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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]], ...)
[[Tullio Levi-Civita]] and [[Gregorio Ricci-Curbastro]] popularised tensors in 1900 – continuing the earlier work of [[Bernhard Riemann]], [[Elwin Bruno Christoffel]], and others – as part of the ''[[absolute differential calculus]]''. The concept enabled an alternative formulation of the intrinsic [[differential geometry]] of a [[manifold]] in the form of the [[Riemann curvature tensor]].<ref name="Kline">
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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
Just as the components of a vector change when we change the [[basis (linear algebra)|basis]] of the vector space, the components of a tensor also change under such a transformation. Each type of tensor comes equipped with a ''transformation law'' that details how the components of the tensor respond to a [[change of basis]]. The components of a vector can respond in two distinct ways to a [[change of basis]] (see ''[[Covariance and contravariance of vectors]]''), where the new [[basis vectors]] <math>\mathbf{\hat{e}}_i</math> are expressed in terms of the old basis vectors <math>\mathbf{e}_j</math> as,
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This discussion motivates the following formal definition:<ref name="Sharpe2000">{{cite book|first=R.W. |last=Sharpe|title=Differential Geometry: Cartan's Generalization of Klein's Erlangen Program|url={{google books |plainurl=y |id=Ytqs4xU5QKAC| page=194}}|date=2000|publisher=Springer |isbn=978-0-387-94732-7| page=194}}</ref><ref>{{citation|chapter-url={{google books |plainurl=y |id=WROiC9st58gC}}|first=Jan Arnoldus|last=Schouten|author-link=Jan Arnoldus Schouten|title=Tensor analysis for physicists|year=1954|publisher=Courier Corporation|isbn=978-0-486-65582-6|chapter=Chapter II|url=https://archive.org/details/isbn_9780486655826}}</ref>
{{
:<math>T^{i_1\dots i_p}_{j_{1}\dots j_{q}}[\mathbf{f}]</math>
to each basis {{math|'''f''' {{=}} ('''e'''<sub>1</sub>, ..., '''e'''<sub>''n''</sub>)}} of an ''n''-dimensional vector space such that, if we apply the change of basis
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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 tensor products ===
{{Main|Tensor (intrinsic definition)}}
For some mathematical applications, a more abstract approach is sometimes useful. This can be achieved by defining tensors in terms of elements of [[tensor product]]s of vector spaces, which in turn are defined through a [[universal property]] as explained [[
A '''type {{math|(''p'', ''q'')}} tensor''' is defined in this context as an element of the tensor product of vector spaces,<ref>{{cite book|last1=Dodson|first1=C.T.J.|title=Tensor geometry: The Geometric Viewpoint and Its Uses |edition=2nd |date=2013 |isbn=9783642105142 |volume=130|orig-year=1991|series=Graduate Texts in Mathematics |publisher=Springer|last2=Poston |first2=T. |page= 105}}</ref><ref>{{Springer|id=a/a011120|title=Affine tensor}}</ref>
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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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T^{i_1\dots i_p}_{j_1\dots j_q}\left(x^1, \ldots, x^n\right).
</math>
== History ==
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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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|year=1892
|issue=16
}}</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.
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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}}</ref>}}
Tensors and [[tensor field]]s were also found to be useful in other fields such as [[continuum mechanics]]. Some well-known examples of tensors in [[differential geometry]] are [[quadratic form]]s such as [[metric tensor]]s, and the [[Riemann curvature tensor]]. The [[exterior algebra]] of [[Hermann Grassmann]], from the middle of the nineteenth century, is itself a tensor theory, and highly geometric, but it was some time before it was seen, with the theory of [[differential form]]s, as naturally unified with tensor calculus. The work of [[Élie Cartan]] made differential forms one of the basic kinds of tensors used in mathematics, and [[Hassler Whitney]] popularized the [[tensor product]].
From about the 1920s onwards, it was realised that tensors play a basic role in [[algebraic topology]] (for example in the [[Künneth theorem]]).<ref name="Spanier2012">{{cite book|first=Edwin H. |last=Spanier|title=Algebraic Topology|url={{google books |plainurl=y |id=iKx3BQAAQBAJ&|page=227}}|date=2012|publisher=Springer |isbn=978-1-4684-9322-1|pages=227|quote=the Künneth formula expressing the homology of the tensor product...}}</ref> Correspondingly there are types of tensors at work in many branches of [[abstract algebra]], particularly in [[homological algebra]] and [[representation theory]]. Multilinear algebra can be developed in greater generality than for scalars coming from a [[field (mathematics)|field]]. For example, scalars can come from a [[ring (mathematics)|ring]]. But the theory is then less geometric and computations more technical and less algorithmic.<ref name="Hungerford2003">{{cite book|first=Thomas W. |last=Hungerford|author-link=Thomas W. Hungerford|title=Algebra|url={{google books |plainurl=y |id=t6N_tOQhafoC|page=168 }}|date=2003|publisher=Springer |isbn=978-0-387-90518-1|page=168 |quote=...the classification (up to isomorphism) of modules over an arbitrary ring is quite difficult...}}</ref> Tensors are generalized within [[category theory]] by means of the concept of [[monoidal category]], from the 1960s.<ref name="MacLane2013">{{cite book|first=Saunders |last=MacLane|author-link=Saunders Mac Lane|title=Categories for the Working Mathematician|url={{google books |plainurl=y |id=6KPSBwAAQBAJ|page=4}}|date=2013|publisher=Springer |isbn=978-1-4612-9839-7|quote=...for example the monoid M ... in the category of abelian groups, × is replaced by the usual tensor product...|page=4}}</ref>
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=== Component-free notation ===
A [[component-free treatment of tensors]] uses notation that emphasises that tensors do not rely on any basis, and is defined in terms of the [[Tensor product
== Operations ==
There are several operations on tensors that again produce a tensor. The linear nature of
=== Tensor product ===
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{{Main|Tensor contraction}}
[[Tensor contraction]] is an operation that reduces a type {{nowrap|(''n'', ''m'')}} tensor to a type {{nowrap|(''n'' − 1, ''m'' − 1)}} tensor, of which the [[Trace (linear algebra)|trace]] is a special case. It thereby reduces the total order of a tensor by two. The operation is achieved by summing components for which one specified contravariant index is the same as one specified covariant index to produce a new component. Components for which those two indices are different are discarded. For example, a {{nowrap|(1, 1)}}-tensor <math>T_i^j</math> can be contracted to a scalar through <math>T_i^i</math>
The contraction is often used in conjunction with the tensor product to contract an index from each tensor.
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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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| edition = 2/e
| year = 2003
| publisher = Westview (Perseus)
| isbn = 978-0-8133-4080-7
}}
* {{cite book
| last = Dimitrienko
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[[Category:Concepts in physics]]
[[Category:
[[Category:Tensors]]
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