) \end$

# Tensor fields

Differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ...
,
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

and
engineering Engineering is the use of 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 range of more speciali ...

must often deal with
tensor field In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
s on
smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's s ...
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

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