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[[File:Components stress tensor.svg|right|thumb|300px|The second-order [[Cauchy stress tensor]] <math>\mathbf{T}</math> describes the stress experienced by a material at a given point. For any unit vector <math>\mathbf{v}</math>, the product <math>\mathbf{T} \cdot \mathbf{v}</math> is a vector, denoted <math>\mathbf{T}(\mathbf{v})</math>, that quantifies the force per area along the plane perpendicular to <math>\mathbf{v}</math>. This image shows, for cube faces perpendicular to <math>\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3</math>, the corresponding stress vectors <math>\mathbf{T}(\mathbf{e}_1), \mathbf{T}(\mathbf{e}_2), \mathbf{T}(\mathbf{e}_3)</math> along those faces. Because the stress tensor takes one vector as input and gives one vector as output, it is a second-order tensor.]]
In [[mathematics]], a '''tensor''' is an [[mathematical object|algebraic object]] that describes a [[Multilinear map|multilinear]] relationship between sets of [[algebraic structure|algebraic objects]]
Tensors have become important in [[physics]] because they provide a concise mathematical framework for formulating and solving physics problems in areas such as [[mechanics]] ([[Stress (mechanics)|stress]], [[elasticity (physics)|elasticity]], [[quantum mechanics]], [[fluid mechanics]], [[moment of inertia]], ...), [[Classical electromagnetism|electrodynamics]] ([[electromagnetic tensor]], [[Maxwell stress tensor|Maxwell tensor]], [[permittivity]], [[magnetic susceptibility]], ...), and [[general relativity]] ([[stress–energy tensor]], [[Riemann curvature tensor|curvature tensor]], ...). In applications, it is common to study situations in which a different tensor can occur at each point of an object; for example the stress within an object may vary from one ___location to another. This leads to the concept of a [[tensor field]]. In some areas, tensor fields are so ubiquitous that they are often simply called "tensors".
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=== As multidimensional arrays ===
A tensor may be represented as a (potentially multidimensional) array. Just as a [[Vector space|vector]] in an {{mvar|n}}-[[dimension (vector space)|dimensional]] space is represented by a [[multidimensional array|one-dimensional]] array with {{mvar|n}} components with respect to a given [[Basis (linear algebra)#Ordered bases and coordinates|basis]], any tensor with respect to a basis is represented by a multidimensional array. For example, a [[linear operator]] is represented in a basis as a two-dimensional square {{math|''n'' × ''n''}} array. The numbers in the multidimensional array are known as the ''components'' of the tensor. They are denoted by indices giving their position in the array, as [[subscript and superscript|subscripts and superscripts]], following the symbolic name of the tensor. For example, the components of an order
The total number of indices ({{mvar|m}}) required to identify each component uniquely is equal to the ''dimension'' or the number of ''ways'' of an array, which is why a tensor is sometimes referred to as an {{mvar|m}}-dimensional array or an {{mvar|m}}-way array. The total number of indices is also called the ''order'', ''degree'' or ''rank'' of a tensor,<ref name=DeLathauwerEtAl2000 >{{cite journal| last1= De Lathauwer |first1= Lieven| last2= De Moor |first2= Bart| last3= Vandewalle |first3= Joos| date=2000|title=A Multilinear Singular Value Decomposition |journal= [[SIAM J. Matrix Anal. Appl.]]|volume=21|issue= 4|pages=1253–1278|doi= 10.1137/S0895479896305696|s2cid= 14344372|url= https://alterlab.org/teaching/BME6780/papers+patents/De_Lathauwer_2000.pdf}}</ref><ref name=Vasilescu2002Tensorfaces >{{cite book |first1=M.A.O. |last1=Vasilescu |first2=D. |last2=Terzopoulos |title=Computer Vision — ECCV 2002 |chapter=Multilinear Analysis of Image Ensembles: TensorFaces |series=Lecture Notes in Computer Science |volume=2350 |pages=447–460 |doi=10.1007/3-540-47969-4_30 |date=2002 |isbn=978-3-540-43745-1 |s2cid=12793247 |chapter-url=http://www.cs.toronto.edu/~maov/tensorfaces/Springer%20ECCV%202002_files/eccv02proceeding_23500447.pdf |access-date=2022-12-29 |archive-date=2022-12-29 |archive-url=https://web.archive.org/web/20221229090931/http://www.cs.toronto.edu/~maov/tensorfaces/Springer%20ECCV%202002_files/eccv02proceeding_23500447.pdf |url-status=dead }}</ref><ref name=KoldaBader2009 >{{cite journal| last1= Kolda |first1= Tamara| last2= Bader |first2= Brett| date=2009|title=Tensor Decompositions and Applications |journal= [[SIAM Review]]|volume=51|issue= 3|pages=455–500|doi= 10.1137/07070111X|bibcode= 2009SIAMR..51..455K|s2cid= 16074195|url= https://www.kolda.net/publication/TensorReview.pdf}}</ref> although the term "rank" generally has [[tensor rank|another meaning]] in the context of matrices and tensors.
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{{Main|Multilinear map}}
A downside to the definition of a tensor using the multidimensional array approach is that it is not apparent from the definition that the defined object is indeed basis independent, as is expected from an intrinsically geometric object. Although it is possible to show that transformation laws indeed ensure independence from the basis, sometimes a more intrinsic definition is preferred. One approach that is common in [[differential geometry]] is to define tensors relative to a fixed (finite-dimensional) vector space ''V'', which is usually taken to be a particular vector space of some geometrical significance like the [[tangent space]] to a manifold.<ref>{{citation|last=Lee|first=John|title=Introduction to smooth manifolds|url={{google books |plainurl=y |id=4sGuQgAACAAJ|page=173}}|page=173|year=2000|publisher=Springer|isbn=978-0-387-95495-0}}</ref> In this approach, a type {{nowrap|(''p'', ''q'')}} tensor ''T'' is defined as a [[multilinear map]],
:<math> T: \underbrace{V^* \times\dots\times V^*}_{p \text{ copies}} \times \underbrace{ V \times\dots\times V}_{q \text{ copies}} \rightarrow \
where ''V''<sup>∗</sup> is the corresponding [[dual space]] of covectors, which is linear in each of its arguments. The above assumes ''V'' is a vector space over the [[real number]]s, {{tmath|\R}}. More generally, ''V'' can be taken over any [[Field (mathematics)|field]] ''F'' (e.g. the [[complex number]]s), with ''F'' replacing {{tmath|\R}} as the codomain of the multilinear maps.
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== Operations ==
There are several operations on tensors that again produce a tensor. The linear nature of
=== Tensor product ===
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===Machine learning===
{{Main|Tensor (machine learning)}}
The properties of
== Generalizations ==
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=== Spinors ===
{{Main|Spinor}}
When changing from one [[orthonormal basis]] (called a ''frame'') to another by a rotation, the components of a tensor transform by that same rotation. This transformation does not depend on the path taken through the space of frames. However, the space of frames is not [[simply connected]] (see [[orientation entanglement]] and [[plate trick]]): there are continuous paths in the space of frames with the same beginning and ending configurations that are not deformable one into the other. It is possible to attach an additional discrete invariant to each frame that incorporates this path dependence, and which turns out (locally) to have values of ±1.<ref>{{cite book|title=The road to reality: a complete guide to the laws of our universe|url={{google books |plainurl=y |id=VWTNCwAAQBAJ|page=203}}|first=Roger|last=Penrose|author-link=Roger Penrose|publisher=Knopf|year=2005|pages=203–206}}</ref> A [[spinor]] is an object that transforms like a tensor under rotations in the frame, apart from a possible sign that is determined by the value of this discrete invariant.<ref>{{cite book|first=E. |last=Meinrenken|title=Clifford Algebras and Lie Theory|chapter=The spin representation|series=Ergebnisse der Mathematik undihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics|volume=58|pages=49–85|doi=10.1007/978-3-642-36216-3_3|publisher=Springer |year=2013|isbn=978-3-642-36215-6}}</ref><ref>{{citation|first=S. H.|last=Dong |title=Wave Equations in Higher Dimensions|chapter=2. Special Orthogonal Group SO(''N'')|publisher=Springer|year=2011|pages=13–38}}</ref>
Spinors are elements of the [[spin representation]] of the rotation group, while tensors are elements of its [[tensor representation]]s. Other [[classical group]]s have tensor representations, and so also tensors that are compatible with the group, but all non-compact classical groups have infinite-dimensional unitary representations as well.
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* {{wiktionary-inline|tensor}}
* [[Array data type]], for tensor storage and manipulation
* [[Bitensor]]
=== Foundational ===
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