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Tensor

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</style><div role="note" class="hatnote navigation-not-searchable">For other uses, see <a href="Tensor_(disambiguation)" class="mw-disambig" title="Tensor (disambiguation)">Tensor (disambiguation)</a>.</div>

<div role="note" class="hatnote navigation-not-searchable">This article is about tensors on a single vector space and is not to be confused with <a href="Vector_field" title="Vector field">Vector field</a> or <a href="Tensor_field" title="Tensor field">Tensor field</a>.</div>

In <a href="Mathematics" title="Mathematics">mathematics</a>, a tensor is an <a href="Mathematical_object" title="Mathematical object">algebraic object</a> that describes a <a href="Multilinear_map" title="Multilinear map">multilinear</a> relationship between sets of <a href="Algebraic_structure" title="Algebraic structure">algebraic objects</a> associated with a <a href="Vector_space" title="Vector space">vector space</a>. Tensors may map between different objects such as <a href="Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">vectors</a>, <a href="Scalar_(mathematics)" title="Scalar (mathematics)">scalars</a>, and even other tensors. There are many types of tensors, including <a href="Scalar_(mathematics)" title="Scalar (mathematics)">scalars</a> and <a href="Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">vectors</a> (which are the simplest tensors), <a href="Dual_vector" class="mw-redirect" title="Dual vector">dual vectors</a>, <a href="Multilinear_map" title="Multilinear map">multilinear maps</a> between vector spaces, and even some operations such as the <a href="Dot_product" title="Dot product">dot product</a>. Tensors are defined <a href="Tensor_(intrinsic_definition)" title="Tensor (intrinsic definition)">independent</a> of any <a href="Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a>, although they are often referred to by their components in a basis related to a particular coordinate system; those components form an array, which can be thought of as a high-dimensional <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a>.

Tensors have become important in <a href="Physics" title="Physics">physics</a> because they provide a concise mathematical framework for formulating and solving physics problems in areas such as <a href="Mechanics" title="Mechanics">mechanics</a> (<a href="Stress_(mechanics)" title="Stress (mechanics)">stress</a>, <a href="Elasticity_(physics)" title="Elasticity (physics)">elasticity</a>, <a href="Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, <a href="Fluid_mechanics" title="Fluid mechanics">fluid mechanics</a>, <a href="Moment_of_inertia" title="Moment of inertia">moment of inertia</a>, ...), <a href="Classical_electromagnetism" title="Classical electromagnetism">electrodynamics</a> (<a href="Electromagnetic_tensor" title="Electromagnetic tensor">electromagnetic tensor</a>, <a href="Maxwell_stress_tensor" title="Maxwell stress tensor">Maxwell tensor</a>, <a href="Permittivity" title="Permittivity">permittivity</a>, <a href="Magnetic_susceptibility" title="Magnetic susceptibility">magnetic susceptibility</a>, ...), and <a href="General_relativity" title="General relativity">general relativity</a> (<a href="Stress%E2%80%93energy_tensor" title="Stress–energy tensor">stress–energy tensor</a>, <a href="Riemann_curvature_tensor" title="Riemann curvature tensor">curvature tensor</a>, ...). 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 <a href="Tensor_field" title="Tensor field">tensor field</a>. In some areas, tensor fields are so ubiquitous that they are often simply called "tensors".

<a href="Tullio_Levi-Civita" title="Tullio Levi-Civita">Tullio Levi-Civita</a> and <a href="Gregorio_Ricci-Curbastro" title="Gregorio Ricci-Curbastro">Gregorio Ricci-Curbastro</a> popularised tensors in 1900 – continuing the earlier work of <a href="Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a>, <a href="Elwin_Bruno_Christoffel" title="Elwin Bruno Christoffel">Elwin Bruno Christoffel</a>, and others – as part of the <a href="Absolute_differential_calculus" class="mw-redirect" title="Absolute differential calculus">absolute differential calculus</a>. The concept enabled an alternative formulation of the intrinsic <a href="Differential_geometry" title="Differential geometry">differential geometry</a> of a <a href="Manifold" title="Manifold">manifold</a> in the form of the <a href="Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a>.<a href="#cite_note-Kline-1">[1]</a>

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

Although seemingly different, the various approaches to defining tensors describe the same geometric concept using different language and at different levels of abstraction.

<div class="mw-heading mw-heading3"><h3 id="As_multidimensional_arrays">As multidimensional arrays</h3></div>

A tensor may be represented as a (potentially multidimensional) array. Just as a <a href="Vector_space" title="Vector space">vector</a> in an n-<a href="Dimension_(vector_space)" title="Dimension (vector space)">dimensional</a> space is represented by a <a href="Multidimensional_array" class="mw-redirect" title="Multidimensional array">one-dimensional</a> array with n components with respect to a given <a href="Basis_(linear_algebra)#Ordered_bases_and_coordinates" title="Basis (linear algebra)">basis</a>, any tensor with respect to a basis is represented by a multidimensional array. For example, a <a href="Linear_operator" class="mw-redirect" title="Linear operator">linear operator</a> is represented in a basis as a two-dimensional square 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 <a href="Subscript_and_superscript" title="Subscript and superscript">subscripts and superscripts</a>, following the symbolic name of the tensor. For example, the components of an order-2 tensor T could be denoted Tij , where i and j are indices running from 1 to n, or also by T i j. Whether an index is displayed as a superscript or subscript depends on the transformation properties of the tensor, described below. Thus while Tij and T i j can both be expressed as n-by-n matrices, and are numerically related via <a href="Raising_and_lowering_indices" class="mw-redirect" title="Raising and lowering indices">index juggling</a>, the difference in their transformation laws indicates it would be improper to add them together.

The total number of indices (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 m-dimensional array or an m-way array. The total number of indices is also called the order, degree or rank of a tensor,<a href="#cite_note-DeLathauwerEtAl2000-2">[2]</a><a href="#cite_note-Vasilescu2002Tensorfaces-3">[3]</a><a href="#cite_note-KoldaBader2009-4">[4]</a> although the term "rank" generally has <a href="Tensor_rank" class="mw-redirect" title="Tensor rank">another meaning</a> in the context of matrices and tensors.

Just as the components of a vector change when we change the <a href="Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> 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 <a href="Change_of_basis" title="Change of basis">change of basis</a>. The components of a vector can respond in two distinct ways to a <a href="Change_of_basis" title="Change of basis">change of basis</a> (see <a href="Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">Covariance and contravariance of vectors</a>), where the new <a href="Basis_vectors" class="mw-redirect" title="Basis vectors">basis vectors</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {\hat {e}} _{i}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {\hat {e}} _{i}}</annotation>

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</math><img src="./f01cfee86a741d3d25c6510aea329ab584019060.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.136ex; height:2.676ex;" alt="{\displaystyle \mathbf {\hat {e}} _{i}}" loading="lazy"> are expressed in terms of the old basis vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {e} _{j}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {e} _{j}}</annotation>

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</math><img src="./4b7742f3851d608848056eab437b32e8b753dd5b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.135ex; height:2.343ex;" alt="{\displaystyle \mathbf {e} _{j}}" loading="lazy"> as,

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {\hat {e}} _{i}=\sum _{j=1}^{n}\mathbf {e} _{j}R_{i}^{j}=\mathbf {e} _{j}R_{i}^{j}.}</annotation>

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</math><img src="./b07436c888a2f0484c09b43a310b9ac84808dde6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:22.339ex; height:7.176ex;" alt="{\displaystyle \mathbf {\hat {e}} _{i}=\sum _{j=1}^{n}\mathbf {e} _{j}R_{i}^{j}=\mathbf {e} _{j}R_{i}^{j}.}" loading="lazy"></dd></dl>

Here R ji are the entries of the change of basis matrix, and in the rightmost expression the <a href="Summation" title="Summation">summation</a> sign was suppressed: this is the <a href="Einstein_summation_convention" class="mw-redirect" title="Einstein summation convention">Einstein summation convention</a>, which will be used throughout this article.<a href="#cite_note-5">[Note 1]</a> The components vi of a column vector v transform with the <a href="Matrix_inverse" class="mw-redirect" title="Matrix inverse">inverse</a> of the matrix R,

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<annotation encoding="application/x-tex">{\displaystyle {\hat {v}}^{i}=\left(R^{-1}\right)_{j}^{i}v^{j},}</annotation>

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</math><img src="./035c66432d301f1e8de949abc84ef69f58220668.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:14.946ex; height:4.343ex;" alt="{\displaystyle {\hat {v}}^{i}=\left(R^{-1}\right)_{j}^{i}v^{j},}" loading="lazy"></dd></dl>

where the hat denotes the components in the new basis. This is called a contravariant transformation law, because the vector components transform by the inverse of the change of basis. In contrast, the components, wi, of a covector (or row vector), w, transform with the matrix R itself,

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<annotation encoding="application/x-tex">{\displaystyle {\hat {w}}_{i}=w_{j}R_{i}^{j}.}</annotation>

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</math><img src="./046255faae00b51cbfb9b5dd9a724ba722a78098.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.457ex; height:3.509ex;" alt="{\displaystyle {\hat {w}}_{i}=w_{j}R_{i}^{j}.}" loading="lazy"></dd></dl>

This is called a covariant transformation law, because the covector components transform by the same matrix as the change of basis matrix. The components of a more general tensor are transformed by some combination of covariant and contravariant transformations, with one transformation law for each index. If the transformation matrix of an index is the inverse matrix of the basis transformation, then the index is called contravariant and is conventionally denoted with an upper index (superscript). If the transformation matrix of an index is the basis transformation itself, then the index is called covariant and is denoted with a lower index (subscript).

As a simple example, the matrix of a linear operator with respect to a basis is a rectangular array <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}">

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<annotation encoding="application/x-tex">{\displaystyle T}</annotation>

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</math><img src="./ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"> that transforms under a change of basis matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=\left(R_{i}^{j}\right)}">

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<annotation encoding="application/x-tex">{\displaystyle R=\left(R_{i}^{j}\right)}</annotation>

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</math><img src="./4936428359e683f4e9d436c8983f84abca774483.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.312ex; height:4.843ex;" alt="{\displaystyle R=\left(R_{i}^{j}\right)}" loading="lazy"> by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {T}}=R^{-1}TR}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {T}}=R^{-1}TR}</annotation>

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</math><img src="./a0775690ec8d09e27d1be70c6d5ae5f456c2585e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.37ex; height:2.843ex;" alt="{\displaystyle {\hat {T}}=R^{-1}TR}" loading="lazy">. For the individual matrix entries, this transformation law has the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {T}}_{j'}^{i'}=\left(R^{-1}\right)_{i}^{i'}T_{j}^{i}R_{j'}^{j}}">

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<annotation encoding="application/x-tex">{\displaystyle {\hat {T}}_{j'}^{i'}=\left(R^{-1}\right)_{i}^{i'}T_{j}^{i}R_{j'}^{j}}</annotation>

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</math><img src="./cd5c9308f2f6238370694e88af4e85ff075587a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:19.597ex; height:4.676ex;" alt="{\displaystyle {\hat {T}}_{j'}^{i'}=\left(R^{-1}\right)_{i}^{i'}T_{j}^{i}R_{j'}^{j}}" loading="lazy"> so the tensor corresponding to the matrix of a linear operator has one covariant and one contravariant index: it is of type (1,1).

Combinations of covariant and contravariant components with the same index allow us to express geometric invariants. For example, the fact that a vector is the same object in different coordinate systems can be captured by the following equations, using the formulas defined above:

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {v} ={\hat {v}}^{i}\,\mathbf {\hat {e}} _{i}=\left(\left(R^{-1}\right)_{j}^{i}{v}^{j}\right)\left(\mathbf {e} _{k}R_{i}^{k}\right)=\left(\left(R^{-1}\right)_{j}^{i}R_{i}^{k}\right){v}^{j}\mathbf {e} _{k}=\delta _{j}^{k}{v}^{j}\mathbf {e} _{k}={v}^{k}\,\mathbf {e} _{k}={v}^{i}\,\mathbf {e} _{i}}</annotation>

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</math><img src="./895e51db53ac41319e0ec18f702940eea2a38024.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:77.438ex; height:4.843ex;" alt="{\displaystyle \mathbf {v} ={\hat {v}}^{i}\,\mathbf {\hat {e}} _{i}=\left(\left(R^{-1}\right)_{j}^{i}{v}^{j}\right)\left(\mathbf {e} _{k}R_{i}^{k}\right)=\left(\left(R^{-1}\right)_{j}^{i}R_{i}^{k}\right){v}^{j}\mathbf {e} _{k}=\delta _{j}^{k}{v}^{j}\mathbf {e} _{k}={v}^{k}\,\mathbf {e} _{k}={v}^{i}\,\mathbf {e} _{i}}" loading="lazy">,</dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{j}^{k}}">

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<annotation encoding="application/x-tex">{\displaystyle \delta _{j}^{k}}</annotation>

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</math><img src="./122c7afde9672b9c44fcc3d6ac639e9f9993a0a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.142ex; height:3.509ex;" alt="{\displaystyle \delta _{j}^{k}}" loading="lazy"> is the <a href="Kronecker_delta" title="Kronecker delta">Kronecker delta</a>, which functions similarly to the <a href="Identity_matrix" title="Identity matrix">identity matrix</a>, and has the effect of renaming indices (j into k in this example). This shows several features of the component notation: the ability to re-arrange terms at will (<a href="Commutativity" class="mw-redirect" title="Commutativity">commutativity</a>), the need to use different indices when working with multiple objects in the same expression, the ability to rename indices, and the manner in which contravariant and covariant tensors combine so that all instances of the transformation matrix and its inverse cancel, so that expressions like <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {v}^{i}\,\mathbf {e} _{i}}">

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<annotation encoding="application/x-tex">{\displaystyle {v}^{i}\,\mathbf {e} _{i}}</annotation>

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</math><img src="./c43a01d5b76448b0e7eaea4a6581fd2079293d91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.339ex; height:3.009ex;" alt="{\displaystyle {v}^{i}\,\mathbf {e} _{i}}" loading="lazy"> can immediately be seen to be geometrically identical in all coordinate systems.

Similarly, a linear operator, viewed as a geometric object, does not actually depend on a basis: it is just a linear map that accepts a vector as an argument and produces another vector. The transformation law for how the matrix of components of a linear operator changes with the basis is consistent with the transformation law for a contravariant vector, so that the action of a linear operator on a contravariant vector is represented in coordinates as the matrix product of their respective coordinate representations. That is, the components <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (Tv)^{i}}">

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<annotation encoding="application/x-tex">{\displaystyle (Tv)^{i}}</annotation>

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</math><img src="./c22233f6363098107713d9e8e330dbc841997bca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.373ex; height:3.176ex;" alt="{\displaystyle (Tv)^{i}}" loading="lazy"> are given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (Tv)^{i}=T_{j}^{i}v^{j}}">

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<annotation encoding="application/x-tex">{\displaystyle (Tv)^{i}=T_{j}^{i}v^{j}}</annotation>

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</math><img src="./805ca7d1fecb0975e8aa67ccdaeed0aaa3f83b17.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:13.028ex; height:3.676ex;" alt="{\displaystyle (Tv)^{i}=T_{j}^{i}v^{j}}" loading="lazy">. These components transform contravariantly, since

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<annotation encoding="application/x-tex">{\displaystyle \left({\widehat {Tv}}\right)^{i'}={\hat {T}}_{j'}^{i'}{\hat {v}}^{j'}=\left[\left(R^{-1}\right)_{i}^{i'}T_{j}^{i}R_{j'}^{j}\right]\left[\left(R^{-1}\right)_{k}^{j'}v^{k}\right]=\left(R^{-1}\right)_{i}^{i'}(Tv)^{i}.}</annotation>

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</math><img src="./a08d1c77a9be55af5d64e3a8cbaede2da77c528b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:63.571ex; height:5.509ex;" alt="{\displaystyle \left({\widehat {Tv}}\right)^{i'}={\hat {T}}_{j'}^{i'}{\hat {v}}^{j'}=\left[\left(R^{-1}\right)_{i}^{i'}T_{j}^{i}R_{j'}^{j}\right]\left[\left(R^{-1}\right)_{k}^{j'}v^{k}\right]=\left(R^{-1}\right)_{i}^{i'}(Tv)^{i}.}" loading="lazy"></dd></dl>

The transformation law for an order p + q tensor with p contravariant indices and q covariant indices is thus given as,

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<annotation encoding="application/x-tex">{\displaystyle {\hat {T}}_{j'_{1},\ldots ,j'_{q}}^{i'_{1},\ldots ,i'_{p}}=\left(R^{-1}\right)_{i_{1}}^{i'_{1}}\cdots \left(R^{-1}\right)_{i_{p}}^{i'_{p}}}</annotation>

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</math><img src="./c1edbef8c88c07b793f46996d3f01dabde02e00e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:30.127ex; height:4.843ex;" alt="{\displaystyle {\hat {T}}_{j'_{1},\ldots ,j'_{q}}^{i'_{1},\ldots ,i'_{p}}=\left(R^{-1}\right)_{i_{1}}^{i'_{1}}\cdots \left(R^{-1}\right)_{i_{p}}^{i'_{p}}}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{j_{1},\ldots ,j_{q}}^{i_{1},\ldots ,i_{p}}}">

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</math><img src="./296d94dee17e05642f438d47df38b20d25feece5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:7.595ex; height:4.343ex;" alt="{\displaystyle T_{j_{1},\ldots ,j_{q}}^{i_{1},\ldots ,i_{p}}}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{j'_{1}}^{j_{1}}\cdots R_{j'_{q}}^{j_{q}}.}">

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<annotation encoding="application/x-tex">{\displaystyle R_{j'_{1}}^{j_{1}}\cdots R_{j'_{q}}^{j_{q}}.}</annotation>

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</math><img src="./f7d01f8a8da490000f6e3bc1257e38a53ff26872.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.102ex; height:4.676ex;" alt="{\displaystyle R_{j'_{1}}^{j_{1}}\cdots R_{j'_{q}}^{j_{q}}.}" loading="lazy"></dd></dl>

Here the primed indices denote components in the new coordinates, and the unprimed indices denote the components in the old coordinates. Such a tensor is said to be of order or type (p, q). The terms "order", "type", "rank", "valence", and "degree" are all sometimes used for the same concept. Here, the term "order" or "total order" will be used for the total dimension of the array (or its generalization in other definitions), p + q in the preceding example, and the term "type" for the pair giving the number of contravariant and covariant indices. A tensor of type (p, q) is also called a (p, q)-tensor for short.

This discussion motivates the following formal definition:<a href="#cite_note-Sharpe2000-6">[5]</a><a href="#cite_note-7">[6]</a>

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</style><blockquote class="templatequote">Definition. A tensor of type (p, q) is an assignment of a multidimensional array

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<annotation encoding="application/x-tex">{\displaystyle T_{j_{1}\dots j_{q}}^{i_{1}\dots i_{p}}[\mathbf {f} ]}</annotation>

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</math><img src="./5c5971d3e9c4311001522ff021018a8210a3b184.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:9.027ex; height:4.343ex;" alt="{\displaystyle T_{j_{1}\dots j_{q}}^{i_{1}\dots i_{p}}[\mathbf {f} ]}" loading="lazy"></dd></dl>

to each basis f = (e1, ..., en) of an n-dimensional vector space such that, if we apply the change of basis

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {f} \mapsto \mathbf {f} \cdot R=\left(\mathbf {e} _{i}R_{1}^{i},\dots ,\mathbf {e} _{i}R_{n}^{i}\right)}</annotation>

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</math><img src="./42ff57d968833574952f87f8fd8981ab4b567f6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.421ex; height:3.176ex;" alt="{\displaystyle \mathbf {f} \mapsto \mathbf {f} \cdot R=\left(\mathbf {e} _{i}R_{1}^{i},\dots ,\mathbf {e} _{i}R_{n}^{i}\right)}" loading="lazy"></dd></dl>

then the multidimensional array obeys the transformation law

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<annotation encoding="application/x-tex">{\displaystyle T_{j'_{1}\dots j'_{q}}^{i'_{1}\dots i'_{p}}[\mathbf {f} \cdot R]=\left(R^{-1}\right)_{i_{1}}^{i'_{1}}\cdots \left(R^{-1}\right)_{i_{p}}^{i'_{p}}}</annotation>

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</math><img src="./e58c6ad944ce4ade82f82a8f5fe78375fcc949d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.786ex; height:4.843ex;" alt="{\displaystyle T_{j'_{1}\dots j'_{q}}^{i'_{1}\dots i'_{p}}[\mathbf {f} \cdot R]=\left(R^{-1}\right)_{i_{1}}^{i'_{1}}\cdots \left(R^{-1}\right)_{i_{p}}^{i'_{p}}}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{j_{1},\ldots ,j_{q}}^{i_{1},\ldots ,i_{p}}[\mathbf {f} ]}">

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<annotation encoding="application/x-tex">{\displaystyle T_{j_{1},\ldots ,j_{q}}^{i_{1},\ldots ,i_{p}}[\mathbf {f} ]}</annotation>

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</math><img src="./7076a7d3a987273f780cb2149bf3b75bf5b88a57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:9.942ex; height:4.343ex;" alt="{\displaystyle T_{j_{1},\ldots ,j_{q}}^{i_{1},\ldots ,i_{p}}[\mathbf {f} ]}" loading="lazy"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{j'_{1}}^{j_{1}}\cdots R_{j'_{q}}^{j_{q}}.}">

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<annotation encoding="application/x-tex">{\displaystyle R_{j'_{1}}^{j_{1}}\cdots R_{j'_{q}}^{j_{q}}.}</annotation>

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</math><img src="./f7d01f8a8da490000f6e3bc1257e38a53ff26872.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.102ex; height:4.676ex;" alt="{\displaystyle R_{j'_{1}}^{j_{1}}\cdots R_{j'_{q}}^{j_{q}}.}" loading="lazy"></dd></dl>

</blockquote>

The definition of a tensor as a multidimensional array satisfying a transformation law traces back to the work of Ricci.<a href="#cite_note-Kline-1">[1]</a>

An equivalent definition of a tensor uses the <a href="Representation_theory" title="Representation theory">representations</a> of the <a href="General_linear_group" title="General linear group">general linear group</a>. There is an <a href="Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">action</a> of the general linear group on the set of all <a href="Ordered_basis" class="mw-redirect" title="Ordered basis">ordered bases</a> of an n-dimensional vector space. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {f} =(\mathbf {f} _{1},\dots ,\mathbf {f} _{n})}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {f} R=\left(\mathbf {f} _{i}R_{1}^{i},\dots ,\mathbf {f} _{i}R_{n}^{i}\right).}</annotation>

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</math><img src="./a86a4c7f8f825de6f117b780d7f10913b7ad6fc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.291ex; height:3.176ex;" alt="{\displaystyle \mathbf {f} R=\left(\mathbf {f} _{i}R_{1}^{i},\dots ,\mathbf {f} _{i}R_{n}^{i}\right).}" loading="lazy"></dd></dl>

Let F be the set of all ordered bases. Then F is a <a href="Principal_homogeneous_space" title="Principal homogeneous space">principal homogeneous space</a> for GL(n). Let W be a vector space and let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

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</math><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> is an <a href="Equivariant_map" title="Equivariant map">equivariant map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T:F\to W}">

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</math><img src="./6a9a961dbdad0c4ae295f26fc0636cf2f1093d30.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.364ex; height:2.176ex;" alt="{\displaystyle T:F\to W}" loading="lazy">. Equivariance here means that

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</math><img src="./a5df2b02629f5b159f94de95e366d9ead1849f6b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.085ex; height:3.343ex;" alt="{\displaystyle T(FR)=\rho \left(R^{-1}\right)T(F).}" loading="lazy"></dd></dl>

When <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

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</math><img src="./1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"> is a <a href="Tensor_representation" title="Tensor representation">tensor representation</a> of the general linear group, this gives the usual definition of tensors as multidimensional arrays. This definition is often used to describe tensors on manifolds,<a href="#cite_note-8">[7]</a> and readily generalizes to other groups.<a href="#cite_note-Sharpe2000-6">[5]</a>

<div class="mw-heading mw-heading3"><h3 id="As_multilinear_maps">As multilinear maps</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Multilinear_map" title="Multilinear map">Multilinear map</a></div>

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 <a href="Differential_geometry" title="Differential geometry">differential geometry</a> 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 <a href="Tangent_space" title="Tangent space">tangent space</a> to a manifold.<a href="#cite_note-9">[8]</a> In this approach, a type (p, q) tensor T is defined as a <a href="Multilinear_map" title="Multilinear map">multilinear map</a>,

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<annotation encoding="application/x-tex">{\displaystyle T:\underbrace {V^{*}\times \dots \times V^{*}} _{p{\text{ copies}}}\times \underbrace {V\times \dots \times V} _{q{\text{ copies}}}\rightarrow \mathbb {R} ,}</annotation>

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</math><img src="./272d7fc02b7d0e37d35c0bad6e3ffce7d9c5c35b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:38.678ex; height:6.009ex;" alt="{\displaystyle T:\underbrace {V^{*}\times \dots \times V^{*}} _{p{\text{ copies}}}\times \underbrace {V\times \dots \times V} _{q{\text{ copies}}}\rightarrow \mathbb {R} ,}" loading="lazy"></dd></dl>

where V∗ is the corresponding <a href="Dual_space" title="Dual space">dual space</a> of covectors, which is linear in each of its arguments. The above assumes V is a vector space over the <a href="Real_number" title="Real number">real numbers</a>, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

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By applying a multilinear map T of type (p, q) to a basis {ej} for V and a canonical cobasis {εi} for V∗,

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<annotation encoding="application/x-tex">{\displaystyle T_{j_{1}\dots j_{q}}^{i_{1}\dots i_{p}}\equiv T\left({\boldsymbol {\varepsilon }}^{i_{1}},\ldots ,{\boldsymbol {\varepsilon }}^{i_{p}},\mathbf {e} _{j_{1}},\ldots ,\mathbf {e} _{j_{q}}\right),}</annotation>

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</math><img src="./ecd553c182b8e39aa394af3a293d68cf8b55fed6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:38.607ex; height:4.843ex;" alt="{\displaystyle T_{j_{1}\dots j_{q}}^{i_{1}\dots i_{p}}\equiv T\left({\boldsymbol {\varepsilon }}^{i_{1}},\ldots ,{\boldsymbol {\varepsilon }}^{i_{p}},\mathbf {e} _{j_{1}},\ldots ,\mathbf {e} _{j_{q}}\right),}" loading="lazy"></dd></dl>

a (p + q)-dimensional array of components can be obtained. A different choice of basis will yield different components. But, because T is linear in all of its arguments, the components satisfy the tensor transformation law used in the multilinear array definition. The multidimensional array of components of T thus form a tensor according to that definition. Moreover, such an array can be realized as the components of some multilinear map T. This motivates viewing multilinear maps as the intrinsic objects underlying tensors.

In viewing a tensor as a multilinear map, it is conventional to identify the <a href="Double_dual" class="mw-redirect" title="Double dual">double dual</a> V∗∗ of the vector space V, i.e., the space of linear functionals on the dual vector space V∗, with the vector space V. There is always a <a href="Dual_space#Injection_into_the_double-dual" title="Dual space">natural linear map</a> from V to its double dual, given by evaluating a linear form in V∗ against a vector in V. This linear mapping is an isomorphism in finite dimensions, and it is often then expedient to identify V with its double dual.

<div class="mw-heading mw-heading3"><h3 id="Using_tensor_products">Using tensor products</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Tensor_(intrinsic_definition)" title="Tensor (intrinsic definition)">Tensor (intrinsic definition)</a></div>

For some mathematical applications, a more abstract approach is sometimes useful. This can be achieved by defining tensors in terms of elements of <a href="Tensor_product" title="Tensor product">tensor products</a> of vector spaces, which in turn are defined through a <a href="Universal_property" title="Universal property">universal property</a> as explained <a href="Tensor_product#Universal_property" title="Tensor product">here</a> and <a href="Tensor_(intrinsic_definition)#Universal_property" title="Tensor (intrinsic definition)">here</a>.

A type (p, q) tensor is defined in this context as an element of the tensor product of vector spaces,<a href="#cite_note-10">[9]</a><a href="#cite_note-11">[10]</a>

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<annotation encoding="application/x-tex">{\displaystyle T\in \underbrace {V\otimes \dots \otimes V} _{p{\text{ copies}}}\otimes \underbrace {V^{*}\otimes \dots \otimes V^{*}} _{q{\text{ copies}}}.}</annotation>

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</math><img src="./5d7fb9b51c9081908e62c2cb323e6540e844d65a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:34.289ex; height:6.176ex;" alt="{\displaystyle T\in \underbrace {V\otimes \dots \otimes V} _{p{\text{ copies}}}\otimes \underbrace {V^{*}\otimes \dots \otimes V^{*}} _{q{\text{ copies}}}.}" loading="lazy"></dd></dl>

A basis vi of V and basis wj of W naturally induce a basis vi ⊗ wj of the tensor product V ⊗ W. The components of a tensor T are the coefficients of the tensor with respect to the basis obtained from a basis {ei} for V and its dual basis {εj}, i.e.

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<annotation encoding="application/x-tex">{\displaystyle T=T_{j_{1}\dots j_{q}}^{i_{1}\dots i_{p}}\;\mathbf {e} _{i_{1}}\otimes \cdots \otimes \mathbf {e} _{i_{p}}\otimes {\boldsymbol {\varepsilon }}^{j_{1}}\otimes \cdots \otimes {\boldsymbol {\varepsilon }}^{j_{q}}.}</annotation>

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</math><img src="./4706355e255ee32eaf71941754356d9ee726f961.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:43.961ex; height:4.343ex;" alt="{\displaystyle T=T_{j_{1}\dots j_{q}}^{i_{1}\dots i_{p}}\;\mathbf {e} _{i_{1}}\otimes \cdots \otimes \mathbf {e} _{i_{p}}\otimes {\boldsymbol {\varepsilon }}^{j_{1}}\otimes \cdots \otimes {\boldsymbol {\varepsilon }}^{j_{q}}.}" loading="lazy"></dd></dl>

Using the properties of the tensor product, it can be shown that these components satisfy the transformation law for a type (p, q) tensor. Moreover, the universal property of the tensor product gives a <a href="Bijection" title="Bijection">one-to-one correspondence</a> between tensors defined in this way and tensors defined as multilinear maps.

This 1 to 1 correspondence can be achieved in the following way, because in the finite-dimensional case there exists a canonical isomorphism between a vector space and its double dual:

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<annotation encoding="application/x-tex">{\displaystyle 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)}</annotation>

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</math><img src="./68af5214ad0aa7894bc63bbde46c1541162fd0d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:60.112ex; height:3.176ex;" alt="{\displaystyle 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)}" loading="lazy"></dd></dl>

The last line is using the universal property of the tensor product, that there is a 1 to 1 correspondence between maps from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Hom} ^{2}\left(U^{*}\times V^{*};\mathbb {F} \right)}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Hom} ^{2}\left(U^{*}\times V^{*};\mathbb {F} \right)}</annotation>

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</math><img src="./2d92c3c9458c935e63712f7724640380fec07a4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.866ex; height:3.176ex;" alt="{\displaystyle \operatorname {Hom} ^{2}\left(U^{*}\times V^{*};\mathbb {F} \right)}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Hom} \left(U^{*}\otimes V^{*};\mathbb {F} \right)}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Hom} \left(U^{*}\otimes V^{*};\mathbb {F} \right)}</annotation>

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</math><img src="./2649691c1eca956b4c545ee18f93e504acd06dbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.812ex; height:2.843ex;" alt="{\displaystyle \operatorname {Hom} \left(U^{*}\otimes V^{*};\mathbb {F} \right)}" loading="lazy">.<a href="#cite_note-12">[11]</a>

Tensor products can be defined in great generality – for example, <a href="Tensor_product_of_modules" title="Tensor product of modules">involving arbitrary modules</a> over a ring. In principle, one could define a "tensor" simply to be an element of any tensor product. However, the mathematics literature usually reserves the term tensor for an element of a tensor product of any number of copies of a single vector space V and its dual, as above.

<div class="mw-heading mw-heading3"><h3 id="Tensors_in_infinite_dimensions">Tensors in infinite dimensions</h3></div>

This discussion of tensors so far assumes finite dimensionality of the spaces involved, where the spaces of tensors obtained by each of these constructions are <a href="Naturally_isomorphic" class="mw-redirect" title="Naturally isomorphic">naturally isomorphic</a>.<a href="#cite_note-13">[Note 2]</a> Constructions of spaces of tensors based on the tensor product and multilinear mappings can be generalized, essentially without modification, to <a href="Vector_bundle" title="Vector bundle">vector bundles</a> or <a href="Coherent_sheaves" class="mw-redirect" title="Coherent sheaves">coherent sheaves</a>.<a href="#cite_note-14">[12]</a> For infinite-dimensional vector spaces, inequivalent topologies lead to inequivalent notions of tensor, and these various isomorphisms may or may not hold depending on what exactly is meant by a tensor (see <a href="Topological_tensor_product" title="Topological tensor product">topological tensor product</a>). In some applications, it is the <a href="Tensor_product_of_Hilbert_spaces" title="Tensor product of Hilbert spaces">tensor product of Hilbert spaces</a> that is intended, whose properties are the most similar to the finite-dimensional case. A more modern view is that it is the tensors' structure as a <a href="Symmetric_monoidal_category" title="Symmetric monoidal category">symmetric monoidal category</a> that encodes their most important properties, rather than the specific models of those categories.<a href="#cite_note-15">[13]</a>

<div class="mw-heading mw-heading3"><h3 id="Tensor_fields">Tensor fields</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Tensor_field" title="Tensor field">Tensor field</a></div>

In many applications, especially in differential geometry and physics, it is natural to consider a tensor with components that are functions of the point in a space. This was the setting of Ricci's original work. In modern mathematical terminology such an object is called a <a href="Tensor_field" title="Tensor field">tensor field</a>, often referred to simply as a tensor.<a href="#cite_note-Kline-1">[1]</a>

In this context, a <a href="Coordinate_basis" class="mw-redirect" title="Coordinate basis">coordinate basis</a> is often chosen for the <a href="Tangent_space" title="Tangent space">tangent vector space</a>. The transformation law may then be expressed in terms of <a href="Partial_derivative" title="Partial derivative">partial derivatives</a> of the coordinate functions,

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<annotation encoding="application/x-tex">{\displaystyle {\bar {x}}^{i}\left(x^{1},\ldots ,x^{n}\right),}</annotation>

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defining a coordinate transformation,<a href="#cite_note-Kline-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle {\hat {T}}_{j'_{1}\dots j'_{q}}^{i'_{1}\dots i'_{p}}\left({\bar {x}}^{1},\ldots ,{\bar {x}}^{n}\right)={\frac {\partial {\bar {x}}^{i'_{1}}}{\partial x^{i_{1}}}}\cdots {\frac {\partial {\bar {x}}^{i'_{p}}}{\partial x^{i_{p}}}}{\frac {\partial x^{j_{1}}}{\partial {\bar {x}}^{j'_{1}}}}\cdots {\frac {\partial x^{j_{q}}}{\partial {\bar {x}}^{j'_{q}}}}T_{j_{1}\dots j_{q}}^{i_{1}\dots i_{p}}\left(x^{1},\ldots ,x^{n}\right).}</annotation>

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</math><img src="./d20599fa48bec81f189d0cc05714f52f5756dcbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:70.589ex; height:6.676ex;" alt="{\displaystyle {\hat {T}}_{j'_{1}\dots j'_{q}}^{i'_{1}\dots i'_{p}}\left({\bar {x}}^{1},\ldots ,{\bar {x}}^{n}\right)={\frac {\partial {\bar {x}}^{i'_{1}}}{\partial x^{i_{1}}}}\cdots {\frac {\partial {\bar {x}}^{i'_{p}}}{\partial x^{i_{p}}}}{\frac {\partial x^{j_{1}}}{\partial {\bar {x}}^{j'_{1}}}}\cdots {\frac {\partial x^{j_{q}}}{\partial {\bar {x}}^{j'_{q}}}}T_{j_{1}\dots j_{q}}^{i_{1}\dots i_{p}}\left(x^{1},\ldots ,x^{n}\right).}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

The concepts of later tensor analysis arose from the work of <a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a> in <a href="Differential_geometry" title="Differential geometry">differential geometry</a>, and the formulation was much influenced by the theory of <a href="Algebraic_form" class="mw-redirect" title="Algebraic form">algebraic forms</a> and invariants developed during the middle of the nineteenth century.<a href="#cite_note-16">[14]</a> The word "tensor" itself was introduced in 1846 by <a href="William_Rowan_Hamilton" title="William Rowan Hamilton">William Rowan Hamilton</a><a href="#cite_note-17">[15]</a> to describe something different from what is now meant by a tensor.<a href="#cite_note-18">[Note 3]</a> Gibbs introduced <a href="Dyadics" title="Dyadics">dyadics</a> and <a href="Polyadic_algebra" title="Polyadic algebra">polyadic algebra</a>, which are also tensors in the modern sense.<a href="#cite_note-auto-19">[16]</a> The contemporary usage was introduced by <a href="Woldemar_Voigt" title="Woldemar Voigt">Woldemar Voigt</a> in 1898.<a href="#cite_note-Voigt1898-20">[17]</a>

Tensor calculus was developed around 1890 by <a href="Gregorio_Ricci-Curbastro" title="Gregorio Ricci-Curbastro">Gregorio Ricci-Curbastro</a> under the title absolute differential calculus, and originally presented in 1892.<a href="#cite_note-21">[18]</a> It was made accessible to many mathematicians by the publication of Ricci-Curbastro and <a href="Tullio_Levi-Civita" title="Tullio Levi-Civita">Tullio Levi-Civita</a>'s 1900 classic text Méthodes de calcul différentiel absolu et leurs applications (Methods of absolute differential calculus and their applications).<a href="#cite_note-FOOTNOTERicciLevi-Civita1900-22">[19]</a> In Ricci's notation, he refers to "systems" with covariant and contravariant components, which are known as tensor fields in the modern sense.<a href="#cite_note-auto-19">[16]</a>

In the 20th century, the subject came to be known as tensor analysis, and achieved broader acceptance with the introduction of <a href="Albert_Einstein" title="Albert Einstein">Albert Einstein</a>'s theory of <a href="General_relativity" title="General relativity">general relativity</a>, around 1915. General relativity is formulated completely in the language of tensors. Einstein had learned about them, with great difficulty, from the geometer <a href="Marcel_Grossmann" title="Marcel Grossmann">Marcel Grossmann</a>.<a href="#cite_note-23">[20]</a> Levi-Civita then initiated a correspondence with Einstein to correct mistakes Einstein had made in his use of tensor analysis. The correspondence lasted 1915–17, and was characterized by mutual respect:

<blockquote class="templatequote">I admire the elegance of your method of computation; it must be nice to ride through these fields upon the horse of true mathematics while the like of us have to make our way laboriously on foot.</blockquote><div class="templatequotecite">— Albert Einstein<a href="#cite_note-Goodstein-24">[21]</a></div>

Tensors and <a href="Tensor_field" title="Tensor field">tensor fields</a> were also found to be useful in other fields such as <a href="Continuum_mechanics" title="Continuum mechanics">continuum mechanics</a>. Some well-known examples of tensors in <a href="Differential_geometry" title="Differential geometry">differential geometry</a> are <a href="Quadratic_form" title="Quadratic form">quadratic forms</a> such as <a href="Metric_tensor" title="Metric tensor">metric tensors</a>, and the <a href="Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a>. The <a href="Exterior_algebra" title="Exterior algebra">exterior algebra</a> of <a href="Hermann_Grassmann" title="Hermann Grassmann">Hermann Grassmann</a>, 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 <a href="Differential_form" title="Differential form">differential forms</a>, as naturally unified with tensor calculus. The work of <a href="%C3%89lie_Cartan" title="Élie Cartan">Élie Cartan</a> made differential forms one of the basic kinds of tensors used in mathematics, and <a href="Hassler_Whitney" title="Hassler Whitney">Hassler Whitney</a> popularized the <a href="Tensor_product" title="Tensor product">tensor product</a>.<a href="#cite_note-auto-19">[16]</a>

From about the 1920s onwards, it was realised that tensors play a basic role in <a href="Algebraic_topology" title="Algebraic topology">algebraic topology</a> (for example in the <a href="K%C3%BCnneth_theorem" title="Künneth theorem">Künneth theorem</a>).<a href="#cite_note-Spanier2012-25">[22]</a> Correspondingly there are types of tensors at work in many branches of <a href="Abstract_algebra" title="Abstract algebra">abstract algebra</a>, particularly in <a href="Homological_algebra" title="Homological algebra">homological algebra</a> and <a href="Representation_theory" title="Representation theory">representation theory</a>. Multilinear algebra can be developed in greater generality than for scalars coming from a <a href="Field_(mathematics)" title="Field (mathematics)">field</a>. For example, scalars can come from a <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a>. But the theory is then less geometric and computations more technical and less algorithmic.<a href="#cite_note-Hungerford2003-26">[23]</a> Tensors are generalized within <a href="Category_theory" title="Category theory">category theory</a> by means of the concept of <a href="Monoidal_category" title="Monoidal category">monoidal category</a>, from the 1960s.<a href="#cite_note-MacLane2013-27">[24]</a>

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<div role="note" class="hatnote navigation-not-searchable">See also: <a href="Dyadic_tensor" class="mw-redirect" title="Dyadic tensor">Dyadic tensor</a></div>

An elementary example of a mapping describable as a tensor is the <a href="Dot_product" title="Dot product">dot product</a>, which maps two vectors to a scalar. A more complex example is the <a href="Cauchy_stress_tensor" title="Cauchy stress tensor">Cauchy stress tensor</a> T, which takes a directional unit vector v as input and maps it to the stress vector T(v), which is the force (per unit area) exerted by material on the negative side of the plane orthogonal to v against the material on the positive side of the plane, thus expressing a relationship between these two vectors, shown in the figure (right). The <a href="Cross_product" title="Cross product">cross product</a>, where two vectors are mapped to a third one, is strictly speaking not a tensor because it changes its sign under those transformations that change the orientation of the coordinate system. The <a href="Levi-Civita_symbol" title="Levi-Civita symbol">totally anti-symmetric symbol</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{ijk}}">

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This table shows important examples of tensors on vector spaces and tensor fields on manifolds. The tensors are classified according to their type (n, m), where n is the number of contravariant indices, m is the number of covariant indices, and n + m gives the total order of the tensor. For example, a <a href="Bilinear_form" title="Bilinear form">bilinear form</a> is the same thing as a (0, 2)-tensor; an <a href="Inner_product" class="mw-redirect" title="Inner product">inner product</a> is an example of a (0, 2)-tensor, but not all (0, 2)-tensors are inner products. In the (0, M)-entry of the table, M denotes the dimensionality of the underlying vector space or manifold because for each dimension of the space, a separate index is needed to select that dimension to get a maximally covariant antisymmetric tensor.

<caption>Example tensors on vector spaces and tensor fields on manifolds

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<td><a href="Scalar_(mathematics)" title="Scalar (mathematics)">scalar</a>, e.g. <a href="Scalar_curvature" title="Scalar curvature">scalar curvature</a>

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<td><a href="Covector" class="mw-redirect" title="Covector">covector</a>, <a href="Linear_functional" class="mw-redirect" title="Linear functional">linear functional</a>, <a href="1-form" class="mw-redirect" title="1-form">1-form</a>, e.g. <a href="Multipole_expansion" title="Multipole expansion">dipole moment</a>, <a href="Gradient" title="Gradient">gradient</a> of a scalar field

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<td><a href="Bilinear_form" title="Bilinear form">bilinear form</a>, e.g. <a href="Inner_product" class="mw-redirect" title="Inner product">inner product</a>, <a href="Quadrupole_moment" class="mw-redirect" title="Quadrupole moment">quadrupole moment</a>, <a href="Metric_tensor" title="Metric tensor">metric tensor</a>, <a href="Ricci_curvature" title="Ricci curvature">Ricci curvature</a>, <a href="2-form" class="mw-redirect" title="2-form">2-form</a>, <a href="Symplectic_form" class="mw-redirect" title="Symplectic form">symplectic form</a>

</td>

<td>3-form e.g. <a href="Multipole_moment" class="mw-redirect" title="Multipole moment">octupole moment</a>

</td>

<td>

</td>

<td>e.g. M-form i.e. <a href="Volume_form" title="Volume form">volume form</a>

</td>

<td>

</td></tr>

<tr>

<th scope="row">1

</th>

<td><a href="Vector" class="mw-disambig" title="Vector">vector</a>

</td>

<td><a href="Linear_transformation" class="mw-redirect" title="Linear transformation">linear transformation</a>,<a href="#cite_note-BambergSternberg1991-28">[25]</a> <a href="Kronecker_delta" title="Kronecker delta">Kronecker delta</a>

</td>

<td>e.g. <a href="Cross_product" title="Cross product">cross product</a> in three dimensions

</td>

<td>e.g. <a href="Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a>

</td>

<td>

</td>

<td>

</td>

<td>

</td></tr>

<tr>

<th scope="row">2

</th>

<td><a href="Bivector" title="Bivector">bivector</a>, e.g. <a href="Poisson_structure" class="mw-redirect" title="Poisson structure">Poisson structure</a>, inverse <a href="Metric_tensor" title="Metric tensor">metric tensor</a>

</td>

<td>

</td>

<td>e.g. <a href="Elasticity_tensor" title="Elasticity tensor">elasticity tensor</a>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td></tr>

<tr>

<th scope="row">⋮

</th>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td></tr>

<tr>

<th scope="row">N

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<td><a href="Multivector" title="Multivector">multivector</a>

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<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td></tr>

<tr>

<th scope="row">⋮

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<td>

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>

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<td>

</td>

<td>

</td></tr></tbody></table>

Raising an index on an (n, m)-tensor produces an (n + 1, m − 1)-tensor; this corresponds to moving diagonally down and to the left on the table. Symmetrically, lowering an index corresponds to moving diagonally up and to the right on the table. <a href="#Contraction">Contraction</a> of an upper with a lower index of an (n, m)-tensor produces an (n − 1, m − 1)-tensor; this corresponds to moving diagonally up and to the left on the table.

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</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:448px;max-width:448px"><div class="trow"><div class="tsingle" style="width:222px;max-width:222px"><div class="thumbimage"></div><div class="thumbcaption">Orientation defined by an ordered set of vectors.</div></div><div class="tsingle" style="width:222px;max-width:222px"><div class="thumbimage"></div><div class="thumbcaption">Reversed orientation corresponds to negating the exterior product.</div></div></div><div class="trow" style="display:flex"><div class="thumbcaption">Geometric interpretation of grade n elements in a real <a href="Exterior_algebra" title="Exterior algebra">exterior algebra</a> for n = 0 (signed point), 1 (directed line segment, or vector), 2 (oriented plane element), 3 (oriented volume). The exterior product of n vectors can be visualized as any n-dimensional shape (e.g. n-<a href="Parallelepiped#Parallelotope" title="Parallelepiped">parallelotope</a>, n-<a href="Ellipsoid" title="Ellipsoid">ellipsoid</a>); with magnitude (<a href="Hypervolume" class="mw-redirect" title="Hypervolume">hypervolume</a>), and <a href="Orientation_(vector_space)" title="Orientation (vector space)">orientation</a> defined by that on its n − 1-dimensional boundary and on which side the interior is.<a href="#cite_note-29">[26]</a><a href="#cite_note-30">[27]</a></div></div></div></div>

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

Assuming a <a href="Basis_of_a_vector_space" class="mw-redirect" title="Basis of a vector space">basis</a> of a real vector space, e.g., a coordinate frame in the ambient space, a tensor can be represented as an organized <a href="Array_data_structure" class="mw-redirect" title="Array data structure">multidimensional array</a> of numerical values with respect to this specific basis. Changing the basis transforms the values in the array in a characteristic way that allows to define tensors as objects adhering to this transformational behavior. For example, there are invariants of tensors that must be preserved under any change of the basis, thereby making only certain multidimensional arrays of numbers a <a class="mw-selflink-fragment" href="#Holors">tensor.</a> Compare this to the array representing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{ijk}}">

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Because the components of vectors and their duals transform differently under the change of their dual bases, there is a <a href="Covariant_transformation" title="Covariant transformation">covariant and/or contravariant transformation law</a> that relates the arrays, which represent the tensor with respect to one basis and that with respect to the other one. The numbers of, respectively, vectors: n (<a href="Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">contravariant</a> indices) and dual vectors: m (<a href="Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">covariant</a> indices) in the input and output of a tensor determine the type (or valence) of the tensor, a pair of natural numbers (n, m), which determine the precise form of the transformation law. The <style data-mw-deduplicate="TemplateStyles:r1238216509">

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</style>order of a tensor is the sum of these two numbers.

The order (also degree or rank) of a tensor is thus the sum of the orders of its arguments plus the order of the resulting tensor. This is also the dimensionality of the array of numbers needed to represent the tensor with respect to a specific basis, or equivalently, the number of indices needed to label each component in that array. For example, in a fixed basis, a standard linear map that maps a vector to a vector, is represented by a matrix (a 2-dimensional array), and therefore is a 2nd-order tensor. A simple vector can be represented as a 1-dimensional array, and is therefore a 1st-order tensor. Scalars are simple numbers and are thus 0th-order tensors. This way the tensor representing the scalar product, taking two vectors and resulting in a scalar has order 2 + 0 = 2, the same as the stress tensor, taking one vector and returning another 1 + 1 = 2. The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{ijk}}">

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</math><img src="./21525193117bdfc0f3ac71b8ec46e3b6d0637daf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.417ex; height:2.343ex;" alt="{\displaystyle \varepsilon _{ijk}}" loading="lazy">-symbol, mapping two vectors to one vector, would have order 2 + 1 = 3.

The collection of tensors on a vector space and its dual forms a <a href="Tensor_algebra" title="Tensor algebra">tensor algebra</a>, which allows products of arbitrary tensors. Simple applications of tensors of order 2, which can be represented as a square matrix, can be solved by clever arrangement of transposed vectors and by applying the rules of matrix multiplication, but the tensor product should not be confused with this.

<div class="mw-heading mw-heading2"><h2 id="Notation">Notation</h2></div>

There are several notational systems that are used to describe tensors and perform calculations involving them.

<div class="mw-heading mw-heading3"><h3 id="Ricci_calculus">Ricci calculus</h3></div>

<a href="Ricci_calculus" title="Ricci calculus">Ricci calculus</a> is the modern formalism and notation for tensor indices: indicating <a href="Inner_product" class="mw-redirect" title="Inner product">inner</a> and <a href="Outer_product" title="Outer product">outer products</a>, <a href="Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">covariance and contravariance</a>, <a href="Summation" title="Summation">summations</a> of tensor components, <a href="Symmetric_tensor" title="Symmetric tensor">symmetry</a> and <a href="Antisymmetric_tensor" title="Antisymmetric tensor">antisymmetry</a>, and <a href="Partial_derivative" title="Partial derivative">partial</a> and <a href="Covariant_derivative" title="Covariant derivative">covariant derivatives</a>.

<div class="mw-heading mw-heading3"><h3 id="Einstein_summation_convention">Einstein summation convention</h3></div>

The <a href="Einstein_summation_convention" class="mw-redirect" title="Einstein summation convention">Einstein summation convention</a> dispenses with writing <a href="Summation_sign" class="mw-redirect" title="Summation sign">summation signs</a>, leaving the summation implicit. Any repeated index symbol is summed over: if the index i is used twice in a given term of a tensor expression, it means that the term is to be summed for all i. Several distinct pairs of indices may be summed this way.

<div class="mw-heading mw-heading3"><h3 id="Penrose_graphical_notation">Penrose graphical notation</h3></div>

<a href="Penrose_graphical_notation" title="Penrose graphical notation">Penrose graphical notation</a> is a diagrammatic notation which replaces the symbols for tensors with shapes, and their indices by lines and curves. It is independent of basis elements, and requires no symbols for the indices.

<div class="mw-heading mw-heading3"><h3 id="Abstract_index_notation">Abstract index notation</h3></div>

The <a href="Abstract_index_notation" title="Abstract index notation">abstract index notation</a> is a way to write tensors such that the indices are no longer thought of as numerical, but rather are <a href="Indeterminate_(variable)" class="mw-redirect" title="Indeterminate (variable)">indeterminates</a>. This notation captures the expressiveness of indices and the basis-independence of index-free notation.

<div class="mw-heading mw-heading3"><h3 id="Component-free_notation">Component-free notation</h3></div>

A <a href="Component-free_treatment_of_tensors" class="mw-redirect" title="Component-free treatment of tensors">component-free treatment of tensors</a> uses notation that emphasises that tensors do not rely on any basis, and is defined in terms of the <a href="Tensor_product" title="Tensor product">tensor product of vector spaces</a>.

<div class="mw-heading mw-heading2"><h2 id="Operations">Operations</h2></div>

There are several operations on tensors that again produce a tensor. The linear nature of tensors implies that two tensors of the same type may be added together, and that tensors may be multiplied by a scalar with results analogous to the <a href="Scalar_multiplication" title="Scalar multiplication">scaling of a vector</a>. On components, these operations are simply performed component-wise. These operations do not change the type of the tensor; but there are also operations that produce a tensor of different type.

<div class="mw-heading mw-heading3"><h3 id="Tensor_product">Tensor product</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Tensor_product" title="Tensor product">Tensor product</a></div>

The <a href="Tensor_product" title="Tensor product">tensor product</a> takes two tensors, S and T, and produces a new tensor, S ⊗ T, whose order is the sum of the orders of the original tensors. When described as multilinear maps, the tensor product simply multiplies the two tensors, i.e.,

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<annotation encoding="application/x-tex">{\displaystyle (S\otimes T)(v_{1},\ldots ,v_{n},v_{n+1},\ldots ,v_{n+m})=S(v_{1},\ldots ,v_{n})T(v_{n+1},\ldots ,v_{n+m}),}</annotation>

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which again produces a map that is linear in all its arguments. On components, the effect is to multiply the components of the two input tensors pairwise, i.e.,

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<annotation encoding="application/x-tex">{\displaystyle (S\otimes T)_{j_{1}\ldots j_{k}j_{k+1}\ldots j_{k+m}}^{i_{1}\ldots i_{l}i_{l+1}\ldots i_{l+n}}=S_{j_{1}\ldots j_{k}}^{i_{1}\ldots i_{l}}T_{j_{k+1}\ldots j_{k+m}}^{i_{l+1}\ldots i_{l+n}}.}</annotation>

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If S is of type (l, k) and T is of type (n, m), then the tensor product S ⊗ T has type (l + n, k + m).

<div class="mw-heading mw-heading3"><h3 id="Contraction">Contraction</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Tensor_contraction" title="Tensor contraction">Tensor contraction</a></div>

<a href="Tensor_contraction" title="Tensor contraction">Tensor contraction</a> is an operation that reduces a type (n, m) tensor to a type (n − 1, m − 1) tensor, of which the <a href="Trace_(linear_algebra)" title="Trace (linear algebra)">trace</a> 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 (1, 1)-tensor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{i}^{j}}">

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</math><img src="./535fdeac0fe85ef501bb527fbe442577c8776bb6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.52ex; height:3.176ex;" alt="{\displaystyle T_{i}^{i}}" loading="lazy">, where the summation is again implied. When the (1, 1)-tensor is interpreted as a linear map, this operation is known as the <a href="Trace_(linear_algebra)" title="Trace (linear algebra)">trace</a>.

The contraction is often used in conjunction with the tensor product to contract an index from each tensor.

The contraction can also be understood using the definition of a tensor as an element of a tensor product of copies of the space V with the space V∗ by first decomposing the tensor into a linear combination of simple tensors, and then applying a factor from V∗ to a factor from V. For example, a tensor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\in V\otimes V\otimes V^{*}}">

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</math><img src="./6054d2fb794e12e39ad72d2e3bb176bfdb4d31fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.703ex; height:2.509ex;" alt="{\displaystyle T\in V\otimes V\otimes V^{*}}" loading="lazy"> can be written as a linear combination

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<annotation encoding="application/x-tex">{\displaystyle T=v_{1}\otimes w_{1}\otimes \alpha _{1}+v_{2}\otimes w_{2}\otimes \alpha _{2}+\cdots +v_{N}\otimes w_{N}\otimes \alpha _{N}.}</annotation>

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</math><img src="./402865f50dd132f866f6805695c306891ce397d9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:57.906ex; height:2.509ex;" alt="{\displaystyle T=v_{1}\otimes w_{1}\otimes \alpha _{1}+v_{2}\otimes w_{2}\otimes \alpha _{2}+\cdots +v_{N}\otimes w_{N}\otimes \alpha _{N}.}" loading="lazy"></dd></dl>

The contraction of T on the first and last slots is then the vector

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<annotation encoding="application/x-tex">{\displaystyle \alpha _{1}(v_{1})w_{1}+\alpha _{2}(v_{2})w_{2}+\cdots +\alpha _{N}(v_{N})w_{N}.}</annotation>

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</math><img src="./1eef4dc7e5eb865d4d1eaa310cdb7bd90ace3cef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.557ex; height:2.843ex;" alt="{\displaystyle \alpha _{1}(v_{1})w_{1}+\alpha _{2}(v_{2})w_{2}+\cdots +\alpha _{N}(v_{N})w_{N}.}" loading="lazy"></dd></dl>

In a vector space with an <a href="Inner_product" class="mw-redirect" title="Inner product">inner product</a> (also known as a <a href="Metric_tensor" title="Metric tensor">metric</a>) g, the term <a href="Tensor_contraction#Metric_contraction" title="Tensor contraction">contraction</a> is used for removing two contravariant or two covariant indices by forming a trace with the metric tensor or its inverse. For example, a (2, 0)-tensor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{ij}}">

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</math><img src="./17b4b8ce4ed17b17367d929290b773eab79b6976.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.197ex; height:2.676ex;" alt="{\displaystyle T^{ij}}" loading="lazy"> can be contracted to a scalar through <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T^{ij}g_{ij}}">

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</math><img src="./d575a13dd7949fd1eacc8b441109324bd42d861f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.783ex; height:3.343ex;" alt="{\displaystyle T^{ij}g_{ij}}" loading="lazy"> (yet again assuming the summation convention).

<div class="mw-heading mw-heading3"><h3 id="Raising_or_lowering_an_index">Raising or lowering an index</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Raising_and_lowering_indices" class="mw-redirect" title="Raising and lowering indices">Raising and lowering indices</a></div>

When a vector space is equipped with a <a href="Nondegenerate_bilinear_form" class="mw-redirect" title="Nondegenerate bilinear form">nondegenerate bilinear form</a> (or <a href="Metric_tensor" title="Metric tensor">metric tensor</a> as it is often called in this context), operations can be defined that convert a contravariant (upper) index into a covariant (lower) index and vice versa. A metric tensor is a (symmetric) (0, 2)-tensor; it is thus possible to contract an upper index of a tensor with one of the lower indices of the metric tensor in the product. This produces a new tensor with the same index structure as the previous tensor, but with lower index generally shown in the same position of the contracted upper index. This operation is quite graphically known as lowering an index.

Conversely, the inverse operation can be defined, and is called raising an index. This is equivalent to a similar contraction on the product with a (2, 0)-tensor. This inverse metric tensor has components that are the matrix inverse of those of the metric tensor.

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Continuum_mechanics">Continuum mechanics</h3></div>

Important examples are provided by <a href="Continuum_mechanics" title="Continuum mechanics">continuum mechanics</a>. The stresses inside a solid body or <a href="Fluid" title="Fluid">fluid</a><a href="#cite_note-31">[28]</a> are described by a tensor field. The <a href="Stress_(mechanics)" title="Stress (mechanics)">stress tensor</a> and <a href="Strain_tensor" class="mw-redirect" title="Strain tensor">strain tensor</a> are both second-order tensor fields, and are related in a general linear elastic material by a fourth-order <a href="Elasticity_tensor" title="Elasticity tensor">elasticity tensor</a> field. In detail, the tensor quantifying stress in a 3-dimensional solid object has components that can be conveniently 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.

If a particular <a href="Volume_form" title="Volume form">surface element</a> 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 <a href="Type_of_a_tensor" class="mw-redirect" title="Type of a tensor">type (2, 0)</a>, in <a href="Linear_elasticity" title="Linear elasticity">linear elasticity</a>, or more precisely by a tensor field of type (2, 0), since the stresses may vary from point to point.

<div class="mw-heading mw-heading3"><h3 id="Other_examples_from_physics">Other examples from physics</h3></div>

Common applications include:

<ul><li><a href="Electromagnetic_tensor" title="Electromagnetic tensor">Electromagnetic tensor</a> (or Faraday tensor) in <a href="Electromagnetism" title="Electromagnetism">electromagnetism</a></li>

<li><a href="Finite_deformation_tensors" class="mw-redirect" title="Finite deformation tensors">Finite deformation tensors</a> for describing deformations and <a href="Strain_tensor" class="mw-redirect" title="Strain tensor">strain tensor</a> for <a href="Strain_(materials_science)" class="mw-redirect" title="Strain (materials science)">strain</a> in <a href="Continuum_mechanics" title="Continuum mechanics">continuum mechanics</a></li>

<li><a href="Permittivity" title="Permittivity">Permittivity</a> and <a href="Electric_susceptibility" title="Electric susceptibility">electric susceptibility</a> are tensors in <a href="Anisotropic" class="mw-redirect" title="Anisotropic">anisotropic</a> media</li>

<li><a href="Four-tensors" class="mw-redirect" title="Four-tensors">Four-tensors</a> in <a href="General_relativity" title="General relativity">general relativity</a> (e.g. <a href="Stress%E2%80%93energy_tensor" title="Stress–energy tensor">stress–energy tensor</a>), used to represent <a href="Momentum" title="Momentum">momentum</a> <a href="Flux" title="Flux">fluxes</a></li>

<li>Spherical tensor operators are the eigenfunctions of the quantum <a href="Angular_momentum_operator" title="Angular momentum operator">angular momentum operator</a> in <a href="Spherical_coordinates" class="mw-redirect" title="Spherical coordinates">spherical coordinates</a></li>

<li>Diffusion tensors, the basis of <a href="Diffusion_tensor_imaging" class="mw-redirect" title="Diffusion tensor imaging">diffusion tensor imaging</a>, represent rates of diffusion in biological environments</li>

<li><a href="Quantum_mechanics" title="Quantum mechanics">Quantum mechanics</a> and <a href="Quantum_computing" title="Quantum computing">quantum computing</a> utilize tensor products for combination of quantum states</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Computer_vision_and_optics">Computer vision and optics</h3></div>

The concept of a tensor of order two is often conflated with that of a matrix. Tensors of higher order do however capture ideas important in science and engineering, as has been shown successively in numerous areas as they develop. This happens, for instance, in the field of <a href="Computer_vision" title="Computer vision">computer vision</a>, with the <a href="Trifocal_tensor" title="Trifocal tensor">trifocal tensor</a> generalizing the <a href="Fundamental_matrix_(computer_vision)" title="Fundamental matrix (computer vision)">fundamental matrix</a>.

The field of <a href="Nonlinear_optics" title="Nonlinear optics">nonlinear optics</a> studies the changes to material <a href="Polarization_density#Relation_between_P_and_E_in_various_materials" title="Polarization density">polarization density</a> under extreme electric fields. The polarization waves generated are related to the generating <a href="Electric_field" title="Electric field">electric fields</a> through the nonlinear susceptibility tensor. If the polarization P is not linearly proportional to the electric field E, the medium is termed nonlinear. To a good approximation (for sufficiently weak fields, assuming no permanent dipole moments are present), P is given by a <a href="Taylor_series" title="Taylor series">Taylor series</a> in E whose coefficients are the nonlinear susceptibilities:

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<annotation encoding="application/x-tex">{\displaystyle {\frac {P_{i}}{\varepsilon _{0}}}=\sum _{j}\chi _{ij}^{(1)}E_{j}+\sum _{jk}\chi _{ijk}^{(2)}E_{j}E_{k}+\sum _{jk\ell }\chi _{ijk\ell }^{(3)}E_{j}E_{k}E_{\ell }+\cdots .\!}</annotation>

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</math><img src="./2d18eca6636943017ede1af7fccbac09a0a20bb1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; margin-right: -0.204ex; width:57.715ex; height:6.676ex;" alt="{\displaystyle {\frac {P_{i}}{\varepsilon _{0}}}=\sum _{j}\chi _{ij}^{(1)}E_{j}+\sum _{jk}\chi _{ijk}^{(2)}E_{j}E_{k}+\sum _{jk\ell }\chi _{ijk\ell }^{(3)}E_{j}E_{k}E_{\ell }+\cdots .\!}" loading="lazy"></dd></dl>

Here <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi ^{(1)}}">

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<annotation encoding="application/x-tex">{\displaystyle \chi ^{(1)}}</annotation>

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</math><img src="./29de1df1cc22c1dd75d35eedba2c66a9feac6eb8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.789ex; height:3.176ex;" alt="{\displaystyle \chi ^{(1)}}" loading="lazy"> is the linear susceptibility, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi ^{(2)}}">

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<annotation encoding="application/x-tex">{\displaystyle \chi ^{(2)}}</annotation>

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</math><img src="./fa36f7f6c02ec32b59cda49e12d59c098738017d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.789ex; height:3.176ex;" alt="{\displaystyle \chi ^{(2)}}" loading="lazy"> gives the <a href="Pockels_effect" title="Pockels effect">Pockels effect</a> and <a href="Second_harmonic_generation" class="mw-redirect" title="Second harmonic generation">second harmonic generation</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \chi ^{(3)}}">

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<annotation encoding="application/x-tex">{\displaystyle \chi ^{(3)}}</annotation>

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</math><img src="./ef39531c7c066f5149d7762021710e548032dd21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.789ex; height:3.176ex;" alt="{\displaystyle \chi ^{(3)}}" loading="lazy"> gives the <a href="Kerr_effect" title="Kerr effect">Kerr effect</a>. This expansion shows the way higher-order tensors arise naturally in the subject matter.

<div class="mw-heading mw-heading3"><h3 id="Machine_learning">Machine learning</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Tensor_(machine_learning)" title="Tensor (machine learning)">Tensor (machine learning)</a></div>

The properties of tensors, especially <a href="Tensor_decomposition" title="Tensor decomposition">tensor decomposition</a>, have enabled their use in <a href="Machine_learning" title="Machine learning">machine learning</a> to embed higher dimensional data in <a href="Artificial_neural_networks" class="mw-redirect" title="Artificial neural networks">artificial neural networks</a>. This notion of tensor differs significantly from that in other areas of mathematics and physics, in the sense that a tensor is the same thing as a multidimensional array. Abstractly, a tensor belongs to tensor product of spaces, each of which has a fixed basis, and the dimensions of the factor spaces can be different. Thus, an example of a tensor in this context is a rectangular matrix. Just as a rectangular matrix has two axes, a horizontal and vertical axis to indicate the position of each entry, a more general tensor has as many axes as there are factors in the tensor product to which it belongs, and an entry of the tensor is referred to be a tuple of integers. The various axes have different dimensions in general.

<div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Tensor_products_of_vector_spaces">Tensor products of vector spaces</h3></div>

The vector spaces of a <a href="Tensor_product" title="Tensor product">tensor product</a> need not be the same, and sometimes the elements of such a more general tensor product are called "tensors". For example, an element of the tensor product space V ⊗ W is a second-order "tensor" in this more general sense,<a href="#cite_note-Maia2011-32">[29]</a> and an order-d tensor may likewise be defined as an element of a tensor product of d different vector spaces.<a href="#cite_note-Hogben2013-33">[30]</a> A type (n, m) tensor, in the sense defined previously, is also a tensor of order n + m in this more general sense. The concept of tensor product <a href="Tensor_product_of_modules" title="Tensor product of modules">can be extended</a> to arbitrary <a href="Module_over_a_ring" class="mw-redirect" title="Module over a ring">modules over a ring</a>.

<div class="mw-heading mw-heading3"><h3 id="Tensors_in_infinite_dimensions_2">Tensors in infinite dimensions</h3></div>

The notion of a tensor can be generalized in a variety of ways to <a href="Dimension_(vector_space)" title="Dimension (vector space)">infinite dimensions</a>. One, for instance, is via the <a href="Tensor_product_of_Hilbert_spaces" title="Tensor product of Hilbert spaces">tensor product</a> of <a href="Hilbert_space" title="Hilbert space">Hilbert spaces</a>.<a href="#cite_note-34">[31]</a> Another way of generalizing the idea of tensor, common in <a href="Nonlinear_system" title="Nonlinear system">nonlinear analysis</a>, is via the <a href="#As_multilinear_maps">multilinear maps definition</a> where instead of using finite-dimensional vector spaces and their <a href="Algebraic_dual" class="mw-redirect" title="Algebraic dual">algebraic duals</a>, one uses infinite-dimensional <a href="Banach_space" title="Banach space">Banach spaces</a> and their <a href="Continuous_dual" class="mw-redirect" title="Continuous dual">continuous dual</a>.<a href="#cite_note-35">[32]</a> Tensors thus live naturally on <a href="Banach_manifold" title="Banach manifold">Banach manifolds</a><a href="#cite_note-36">[33]</a> and <a href="Fr%C3%A9chet_manifold" title="Fréchet manifold">Fréchet manifolds</a>.

<div class="mw-heading mw-heading3"><h3 id="Tensor_densities">Tensor densities</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Tensor_density" title="Tensor density">Tensor density</a></div>

Suppose that a homogeneous medium fills R3, so that the density of the medium is described by a single <a href="Scalar_(physics)" title="Scalar (physics)">scalar</a> value ρ in kg⋅m−3. The mass, in kg, of a region Ω is obtained by multiplying ρ by the volume of the region Ω, or equivalently integrating the constant ρ over the region:

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<annotation encoding="application/x-tex">{\displaystyle m=\int _{\Omega }\rho \,dx\,dy\,dz,}</annotation>

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</math><img src="./d94c63e42ee914af1d00cfafa0ca46499ea55cb4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.468ex; height:5.676ex;" alt="{\displaystyle m=\int _{\Omega }\rho \,dx\,dy\,dz,}" loading="lazy"></dd></dl>

where the Cartesian coordinates x, y, z are measured in m. If the units of length are changed into cm, then the numerical values of the coordinate functions must be rescaled by a factor of 100:

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<annotation encoding="application/x-tex">{\displaystyle x'=100x,\quad y'=100y,\quad z'=100z.}</annotation>

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</math><img src="./07e2a7f4c73cd4dc72b209bb311cffb9fa0d39e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:36.325ex; height:2.843ex;" alt="{\displaystyle x'=100x,\quad y'=100y,\quad z'=100z.}" loading="lazy"></dd></dl>

The numerical value of the density ρ must then also transform by 100−3 m3/cm3 to compensate, so that the numerical value of the mass in kg is still given by integral of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho \,dx\,dy\,dz}">

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<annotation encoding="application/x-tex">{\displaystyle \rho \,dx\,dy\,dz}</annotation>

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</math><img src="./cbdb72af33717b53fcf61af311615358c1b6473e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.584ex; height:2.676ex;" alt="{\displaystyle \rho \,dx\,dy\,dz}" loading="lazy">. Thus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho '=100^{-3}\rho }">

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<annotation encoding="application/x-tex">{\displaystyle \rho '=100^{-3}\rho }</annotation>

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</math><img src="./e98c0a36157083b8ab8789f7029b0e7e9f333f0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.007ex; height:3.176ex;" alt="{\displaystyle \rho '=100^{-3}\rho }" loading="lazy"> (in units of kg⋅cm−3).

More generally, if the Cartesian coordinates x, y, z undergo a linear transformation, then the numerical value of the density ρ must change by a factor of the reciprocal of the absolute value of the <a href="Determinant" title="Determinant">determinant</a> of the coordinate transformation, so that the integral remains invariant, by the <a href="Change_of_variables_formula" class="mw-redirect" title="Change of variables formula">change of variables formula</a> for integration. Such a quantity that scales by the reciprocal of the absolute value of the determinant of the coordinate transition map is called a <a href="Scalar_density" class="mw-redirect" title="Scalar density">scalar density</a>. To model a non-constant density, ρ is a function of the variables x, y, z (a <a href="Scalar_field" title="Scalar field">scalar field</a>), and under a <a href="Curvilinear_coordinates" title="Curvilinear coordinates">curvilinear</a> change of coordinates, it transforms by the reciprocal of the <a href="Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a> of the coordinate change. For more on the intrinsic meaning, see <a href="Density_on_a_manifold" title="Density on a manifold">Density on a manifold</a>.

A tensor density transforms like a tensor under a coordinate change, except that it in addition picks up a factor of the absolute value of the determinant of the coordinate transition:<a href="#cite_note-37">[34]</a>

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<annotation encoding="application/x-tex">{\displaystyle T_{j'_{1}\dots j'_{q}}^{i'_{1}\dots i'_{p}}[\mathbf {f} \cdot R]=\left|\det R\right|^{-w}\left(R^{-1}\right)_{i_{1}}^{i'_{1}}\cdots \left(R^{-1}\right)_{i_{p}}^{i'_{p}}T_{j_{1},\ldots ,j_{q}}^{i_{1},\ldots ,i_{p}}[\mathbf {f} ]R_{j'_{1}}^{j_{1}}\cdots R_{j'_{q}}^{j_{q}}.}</annotation>

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</math><img src="./5f140bbb8af061851d8d4b060ee3907e0455ce5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:65.19ex; height:4.843ex;" alt="{\displaystyle T_{j'_{1}\dots j'_{q}}^{i'_{1}\dots i'_{p}}[\mathbf {f} \cdot R]=\left|\det R\right|^{-w}\left(R^{-1}\right)_{i_{1}}^{i'_{1}}\cdots \left(R^{-1}\right)_{i_{p}}^{i'_{p}}T_{j_{1},\ldots ,j_{q}}^{i_{1},\ldots ,i_{p}}[\mathbf {f} ]R_{j'_{1}}^{j_{1}}\cdots R_{j'_{q}}^{j_{q}}.}" loading="lazy"></dd></dl>

Here w is called the weight. In general, any tensor multiplied by a power of this function or its absolute value is called a tensor density, or a weighted tensor.<a href="#cite_note-38">[35]</a><a href="#cite_note-FOOTNOTEKay198827-39">[36]</a> An example of a tensor density is the <a href="Current_density" title="Current density">current density</a> of <a href="Electromagnetism" title="Electromagnetism">electromagnetism</a>.

Under an affine transformation of the coordinates, a tensor transforms by the linear part of the transformation itself (or its inverse) on each index. These come from the <a href="Rational_representation" title="Rational representation">rational representations</a> of the general linear group. But this is not quite the most general linear transformation law that such an object may have: tensor densities are non-rational, but are still <a href="Semisimple" class="mw-redirect" title="Semisimple">semisimple</a> representations. A further class of transformations come from the logarithmic representation of the general linear group, a reducible but not semisimple representation,<a href="#cite_note-40">[37]</a> consisting of an (x, y) ∈ R2 with the transformation law

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<annotation encoding="application/x-tex">{\displaystyle (x,y)\mapsto (x+y\log \left|\det R\right|,y).}</annotation>

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</math><img src="./ea7dfa92f0d9c2eb01a6b1764250c04bac5efcfb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.947ex; height:2.843ex;" alt="{\displaystyle (x,y)\mapsto (x+y\log \left|\det R\right|,y).}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Geometric_objects">Geometric objects</h3></div>

The transformation law for a tensor behaves as a <a href="Functor" title="Functor">functor</a> on the category of admissible coordinate systems, under general linear transformations (or, other transformations within some class, such as <a href="Local_diffeomorphism" title="Local diffeomorphism">local diffeomorphisms</a>). This makes a tensor a special case of a geometrical object, in the technical sense that it is a function of the coordinate system transforming functorially under coordinate changes.<a href="#cite_note-41">[38]</a> Examples of objects obeying more general kinds of transformation laws are <a href="Jet_(mathematics)" title="Jet (mathematics)">jets</a> and, more generally still, <a href="Natural_bundle" title="Natural bundle">natural bundles</a>.<a href="#cite_note-42">[39]</a><a href="#cite_note-43">[40]</a>

<div class="mw-heading mw-heading3"><h3 id="Spinors">Spinors</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Spinor" title="Spinor">Spinor</a></div>

When changing from one <a href="Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a> (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 <a href="Simply_connected" class="mw-redirect" title="Simply connected">simply connected</a> (see <a href="Orientation_entanglement" title="Orientation entanglement">orientation entanglement</a> and <a href="Plate_trick" title="Plate trick">plate trick</a>): 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.<a href="#cite_note-44">[41]</a> A <a href="Spinor" title="Spinor">spinor</a> 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.<a href="#cite_note-45">[42]</a><a href="#cite_note-46">[43]</a>

Spinors are elements of the <a href="Spin_representation" title="Spin representation">spin representation</a> of the rotation group, while tensors are elements of its <a href="Tensor_representation" title="Tensor representation">tensor representations</a>. Other <a href="Classical_group" title="Classical group">classical groups</a> 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.

<ul><li> The dictionary definition of <a href="https://en.wiktionary.org/wiki/tensor" class="extiw external" title="wiktionary:tensor">tensor</a> at Wiktionary</li>

<li><a href="Array_data_type" class="mw-redirect" title="Array data type">Array data type</a>, for tensor storage and manipulation</li>

<li><a href="Bitensor" title="Bitensor">Bitensor</a></li></ul>

<div class="mw-heading mw-heading3"><h3 id="Foundational">Foundational</h3></div>

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<ul><li><a href="Cartesian_tensor" title="Cartesian tensor">Cartesian tensor</a></li>

<li><a href="Fibre_bundle" class="mw-redirect" title="Fibre bundle">Fibre bundle</a></li>

<li><a href="Glossary_of_tensor_theory" title="Glossary of tensor theory">Glossary of tensor theory</a></li>

<li><a href="Multilinear_subspace_learning#Multilinear_projection" title="Multilinear subspace learning">Multilinear projection</a></li>

<li><a href="Tensor_product_of_modules" title="Tensor product of modules">Tensor product of modules</a></li></ul>

</div>

<div class="mw-heading mw-heading3"><h3 id="Applications_2">Applications</h3></div>

<ul><li><a href="Application_of_tensor_theory_in_engineering" class="mw-redirect" title="Application of tensor theory in engineering">Application of tensor theory in engineering</a></li>

<li><a href="Continuum_mechanics" title="Continuum mechanics">Continuum mechanics</a></li>

<li><a href="Covariant_derivative" title="Covariant derivative">Covariant derivative</a></li>

<li><a href="Curvature" title="Curvature">Curvature</a></li>

<li><a href="Diffusion_MRI" class="mw-redirect" title="Diffusion MRI">Diffusion tensor MRI</a></li>

<li><a href="Einstein_field_equations" title="Einstein field equations">Einstein field equations</a></li>

<li><a href="Fluid_mechanics" title="Fluid mechanics">Fluid mechanics</a></li>

<li><a href="Gravity" title="Gravity">Gravity</a></li>

<li><a href="Multilinear_subspace_learning" title="Multilinear subspace learning">Multilinear subspace learning</a></li>

<li><a href="Riemannian_geometry" title="Riemannian geometry">Riemannian geometry</a></li>

<li><a href="Structure_tensor" title="Structure tensor">Structure tensor</a></li>

<li><a href="Tensor_Contraction_Engine" title="Tensor Contraction Engine">Tensor Contraction Engine</a></li>

<li><a href="Tensor_decomposition" title="Tensor decomposition">Tensor decomposition</a></li>

<li><a href="Tensor_derivative" class="mw-redirect" title="Tensor derivative">Tensor derivative</a></li>

<li><a href="Tensor_software" title="Tensor software">Tensor software</a></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="Explanatory_notes">Explanatory notes</h2></div>

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<li id="cite_note-5"><a href="#cite_ref-5">^</a> The Einstein summation convention, in brief, requires the sum to be taken over all values of the index whenever the same symbol appears as a subscript and superscript in the same term. For example, under this convention <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{i}C^{i}=B_{1}C^{1}+B_{2}C^{2}+\cdots +B_{n}C^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle B_{i}C^{i}=B_{1}C^{1}+B_{2}C^{2}+\cdots +B_{n}C^{n}}</annotation>

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</math><img src="./cce1598317584a5e82687c6ac38a0929c257219c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:36.843ex; height:3.009ex;" alt="{\displaystyle B_{i}C^{i}=B_{1}C^{1}+B_{2}C^{2}+\cdots +B_{n}C^{n}}" loading="lazy">

</li>

<li id="cite_note-13"><a href="#cite_ref-13">^</a> The <a href="Dual_space#Injection_into_the_double-dual" title="Dual space">double duality isomorphism</a>, for instance, is used to identify V with the double dual space V∗∗, which consists of multilinear forms of degree one on V∗. It is typical in linear algebra to identify spaces that are naturally isomorphic, treating them as the same space.

</li>

<li id="cite_note-18"><a href="#cite_ref-18">^</a> Namely, the <a href="Norm_(mathematics)" title="Norm (mathematics)">norm operation</a> in a vector space.

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Specific">Specific</h3></div>

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</style><cite id="CITEREFKline1990" class="citation book cs1">Kline, Morris (1990). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-OsRDAAAQBAJ">Mathematical Thought From Ancient to Modern Times</a>. Vol. 3. Oxford University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-19-506137-6</bdi>.</cite>

</li>

<li id="cite_note-DeLathauwerEtAl2000-2"><a href="#cite_ref-DeLathauwerEtAl2000_2-0">^</a> <cite id="CITEREFDe_LathauwerDe_MoorVandewalle2000" class="citation journal cs1">De Lathauwer, Lieven; De Moor, Bart; Vandewalle, Joos (2000). <a rel="nofollow" class="external text" href="https://alterlab.org/teaching/BME6780/papers+patents/De_Lathauwer_2000.pdf">"A Multilinear Singular Value Decomposition"</a> (PDF). <a href="SIAM_J._Matrix_Anal._Appl." class="mw-redirect" title="SIAM J. Matrix Anal. Appl.">SIAM J. Matrix Anal. Appl.</a> 21 (4): 1253–1278. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2FS0895479896305696">10.1137/S0895479896305696</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:14344372">14344372</a>.</cite>

</li>

<li id="cite_note-Vasilescu2002Tensorfaces-3"><a href="#cite_ref-Vasilescu2002Tensorfaces_3-0">^</a> <cite id="CITEREFVasilescuTerzopoulos2002" class="citation book cs1">Vasilescu, M.A.O.; Terzopoulos, D. (2002). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20221229090931/http://www.cs.toronto.edu/~maov/tensorfaces/Springer%20ECCV%202002_files/eccv02proceeding_23500447.pdf">"Multilinear Analysis of Image Ensembles: TensorFaces"</a> (PDF). Computer Vision — ECCV 2002. Lecture Notes in Computer Science. Vol. 2350. pp. 447–460. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F3-540-47969-4_30">10.1007/3-540-47969-4_30</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-43745-1</bdi>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:12793247">12793247</a>. Archived from <a rel="nofollow" class="external text" href="http://www.cs.toronto.edu/~maov/tensorfaces/Springer%20ECCV%202002_files/eccv02proceeding_23500447.pdf">the original</a> (PDF) on 2022-12-29. Retrieved 2022-12-29.</cite>

</li>

<li id="cite_note-KoldaBader2009-4"><a href="#cite_ref-KoldaBader2009_4-0">^</a> <cite id="CITEREFKoldaBader2009" class="citation journal cs1">Kolda, Tamara; Bader, Brett (2009). <a rel="nofollow" class="external text" href="https://www.kolda.net/publication/TensorReview.pdf">"Tensor Decompositions and Applications"</a> (PDF). <a href="SIAM_Review" class="mw-redirect" title="SIAM Review">SIAM Review</a>. 51 (3): 455–500. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009SIAMR..51..455K">2009SIAMR..51..455K</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1137%2F07070111X">10.1137/07070111X</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:16074195">16074195</a>.</cite>

</li>

<li id="cite_note-Sharpe2000-6">^ <a href="#cite_ref-Sharpe2000_6-0">a</a> <a href="#cite_ref-Sharpe2000_6-1">b</a> <cite id="CITEREFSharpe2000" class="citation book cs1">Sharpe, R.W. (2000). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Ytqs4xU5QKAC&pg=PA194">Differential Geometry: Cartan's Generalization of Klein's Erlangen Program</a>. Springer. p. 194. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-94732-7</bdi>.</cite>

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFSchouten1954" class="citation cs2"><a href="Jan_Arnoldus_Schouten" title="Jan Arnoldus Schouten">Schouten, Jan Arnoldus</a> (1954), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=WROiC9st58gC">"Chapter II"</a>, <a rel="nofollow" class="external text" href="https://archive.org/details/isbn_9780486655826">Tensor analysis for physicists</a>, Courier Corporation, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-65582-6</bdi></cite> <code class="cs1-code">{{citation}}</code>: ISBN / Date incompatibility (help)

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite id="CITEREFKobayashiNomizu1996" class="citation cs2">Kobayashi, Shoshichi; Nomizu, Katsumi (1996), <a href="Foundations_of_Differential_Geometry" title="Foundations of Differential Geometry">Foundations of Differential Geometry</a>, vol. 1 (New ed.), <a href="Wiley_Interscience" class="mw-redirect" title="Wiley Interscience">Wiley Interscience</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-15733-5</bdi></cite>

</li>

<li id="cite_note-9"><a href="#cite_ref-9">^</a> <cite id="CITEREFLee2000" class="citation cs2">Lee, John (2000), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=4sGuQgAACAAJ&pg=PA173">Introduction to smooth manifolds</a>, Springer, p. 173, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-95495-0</bdi></cite>

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> <cite id="CITEREFDodsonPoston2013" class="citation book cs1">Dodson, C.T.J.; Poston, T. (2013) [1991]. Tensor geometry: The Geometric Viewpoint and Its Uses. Graduate Texts in Mathematics. Vol. 130 (2nd ed.). Springer. p. 105. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9783642105142</bdi>.</cite>

</li>

<li id="cite_note-11"><a href="#cite_ref-11">^</a> <cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Affine_tensor">"Affine tensor"</a>, <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite>

</li>

<li id="cite_note-12"><a href="#cite_ref-12">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://math.stackexchange.com/q/4163471">"Why are Tensors (Vectors of the form a⊗b...⊗z) multilinear maps?"</a>. Mathematics Stackexchange. June 5, 2021.</cite>

</li>

<li id="cite_note-14"><a href="#cite_ref-14">^</a> <cite id="CITEREFBourbaki1998" class="citation book cs1">Bourbaki, N. (1998). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=STS9aZ6F204C">"3"</a>. Algebra I: Chapters 1-3. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-64243-5</bdi>.</cite> where the case of finitely generated projective modules is treated. The global sections of sections of a vector bundle over a compact space form a projective module over the ring of smooth functions. All statements for coherent sheaves are true locally.

</li>

<li id="cite_note-15"><a href="#cite_ref-15">^</a> <cite id="CITEREFJoyalStreet1993" class="citation cs2">Joyal, André; Street, Ross (1993), "Braided tensor categories", <a href="Advances_in_Mathematics" title="Advances in Mathematics">Advances in Mathematics</a>, 102: 20–78, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Faima.1993.1055">10.1006/aima.1993.1055</a></cite>

</li>

<li id="cite_note-16"><a href="#cite_ref-16">^</a> <cite id="CITEREFReich1994" class="citation book cs1">Reich, Karin (1994). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=O6lixBzbc0gC">Die Entwicklung des Tensorkalküls</a>. Science networks historical studies. Vol. 11. Birkhäuser. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-7643-2814-6</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/31468174">31468174</a>.</cite>

</li>

<li id="cite_note-17"><a href="#cite_ref-17">^</a> <cite id="CITEREFHamilton1854–1855" class="citation journal cs1">Hamilton, William Rowan (1854–1855). Wilkins, David R. (ed.). <a rel="nofollow" class="external text" href="http://www.emis.de/classics/Hamilton/ExtQuat.pdf">"On some Extensions of Quaternions"</a> (PDF). <a href="Philosophical_Magazine" title="Philosophical Magazine">Philosophical Magazine</a> (7–9): 492–9, 125–137, 261–9, 46–51, 280–290. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0302-7597">0302-7597</a>.</cite> 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, ... "

</li>

<li id="cite_note-auto-19">^ <a href="#cite_ref-auto_19-0">a</a> <a href="#cite_ref-auto_19-1">b</a> <a href="#cite_ref-auto_19-2">c</a> <cite id="CITEREFGuo2021" class="citation book cs1">Guo, Hongyu (2021-06-16). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=5dM3EAAAQBAJ&q=array+vector+matrix+tensor">What Are Tensors Exactly?</a>. World Scientific. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-981-12-4103-1</bdi>.</cite>

</li>

<li id="cite_note-Voigt1898-20"><a href="#cite_ref-Voigt1898_20-0">^</a> <cite id="CITEREFVoigt1898" class="citation book cs1">Voigt, Woldemar (1898). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=QhBDAAAAIAAJ&pg=PA20">Die fundamentalen physikalischen Eigenschaften der Krystalle in elementarer Darstellung</a> [The fundamental physical properties of crystals in an elementary presentation]. Von Veit. pp. 20–. <q>Wir wollen uns deshalb nur darauf stützen, dass Zustände der geschilderten Art bei Spannungen und Dehnungen nicht starrer Körper auftreten, und sie deshalb tensorielle, die für sie charakteristischen physikalischen Grössen aber Tensoren nennen. [We therefore want [our presentation] to be based only on [the assumption that] conditions of the type described occur during stresses and strains of non-rigid bodies, and therefore call them "tensorial" but call the characteristic physical quantities for them "tensors".]</q></cite>

</li>

<li id="cite_note-21"><a href="#cite_ref-21">^</a> <cite id="CITEREFRicci_Curbastro1892" class="citation journal cs1">Ricci Curbastro, G. (1892). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=1bGdAQAACAAJ">"Résumé de quelques travaux sur les systèmes variables de fonctions associés à une forme différentielle quadratique"</a>. Bulletin des Sciences Mathématiques. 2 (16): 167–189.</cite>

</li>

<li id="cite_note-FOOTNOTERicciLevi-Civita1900-22"><a href="#cite_ref-FOOTNOTERicciLevi-Civita1900_22-0">^</a> <a href="#CITEREFRicciLevi-Civita1900">Ricci & Levi-Civita 1900</a>.

</li>

<li id="cite_note-23"><a href="#cite_ref-23">^</a> <cite id="CITEREFPais2005" class="citation book cs1">Pais, Abraham (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=U2mO4nUunuwC">Subtle Is the Lord: The Science and the Life of Albert Einstein</a>. Oxford University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-19-280672-7</bdi>.</cite>

</li>

<li id="cite_note-Goodstein-24"><a href="#cite_ref-Goodstein_24-0">^</a> <cite id="CITEREFGoodstein1982" class="citation journal cs1"><a href="Judith_R._Goodstein" title="Judith R. Goodstein">Goodstein, Judith R.</a> (1982). "The Italian Mathematicians of Relativity". Centaurus. 26 (3): 241–261. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1982Cent...26..241G">1982Cent...26..241G</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1600-0498.1982.tb00665.x">10.1111/j.1600-0498.1982.tb00665.x</a>.</cite>

</li>

<li id="cite_note-Spanier2012-25"><a href="#cite_ref-Spanier2012_25-0">^</a> <cite id="CITEREFSpanier2012" class="citation book cs1">Spanier, Edwin H. (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=iKx3BQAAQBAJ&&pg=PA227">Algebraic Topology</a>. Springer. p. 227. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4684-9322-1</bdi>. <q>the Künneth formula expressing the homology of the tensor product...</q></cite>

</li>

<li id="cite_note-Hungerford2003-26"><a href="#cite_ref-Hungerford2003_26-0">^</a> <cite id="CITEREFHungerford2003" class="citation book cs1"><a href="Thomas_W._Hungerford" title="Thomas W. Hungerford">Hungerford, Thomas W.</a> (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=t6N_tOQhafoC&pg=PA168">Algebra</a>. Springer. p. 168. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-90518-1</bdi>. <q>...the classification (up to isomorphism) of modules over an arbitrary ring is quite difficult...</q></cite>

</li>

<li id="cite_note-MacLane2013-27"><a href="#cite_ref-MacLane2013_27-0">^</a> <cite id="CITEREFMacLane2013" class="citation book cs1"><a href="Saunders_Mac_Lane" title="Saunders Mac Lane">MacLane, Saunders</a> (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=6KPSBwAAQBAJ&pg=PA4">Categories for the Working Mathematician</a>. Springer. p. 4. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4612-9839-7</bdi>. <q>...for example the monoid M ... in the category of abelian groups, × is replaced by the usual tensor product...</q></cite>

</li>

<li id="cite_note-BambergSternberg1991-28"><a href="#cite_ref-BambergSternberg1991_28-0">^</a> <cite id="CITEREFBambergSternberg1991" class="citation book cs1">Bamberg, Paul; Sternberg, Shlomo (1991). A Course in Mathematics for Students of Physics. Vol. 2. Cambridge University Press. p. 669. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-40650-5</bdi>.</cite>

</li>

<li id="cite_note-29"><a href="#cite_ref-29">^</a> <cite id="CITEREFPenrose2007" class="citation book cs1">Penrose, R. (2007). <a href="The_Road_to_Reality" title="The Road to Reality">The Road to Reality</a>. Vintage. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-679-77631-4</bdi>.</cite>

</li>

<li id="cite_note-30"><a href="#cite_ref-30">^</a> <cite id="CITEREFWheelerMisnerThorne1973" class="citation book cs1">Wheeler, J.A.; Misner, C.; Thorne, K.S. (1973). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=w4Gigq3tY1kC">Gravitation</a>. W.H. Freeman. p. 83. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-7167-0344-0</bdi>.</cite>

</li>

<li id="cite_note-31"><a href="#cite_ref-31">^</a> <cite id="CITEREFSchobeiri2021" class="citation book cs1">Schobeiri, Meinhard T. (2021). "Vector and Tensor Analysis, Applications to Fluid Mechanics". Fluid Mechanics for Engineers. Springer. pp. 11–29.</cite>

</li>

<li id="cite_note-Maia2011-32"><a href="#cite_ref-Maia2011_32-0">^</a> <cite id="CITEREFMaia2011" class="citation book cs1">Maia, M. D. (2011). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=wEWw_vGBDW8C&pg=PA48">Geometry of the Fundamental Interactions: On Riemann's Legacy to High Energy Physics and Cosmology</a>. Springer. p. 48. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4419-8273-5</bdi>.</cite>

</li>

<li id="cite_note-Hogben2013-33"><a href="#cite_ref-Hogben2013_33-0">^</a> <cite id="CITEREFHogben2013" class="citation book cs1"><a href="Leslie_Hogben" title="Leslie Hogben">Hogben, Leslie</a>, ed. (2013). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Er7MBQAAQBAJ&pg=PA7">Handbook of Linear Algebra</a> (2nd ed.). CRC Press. pp. 15–7. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4665-0729-6</bdi>.</cite>

</li>

<li id="cite_note-34"><a href="#cite_ref-34">^</a> <cite id="CITEREFSegal1956" class="citation journal cs1">Segal, I. E. (January 1956). <a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1992855">"Tensor Algebras Over Hilbert Spaces. I"</a>. <a href="Transactions_of_the_American_Mathematical_Society" title="Transactions of the American Mathematical Society">Transactions of the American Mathematical Society</a>. 81 (1): 106–134. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1992855">10.2307/1992855</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1992855">1992855</a>.</cite>

</li>

<li id="cite_note-35"><a href="#cite_ref-35">^</a> <cite id="CITEREFAbrahamMarsdenRatiu1988" class="citation book cs1">Abraham, Ralph; Marsden, Jerrold E.; Ratiu, Tudor S. (February 1988). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=dWHet_zgyCAC">"5. Tensors"</a>. Manifolds, Tensor Analysis and Applications. Applied Mathematical Sciences. Vol. 75 (2nd ed.). Springer. pp. 338–9. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-96790-5</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/18562688">18562688</a>. <q>Elements of Trs are called tensors on E, [...].</q></cite>

</li>

<li id="cite_note-36"><a href="#cite_ref-36">^</a> <cite id="CITEREFLang1972" class="citation book cs1"><a href="Serge_Lang" title="Serge Lang">Lang, Serge</a> (1972). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=dn7rBwAAQBAJ">Differential manifolds</a>. <a href="Addison-Wesley" title="Addison-Wesley">Addison-Wesley</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-201-04166-8</bdi>.</cite>

</li>

<li id="cite_note-37"><a href="#cite_ref-37">^</a> <cite id="CITEREFSchouten" class="citation cs2"><a href="Jan_Arnoldus_Schouten" title="Jan Arnoldus Schouten">Schouten, Jan Arnoldus</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=WROiC9st58gC">"§II.8: Densities"</a>, Tensor analysis for physicists</cite>

</li>

<li id="cite_note-38"><a href="#cite_ref-38">^</a> <cite id="CITEREFMcConnell2014" class="citation book cs1">McConnell, A.J. (2014) [1957]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ZCP0AwAAQBAJ">Applications of tensor analysis</a>. Dover. p. 28. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780486145020</bdi>.</cite>

</li>

<li id="cite_note-FOOTNOTEKay198827-39"><a href="#cite_ref-FOOTNOTEKay198827_39-0">^</a> <a href="#CITEREFKay1988">Kay 1988</a>, p. 27.

</li>

<li id="cite_note-40"><a href="#cite_ref-40">^</a> <cite id="CITEREFOlver1995" class="citation cs2">Olver, Peter (1995), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=YuTzf61HILAC&pg=PA77">Equivalence, invariants, and symmetry</a>, Cambridge University Press, p. 77, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780521478113</bdi></cite>

</li>

<li id="cite_note-41"><a href="#cite_ref-41">^</a> <cite id="CITEREFHaantjesLaman1953" class="citation journal cs1">Haantjes, J.; <a href="Gerard_Laman" title="Gerard Laman">Laman, G.</a> (1953). "On the definition of geometric objects. I". Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen: Series A: Mathematical Sciences. 56 (3): 208–215.</cite>

</li>

<li id="cite_note-42"><a href="#cite_ref-42">^</a> <cite id="CITEREFNijenhuis1960" class="citation cs2"><a href="Albert_Nijenhuis" title="Albert Nijenhuis">Nijenhuis, Albert</a> (1960), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20171027025011/http://www.mathunion.org/ICM/ICM1958/Main/icm1958.0463.0469.ocr.pdf">"Geometric aspects of formal differential operations on tensor fields"</a> (PDF), Proc. Internat. Congress Math.(Edinburgh, 1958), Cambridge University Press, pp. 463–9, archived from <a rel="nofollow" class="external text" href="http://www.mathunion.org/ICM/ICM1958/Main/icm1958.0463.0469.ocr.pdf">the original</a> (PDF) on 2017-10-27, retrieved 2017-10-26</cite>.

</li>

<li id="cite_note-43"><a href="#cite_ref-43">^</a> <cite id="CITEREFSalviori1972" class="citation cs2">Salviori, Sarah (1972), <a rel="nofollow" class="external text" href="https://projecteuclid.org/download/pdf_1/euclid.jdg/1214430830">"On the theory of geometric objects"</a>, <a href="Journal_of_Differential_Geometry" title="Journal of Differential Geometry">Journal of Differential Geometry</a>, 7 (1–2): 257–278, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4310%2Fjdg%2F1214430830">10.4310/jdg/1214430830</a></cite>.

</li>

<li id="cite_note-44"><a href="#cite_ref-44">^</a> <cite id="CITEREFPenrose2005" class="citation book cs1"><a href="Roger_Penrose" title="Roger Penrose">Penrose, Roger</a> (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=VWTNCwAAQBAJ&pg=PA203">The road to reality: a complete guide to the laws of our universe</a>. Knopf. pp. 203–206.</cite>

</li>

<li id="cite_note-45"><a href="#cite_ref-45">^</a> <cite id="CITEREFMeinrenken2013" class="citation book cs1">Meinrenken, E. (2013). "The spin representation". Clifford Algebras and Lie Theory. Ergebnisse der Mathematik undihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics. Vol. 58. Springer. pp. 49–85. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-36216-3_3">10.1007/978-3-642-36216-3_3</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-642-36215-6</bdi>.</cite>

</li>

<li id="cite_note-46"><a href="#cite_ref-46">^</a> <cite id="CITEREFDong2011" class="citation cs2">Dong, S. H. (2011), "2. Special Orthogonal Group SO(N)", Wave Equations in Higher Dimensions, Springer, pp. 13–38</cite>

</li>

</ol></div>

<div class="mw-heading mw-heading3"><h3 id="General">General</h3></div>

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<ul><li><cite id="CITEREFBishopSamuel_I._Goldberg1980" class="citation book cs1"><a href="Richard_L._Bishop" title="Richard L. Bishop">Bishop, Richard L.</a>; Samuel I. Goldberg (1980) [1968]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ePFIAwAAQBAJ">Tensor Analysis on Manifolds</a>. Dover. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-64039-6</bdi>.</cite></li>

<li><cite id="CITEREFDanielson2003" class="citation book cs1"><a href="Donald_A._Danielson" title="Donald A. Danielson">Danielson, Donald A.</a> (2003). Vectors and Tensors in Engineering and Physics (2/e ed.). Westview (Perseus). <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8133-4080-7</bdi>.</cite></li>

<li><cite id="CITEREFDimitrienko2002" class="citation book cs1">Dimitrienko, Yuriy (2002). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=7UMYToTiYDsC">Tensor Analysis and Nonlinear Tensor Functions</a>. Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4020-1015-6</bdi>.</cite></li>

<li><cite id="CITEREFJeevanjee2011" class="citation book cs1">Jeevanjee, Nadir (2011). <a rel="nofollow" class="external text" href="https://www.springer.com/new+%26+forthcoming+titles+(default)/book/978-0-8176-4714-8">An Introduction to Tensors and Group Theory for Physicists</a>. Birkhauser. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8176-4714-8</bdi>.</cite></li>

<li><cite id="CITEREFLawden2003" class="citation book cs1">Lawden, D. F. (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=rJYoAwAAQBAJ">Introduction to Tensor Calculus, Relativity and Cosmology</a> (3/e ed.). Dover. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-42540-5</bdi>.</cite></li>

<li><cite id="CITEREFLebedevCloud2003" class="citation book cs1">Lebedev, Leonid P.; Cloud, Michael J. (2003). Tensor Analysis. World Scientific. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-981-238-360-0</bdi>.</cite></li>

<li><cite id="CITEREFLovelockRund1989" class="citation book cs1">Lovelock, David; Rund, Hanno (1989) [1975]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Tl3dCgAAQBAJ">Tensors, Differential Forms, and Variational Principles</a>. Dover. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-65840-7</bdi>.</cite></li>

<li><cite id="CITEREFMunkres1997" class="citation book cs1">Munkres, James R. (1997). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=tGT6K6HdFfwC">Analysis On Manifolds</a>. Avalon. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8133-4548-2</bdi>.</cite> Chapter six gives a "from scratch" introduction to covariant tensors.</li>

<li><cite id="CITEREFRicciLevi-Civita1900" class="citation journal cs1"><a href="Gregorio_Ricci-Curbastro" title="Gregorio Ricci-Curbastro">Ricci, Gregorio</a>; Levi-Civita, Tullio (March 1900). <a rel="nofollow" class="external text" href="https://zenodo.org/record/1428270">"Méthodes de calcul différentiel absolu et leurs applications"</a>. <a href="Mathematische_Annalen" title="Mathematische Annalen">Mathematische Annalen</a>. 54 (1–2): 125–201. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01454201">10.1007/BF01454201</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120009332">120009332</a>.</cite></li>

<li><cite id="CITEREFKay1988" class="citation book cs1">Kay, David C (1988-04-01). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=6tUU3KruG14C">Schaum's Outline of Tensor Calculus</a>. McGraw-Hill. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-033484-7</bdi>.</cite></li>

<li><cite id="CITEREFSchutz1980" class="citation book cs1">Schutz, Bernard F. (28 January 1980). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=HAPMB2e643kC">Geometrical Methods of Mathematical Physics</a>. Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-29887-2</bdi>.</cite></li>

<li><cite id="CITEREFSyngeSchild1969" class="citation book cs1">Synge, John Lighton; Schild, Alfred (1969). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=8vlGhlxqZjsC">Tensor Calculus</a>. Courier Corporation. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-63612-2</bdi>.</cite></li></ul>

</div>

<ul><li>This article incorporates material from tensor on <a href="PlanetMath" title="PlanetMath">PlanetMath</a>, which is licensed under the Creative Commons Attribution/Share-Alike License.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

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<div class="side-box-text plainlist">Wikimedia Commons has media related to <a href="https://commons.wikimedia.org/wiki/Category:Tensors" class="extiw external" title="commons:Category:Tensors">Tensors</a>.</div></div>

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<ul><li><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Tensor.html">"Tensor"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite></li>

<li><cite id="CITEREFBowenWang1976" class="citation book cs1"><a href="Ray_M._Bowen" title="Ray M. Bowen">Bowen, Ray M.</a>; Wang, C.C. (1976). Linear and Multilinear Algebra. Introduction to Vectors and Tensors. Vol. 1. Plenum Press. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/1969.1%2F2502">1969.1/2502</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780306375088</bdi>.</cite></li>

<li><cite id="CITEREFBowenWang2006" class="citation book cs1">Bowen, Ray M.; Wang, C.C. (2006). Vector and Tensor Analysis. Introduction to Vectors and Tensors. Vol. 2. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/1969.1%2F3609">1969.1/3609</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780306375095</bdi>.</cite></li>

<li><cite id="CITEREFKolecki2002" class="citation web cs1">Kolecki, Joseph C. (2002). <a rel="nofollow" class="external text" href="https://ntrs.nasa.gov/citations/20020083040">"An Introduction to Tensors for Students of Physics and Engineering"</a>. Cleveland, Ohio: <a href="NASA" title="NASA">NASA</a> Glenn Research Center. 20020083040.</cite></li>

<li><cite id="CITEREFKolecki2005" class="citation web cs1">Kolecki, Joseph C. (2005). <a rel="nofollow" class="external text" href="https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20050175884.pdf">"Foundations of Tensor Analysis for Students of Physics and Engineering With an Introduction to the Theory of Relativity"</a> (PDF). Cleveland, Ohio: NASA Glenn Research Center. 20050175884.</cite></li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20051104201543/http://nrich.maths.org/askedNRICH/edited/2604.html">A discussion of the various approaches to teaching tensors, and recommendations of textbooks</a></li>

<li><cite id="CITEREFSharipov2004" class="citation arxiv cs1">Sharipov, Ruslan (2004). "Quick introduction to tensor analysis". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math.HO/0403252">math.HO/0403252</a>.</cite></li>

<li><cite id="CITEREFFeynman1964–2013" class="citation web cs1"><a href="Richard_Feynman" title="Richard Feynman">Feynman, Richard</a> (1964–2013). <a rel="nofollow" class="external text" href="https://feynmanlectures.caltech.edu/II_31.html">"31. Tensors"</a>. The Feynman Lectures. California Institute of Technology.</cite></li></ul>

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</style><div id="Tensors176" style="font-size:114%;margin:0 4em"></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div><a href="Glossary_of_tensor_theory" title="Glossary of tensor theory">Glossary of tensor theory</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Scope</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Mathematics" title="Mathematics">Mathematics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Coordinate_system" title="Coordinate system">Coordinate system</a></li>

<li><a href="Differential_geometry" title="Differential geometry">Differential geometry</a></li>

<li><a href="Dyadics" title="Dyadics">Dyadic algebra</a></li>

<li><a href="Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a></li>

<li><a href="Exterior_calculus" class="mw-redirect" title="Exterior calculus">Exterior calculus</a></li>

<li><a href="Multilinear_algebra" title="Multilinear algebra">Multilinear algebra</a></li>

<li><a href="Tensor_algebra" title="Tensor algebra">Tensor algebra</a></li>

<li><a href="Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor calculus</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><div class="hlist"><ul><li><a href="Physics" title="Physics">Physics</a></li><li><a href="Engineering" title="Engineering">Engineering</a></li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Computer_vision" title="Computer vision">Computer vision</a></li>

<li><a href="Continuum_mechanics" title="Continuum mechanics">Continuum mechanics</a></li>

<li><a href="Electromagnetism" title="Electromagnetism">Electromagnetism</a></li>

<li><a href="General_relativity" title="General relativity">General relativity</a></li>

<li><a href="Transport_phenomena" title="Transport phenomena">Transport phenomena</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Notation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Abstract_index_notation" title="Abstract index notation">Abstract index notation</a></li>

<li><a href="Einstein_notation" title="Einstein notation">Einstein notation</a></li>

<li><a href="Index_notation" title="Index notation">Index notation</a></li>

<li><a href="Multi-index_notation" title="Multi-index notation">Multi-index notation</a></li>

<li><a href="Penrose_graphical_notation" title="Penrose graphical notation">Penrose graphical notation</a></li>

<li><a href="Ricci_calculus" title="Ricci calculus">Ricci calculus</a></li>

<li><a href="Tetrad_(index_notation)" class="mw-redirect" title="Tetrad (index notation)">Tetrad (index notation)</a></li>

<li><a href="Van_der_Waerden_notation" title="Van der Waerden notation">Van der Waerden notation</a></li>

<li><a href="Voigt_notation" title="Voigt notation">Voigt notation</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Tensor definitions</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Tensor_(intrinsic_definition)" title="Tensor (intrinsic definition)">Tensor (intrinsic definition)</a></li>

<li><a href="Tensor_field" title="Tensor field">Tensor field</a></li>

<li><a href="Tensor_density" title="Tensor density">Tensor density</a></li>

<li><a href="Tensors_in_curvilinear_coordinates" title="Tensors in curvilinear coordinates">Tensors in curvilinear coordinates</a></li>

<li><a href="Mixed_tensor" title="Mixed tensor">Mixed tensor</a></li>

<li><a href="Antisymmetric_tensor" title="Antisymmetric tensor">Antisymmetric tensor</a></li>

<li><a href="Symmetric_tensor" title="Symmetric tensor">Symmetric tensor</a></li>

<li><a href="Tensor_operator" title="Tensor operator">Tensor operator</a></li>

<li><a href="Tensor_bundle" title="Tensor bundle">Tensor bundle</a></li>

<li><a href="Two-point_tensor" title="Two-point tensor">Two-point tensor</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Operation_(mathematics)" title="Operation (mathematics)">Operations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Covariant_derivative" title="Covariant derivative">Covariant derivative</a></li>

<li><a href="Exterior_covariant_derivative" title="Exterior covariant derivative">Exterior covariant derivative</a></li>

<li><a href="Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li>

<li><a href="Exterior_product" class="mw-redirect" title="Exterior product">Exterior product</a></li>

<li><a href="Hodge_star_operator" title="Hodge star operator">Hodge star operator</a></li>

<li><a href="Lie_derivative" title="Lie derivative">Lie derivative</a></li>

<li><a href="Raising_and_lowering_indices" class="mw-redirect" title="Raising and lowering indices">Raising and lowering indices</a></li>

<li><a href="Symmetrization" title="Symmetrization">Symmetrization</a></li>

<li><a href="Tensor_contraction" title="Tensor contraction">Tensor contraction</a></li>

<li><a href="Tensor_product" title="Tensor product">Tensor product</a></li>

<li><a href="Transpose" title="Transpose">Transpose</a> (2nd-order tensors)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related abstractions</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Affine_connection" title="Affine connection">Affine connection</a></li>

<li><a href="Basis_(linear_algebra)" title="Basis (linear algebra)">Basis</a></li>

<li><a href="Cartan_formalism_(physics)" class="mw-redirect" title="Cartan formalism (physics)">Cartan formalism (physics)</a></li>

<li><a href="Connection_form" title="Connection form">Connection form</a></li>

<li><a href="Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">Covariance and contravariance of vectors</a></li>

<li><a href="Differential_form" title="Differential form">Differential form</a></li>

<li><a href="Dimension" title="Dimension">Dimension</a></li>

<li><a href="Exterior_form" class="mw-redirect" title="Exterior form">Exterior form</a></li>

<li><a href="Fiber_bundle" title="Fiber bundle">Fiber bundle</a></li>

<li><a href="Geodesic" title="Geodesic">Geodesic</a></li>

<li><a href="Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita connection</a></li>

<li><a href="Linear_map" title="Linear map">Linear map</a></li>

<li><a href="Manifold" title="Manifold">Manifold</a></li>

<li><a href="Matrix_(mathematics)" title="Matrix (mathematics)">Matrix</a></li>

<li><a href="Multivector" title="Multivector">Multivector</a></li>

<li><a href="Pseudotensor" title="Pseudotensor">Pseudotensor</a></li>

<li><a href="Spinor" title="Spinor">Spinor</a></li>

<li><a href="Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">Vector</a></li>

<li><a href="Vector_space" title="Vector space">Vector space</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Notable tensors</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Mathematics</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Kronecker_delta" title="Kronecker delta">Kronecker delta</a></li>

<li><a href="Levi-Civita_symbol" title="Levi-Civita symbol">Levi-Civita symbol</a></li>

<li><a href="Metric_tensor" title="Metric tensor">Metric tensor</a></li>

<li><a href="Nonmetricity_tensor" title="Nonmetricity tensor">Nonmetricity tensor</a></li>

<li><a href="Ricci_curvature" title="Ricci curvature">Ricci curvature</a></li>

<li><a href="Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a></li>

<li><a href="Torsion_tensor" title="Torsion tensor">Torsion tensor</a></li>

<li><a href="Weyl_tensor" title="Weyl tensor">Weyl tensor</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Physics</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Moment_of_inertia#Inertia_tensor" title="Moment of inertia">Moment of inertia</a></li>

<li><a href="Angular_momentum#Angular_momentum_in_relativistic_mechanics" title="Angular momentum">Angular momentum tensor</a></li>

<li><a href="Spin_tensor" title="Spin tensor">Spin tensor</a></li>

<li><a href="Cauchy_stress_tensor" title="Cauchy stress tensor">Cauchy stress tensor</a></li>

<li><a href="Stress%E2%80%93energy_tensor" title="Stress–energy tensor">stress–energy tensor</a></li>

<li><a href="Einstein_tensor" title="Einstein tensor">Einstein tensor</a></li>

<li><a href="Electromagnetic_tensor" title="Electromagnetic tensor">EM tensor</a></li>

<li><a href="Gluon_field_strength_tensor" title="Gluon field strength tensor">Gluon field strength tensor</a></li>

<li><a href="Metric_tensor_(general_relativity)" title="Metric tensor (general relativity)">Metric tensor (GR)</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Mathematician" title="Mathematician">Mathematicians</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="%C3%89lie_Cartan" title="Élie Cartan">Élie Cartan</a></li>

<li><a href="Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a></li>

<li><a href="Elwin_Bruno_Christoffel" title="Elwin Bruno Christoffel">Elwin Bruno Christoffel</a></li>

<li><a href="Albert_Einstein" title="Albert Einstein">Albert Einstein</a></li>

<li><a href="Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li>

<li><a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a></li>

<li><a href="Hermann_Grassmann" title="Hermann Grassmann">Hermann Grassmann</a></li>

<li><a href="Tullio_Levi-Civita" title="Tullio Levi-Civita">Tullio Levi-Civita</a></li>

<li><a href="Gregorio_Ricci-Curbastro" title="Gregorio Ricci-Curbastro">Gregorio Ricci-Curbastro</a></li>

<li><a href="Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a></li>

<li><a href="Jan_Arnoldus_Schouten" title="Jan Arnoldus Schouten">Jan Arnoldus Schouten</a></li>

<li><a href="Woldemar_Voigt" title="Woldemar Voigt">Woldemar Voigt</a></li>

<li><a href="Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a></li></ul>

</div></td></tr></tbody></table></div>

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