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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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then the multidimensional array obeys the transformation law
:<math>
</math> <math>
T^{i_1, \ldots, i_p}_{j_1, \ldots, j_q}[\mathbf{f}]
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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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Using the properties of the tensor product, it can be shown that these components satisfy the transformation law for a type {{math|(''p'', ''q'')}} tensor. Moreover, the universal property of the tensor product gives a [[bijection|one-to-one correspondence]] between tensors defined in this way and tensors defined as multilinear maps.
This 1 to 1 correspondence can be
:<math>U \otimes V \cong\left(U^{* *}\right) \otimes\left(V^{* *}\right) \cong\left(U^{*} \otimes V^{*}\right)^{*} \cong \operatorname{Hom}^{2}\left(U^{*} \times V^{*} ; \mathbb{F}\right)</math>
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|issue=7–9
|issn=0302-7597
}} From p. 498: "And if we agree to call the ''square root'' (taken with a suitable sign) of this scalar product of two conjugate polynomes, P and KP, the common TENSOR of each, ... "</ref> to describe something different from what is now meant by a tensor.<ref group=Note>Namely, the [[norm (mathematics)|norm operation]] in a vector space.</ref> Gibbs introduced [[
Tensor calculus was developed around 1890 by [[Gregorio Ricci-Curbastro]] under the title ''absolute differential calculus'', and originally presented
|first=G. |last=Ricci Curbastro
|title=Résumé de quelques travaux sur les systèmes variables de fonctions associés à une forme différentielle quadratique
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}}</ref> It was made accessible to many mathematicians by the publication of Ricci-Curbastro and [[Tullio Levi-Civita]]'s 1900 classic text ''Méthodes de calcul différentiel absolu et leurs applications'' (Methods of absolute differential calculus and their applications).{{sfn|Ricci|Levi-Civita|1900}} In Ricci's notation, he refers to "systems" with covariant and contravariant components, which are known as tensor fields in the modern sense.<ref name="auto"/>
In the 20th century, the subject came to be known as ''tensor analysis'', and achieved broader acceptance with the introduction of [[Albert
|first=Abraham |last=Pais
|title=Subtle Is the Lord: The Science and the Life of Albert Einstein
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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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=== Continuum mechanics ===
Important examples are provided by [[continuum mechanics]]. A tensor field describes the stresses inside a solid body or [[fluid]].<ref>{{cite book |last1=Schobeiri |first1=Meinhard T. |date=2021 |title=Fluid Mechanics for Engineers |publisher=Springer |pages=11–29 |chapter=Vector and Tensor Analysis, Applications to Fluid Mechanics}}</ref> The [[Stress (mechanics)|stress tensor]] and [[strain tensor]] are both second-order tensor fields, and are related in a general linear elastic material by a fourth-order [[elasticity tensor]] field. In detail, the tensor quantifying stress in a 3-dimensional solid object has components that can be represented as a 3 × 3 array. The three faces of a cube-shaped infinitesimal volume segment of the solid are each subject to some given force. The force's vector components are also three in number. Thus, 3 × 3, or 9 components are required to describe the stress at this cube-shaped infinitesimal segment. Within the bounds of this solid is a whole mass of varying stress quantities, each requiring 9 quantities to describe. Thus, a second-order tensor is needed.▼
▲Important examples are provided by [[continuum mechanics]].
If a particular [[Volume form|surface element]] inside the material is singled out, the material on one side of the surface will apply a force on the other side. In general, this force will not be orthogonal to the surface, but it will depend on the orientation of the surface in a linear manner. This is described by a tensor of [[type of a tensor|type {{nowrap|(2, 0)}}]], in [[linear elasticity]], or more precisely by a tensor field of type {{nowrap|(2, 0)}}, since the stresses may vary from point to point.▼
▲If a particular [[Volume form|surface element]] inside the material is singled out, the material on one side of the surface will apply a force on the other side. In general, this force will not be orthogonal to the surface, but it will depend on the orientation of the surface in a linear manner. This is described by a tensor of [[type of a tensor|type {{nowrap|(2, 0)}}]], in [[linear elasticity]], or more precisely by a tensor field of type {{nowrap|(2, 0)}}, since the stresses may vary from point to point.
=== Other examples from physics ===
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===Machine learning===
{{Main|Tensor (machine learning)}}
The properties of
== Generalizations ==
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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>
== See also ==
* {{wiktionary-inline|tensor}}
* [[Array data type]], for tensor storage and manipulation
* [[Bitensor]]
=== Foundational ===
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[[Category:Concepts in physics]]
[[Category:
[[Category:Tensors]]
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