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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Tensor field</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with the <a href="Tensor_product_of_fields" title="Tensor product of fields">Tensor product of fields</a>.</div>

</p><p>In <a href="Mathematics" title="Mathematics">mathematics</a> and <a href="Physics" title="Physics">physics</a>, a <b>tensor field</b> is a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> assigning a <a href="Tensor" title="Tensor">tensor</a> to each point of a <a href="Region_(mathematics)" class="mw-redirect" title="Region (mathematics)">region</a> of a <a href="Mathematical_space" class="mw-redirect" title="Mathematical space">mathematical space</a> (typically a <a href="Euclidean_space" title="Euclidean space">Euclidean space</a> or <a href="Manifold" title="Manifold">manifold</a>) or of the <a href="Physical_space" class="mw-redirect" title="Physical space">physical space</a>. Tensor fields are used in <a href="Differential_geometry" title="Differential geometry">differential geometry</a>, <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, <a href="General_relativity" title="General relativity">general relativity</a>, in the analysis of <a href="Stress_(physics)" class="mw-redirect" title="Stress (physics)">stress</a> and <a href="Strain_tensor" class="mw-redirect" title="Strain tensor">strain</a> in material object, and in numerous applications in the <a href="Physical_sciences" class="mw-redirect" title="Physical sciences">physical sciences</a>. As a tensor is a generalization of a <a href="Scalar_(physics)" title="Scalar (physics)">scalar</a> (a pure number representing a value, for example speed) and a <a href="Vector_(physics)" class="mw-redirect" title="Vector (physics)">vector</a> (a magnitude and a direction, like velocity), a tensor field is a generalization of a <i><a href="Scalar_field" title="Scalar field">scalar field</a></i> and a <i><a href="Vector_field" title="Vector field">vector field</a></i> that assigns, respectively, a scalar or vector to each point of space. If a tensor <span class="texhtml mvar" style="font-style:italic;">A</span> is defined on a vector fields set <span class="texhtml mvar" style="font-style:italic;">X(M)</span> over a module <span class="texhtml mvar" style="font-style:italic;">M</span>, we call <span class="texhtml mvar" style="font-style:italic;">A</span> a tensor field on <span class="texhtml mvar" style="font-style:italic;">M</span>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

A tensor field, in common usage, is often referred to in the shorter form "tensor". For example, the <i><a href="Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a></i> refers a tensor <i>field</i>, as it associates a tensor to each point of a <a href="Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a>, a <a href="Topological_space" title="Topological space">topological space</a>.

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

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

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</math></span><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> be a <a href="Manifold" title="Manifold">manifold</a>, for instance the <a href="Euclidean_space" title="Euclidean space">Euclidean space</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

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</style><blockquote class="templatequote"><p><b>Definition.</b> A <b>tensor field</b> of type <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p,q)}">

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

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</math></span><img src="./9769c58523b9b639866a2d48e657d9c26911143a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.082ex; height:2.843ex;" alt="{\displaystyle (p,q)}" loading="lazy"></span> is a section

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\ \in \ \Gamma (M,V^{\otimes p}\otimes (V^{*})^{\otimes q})}">

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where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=TM}">

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</math></span><img src="./3208a44e5a906e7ebeef28f3071a666fa52ba7df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.964ex; height:2.176ex;" alt="{\displaystyle V=TM}" loading="lazy"></span> to be the <a href="Tangent_bundle" title="Tangent bundle">tangent bundle</a> of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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

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</math></span><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> (whose sections are called vector fields or contra variant vector fields in Physics) and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V^{*}=T^{*}M}">

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

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</math></span><img src="./b30eef5516de4cdb5041b2641ab405f6eeadf218.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.286ex; height:2.343ex;" alt="{\displaystyle V^{*}=T^{*}M}" loading="lazy"></span> is its dual bundle, the cotangent space (whose sections are called 1 forms, or covariant vector fields in Physics), and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \otimes }">

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

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</math></span><img src="./de29098f5a34ee296a505681a0d5e875070f2aea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \otimes }" loading="lazy"></span> is the <a href="Tensor_product_bundle" title="Tensor product bundle">tensor product</a> of vector bundles.</p></blockquote>

<p>Equivalently, a tensor field is a collection of elements <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{x}\in V_{x}^{\otimes p}\otimes (V_{x}^{*})^{\otimes q}}">

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<annotation encoding="application/x-tex">{\displaystyle T_{x}\in V_{x}^{\otimes p}\otimes (V_{x}^{*})^{\otimes q}}</annotation>

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</math></span><img src="./10eb6983cdc37092efe9aa6ffc588de5b107e447.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.513ex; height:3.176ex;" alt="{\displaystyle T_{x}\in V_{x}^{\otimes p}\otimes (V_{x}^{*})^{\otimes q}}" loading="lazy"></span> for every point <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in M}">

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

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</math></span><img src="./9df57d73e9532bb93a1439890bcddbc2806f5859.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.613ex; height:2.176ex;" alt="{\displaystyle x\in M}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \otimes }">

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

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</math></span><img src="./de29098f5a34ee296a505681a0d5e875070f2aea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \otimes }" loading="lazy"></span> now denotes the tensor product of vectors spaces, such that it constitutes a smooth map <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T:M\rightarrow V^{\otimes p}\otimes (V^{*})^{\otimes q}}">

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<annotation encoding="application/x-tex">{\displaystyle T:M\rightarrow V^{\otimes p}\otimes (V^{*})^{\otimes q}}</annotation>

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</math></span><img src="./29c396fd78d8d9e800bc4d5d0b3bf48e8220955e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.772ex; height:3.009ex;" alt="{\displaystyle T:M\rightarrow V^{\otimes p}\otimes (V^{*})^{\otimes q}}" loading="lazy"></span>. The elements <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{x}}">

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

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</math></span><img src="./b4339a8d96e6bf8706d6ba8b56c5349417b712ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.53ex; height:2.509ex;" alt="{\displaystyle T_{x}}" loading="lazy"></span> are called <a href="Tensor" title="Tensor">tensors</a>.

</p><p>Locally in a coordinate neighbourhood <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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

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</math></span><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"></span> with coordinates <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{1},\ldots x^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle x^{1},\ldots x^{n}}</annotation>

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</math></span><img src="./07dff718862076900a3aa5d4eb2e1faa8427ede0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.076ex; height:3.009ex;" alt="{\displaystyle x^{1},\ldots x^{n}}" loading="lazy"></span> we have a local basis (Vielbein) of vector fields <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{1}={\frac {\partial }{\partial x^{n}}}\ldots \partial _{n}={\frac {\partial }{\partial x_{n}}}}">

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<annotation encoding="application/x-tex">{\displaystyle \partial _{1}={\frac {\partial }{\partial x^{n}}}\ldots \partial _{n}={\frac {\partial }{\partial x_{n}}}}</annotation>

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</math></span><img src="./4a5a92cd4e7ab815de56e5a77e3b995559a9ed05.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.841ex; height:5.843ex;" alt="{\displaystyle \partial _{1}={\frac {\partial }{\partial x^{n}}}\ldots \partial _{n}={\frac {\partial }{\partial x_{n}}}}" loading="lazy"></span>, and a dual basis of 1 forms <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dx^{1},\ldots dx^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle dx^{1},\ldots dx^{n}}</annotation>

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</math></span><img src="./b932e8f6ac4ea73a63fc7a2c49a4ac6e9770a300.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.508ex; height:3.009ex;" alt="{\displaystyle dx^{1},\ldots dx^{n}}" loading="lazy"></span> so that

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<annotation encoding="application/x-tex">{\displaystyle dx^{i}(\partial _{j})=\partial _{j}x^{i}=\delta _{j}^{i}}</annotation>

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</math></span><img src="./757c77d935c30eea9666e5d1668b2265ff118276.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:19.711ex; height:3.676ex;" alt="{\displaystyle dx^{i}(\partial _{j})=\partial _{j}x^{i}=\delta _{j}^{i}}" loading="lazy"></span>. In the coordinate neighbourhood <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

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

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</math></span><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"></span> we then have

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<annotation encoding="application/x-tex">{\displaystyle T_{x}=T_{j_{1},\ldots ,j_{q}}^{i_{1},\ldots i_{p}}(x^{1},\ldots ,x^{n})\partial _{i_{1}}\otimes \cdots \otimes \partial _{i_{p}}\otimes dx^{j_{1}}\otimes \cdots \otimes dx^{j_{q}}}</annotation>

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where here and below we use Einstein summation conventions. Note that if we choose different coordinate system <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y^{1}\ldots y^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle y^{1}\ldots y^{n}}</annotation>

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</math></span><img src="./8b00c67936071c0d35367cb2d23e370951e0ef54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.091ex; height:3.009ex;" alt="{\displaystyle y^{1}\ldots y^{n}}" loading="lazy"></span> then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial }{\partial x^{i}}}={\frac {\partial y^{k}}{\partial x^{i}}}{\frac {\partial }{\partial y^{k}}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial x^{i}}}={\frac {\partial y^{k}}{\partial x^{i}}}{\frac {\partial }{\partial y^{k}}}}</annotation>

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</math></span><img src="./2dbf915bd8509b1bf256d5fba0f89d7e506913ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:16.189ex; height:6.676ex;" alt="{\displaystyle {\frac {\partial }{\partial x^{i}}}={\frac {\partial y^{k}}{\partial x^{i}}}{\frac {\partial }{\partial y^{k}}}}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dx^{j}={\frac {\partial x^{j}}{\partial y^{\ell }}}dy^{\ell }}">

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<annotation encoding="application/x-tex">{\displaystyle dx^{j}={\frac {\partial x^{j}}{\partial y^{\ell }}}dy^{\ell }}</annotation>

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</math></span><img src="./926fb161353226dad89749d31304e74de0595e98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:14.242ex; height:6.509ex;" alt="{\displaystyle dx^{j}={\frac {\partial x^{j}}{\partial y^{\ell }}}dy^{\ell }}" loading="lazy"></span> where the coordinates <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x^{1},\ldots ,x^{n})}">

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

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</math></span><img src="./28f6821ca603343f222e4b438613cf323b4abad1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.92ex; height:3.176ex;" alt="{\displaystyle (x^{1},\ldots ,x^{n})}" loading="lazy"></span> can be expressed in the coordinates <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (y^{1},\ldots y^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle (y^{1},\ldots y^{n}}</annotation>

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</math></span><img src="./cc9e86fb53e046c954861aab808d6c442b330d9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.643ex; height:3.176ex;" alt="{\displaystyle (y^{1},\ldots y^{n}}" loading="lazy"></span> and vice versa, so that

</p><p><span class="mwe-math-element mwe-math-element-block"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}T_{x}&=T_{j_{1},\ldots ,j_{q}}^{i_{1},\ldots i_{p}}(x^{1},\ldots ,x^{n}){\frac {\partial }{\partial x^{i_{1}}}}\otimes \cdots \otimes {\frac {\partial }{\partial x^{i_{p}}}}\otimes dx^{j_{1}}\otimes \cdots \otimes dx^{j_{q}}\\&=T_{j_{1},\ldots ,j_{q}}^{i_{1},\ldots i_{p}}(x^{1},\ldots ,x^{n}){\frac {\partial y^{k_{1}}}{\partial x^{i_{1}}}}\cdots {\frac {\partial y^{k_{p}}}{\partial x^{i_{p}}}}{\frac {\partial x^{j_{1}}}{\partial y^{\ell _{1}}}}\cdots {\frac {\partial x^{j_{q}}}{\partial y^{\ell _{q}}}}{\frac {\partial }{\partial y^{k_{1}}}}\otimes \cdots \otimes {\frac {\partial }{\partial y^{k_{p}}}}\otimes dy^{\ell _{1}}\otimes \cdots \otimes dy^{\ell _{q}}\\&=T_{\ell _{1},\cdots \ell _{q}}^{k_{1},\ldots ,k_{p}}(y^{1},\ldots y^{n}){\frac {\partial }{\partial y^{k_{1}}}}\otimes \cdots \otimes {\frac {\partial }{\partial y^{k_{p}}}}\otimes dy^{\ell _{1}}\otimes \cdots \otimes dy^{\ell _{q}}\\\end{aligned}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}T_{x}&=T_{j_{1},\ldots ,j_{q}}^{i_{1},\ldots i_{p}}(x^{1},\ldots ,x^{n}){\frac {\partial }{\partial x^{i_{1}}}}\otimes \cdots \otimes {\frac {\partial }{\partial x^{i_{p}}}}\otimes dx^{j_{1}}\otimes \cdots \otimes dx^{j_{q}}\\&=T_{j_{1},\ldots ,j_{q}}^{i_{1},\ldots i_{p}}(x^{1},\ldots ,x^{n}){\frac {\partial y^{k_{1}}}{\partial x^{i_{1}}}}\cdots {\frac {\partial y^{k_{p}}}{\partial x^{i_{p}}}}{\frac {\partial x^{j_{1}}}{\partial y^{\ell _{1}}}}\cdots {\frac {\partial x^{j_{q}}}{\partial y^{\ell _{q}}}}{\frac {\partial }{\partial y^{k_{1}}}}\otimes \cdots \otimes {\frac {\partial }{\partial y^{k_{p}}}}\otimes dy^{\ell _{1}}\otimes \cdots \otimes dy^{\ell _{q}}\\&=T_{\ell _{1},\cdots \ell _{q}}^{k_{1},\ldots ,k_{p}}(y^{1},\ldots y^{n}){\frac {\partial }{\partial y^{k_{1}}}}\otimes \cdots \otimes {\frac {\partial }{\partial y^{k_{p}}}}\otimes dy^{\ell _{1}}\otimes \cdots \otimes dy^{\ell _{q}}\\\end{aligned}}}</annotation>

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i.e.

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<annotation encoding="application/x-tex">{\displaystyle T_{\ell _{1},\cdots \ell _{q}}^{k_{1},\ldots ,k_{p}}(y^{1},\ldots y^{n})=T_{j_{1},\ldots ,j_{q}}^{i_{1},\ldots i_{p}}(x^{1},\ldots ,x^{n}){\frac {\partial y^{k_{1}}}{\partial x^{i_{1}}}}\cdots {\frac {\partial y^{k_{p}}}{\partial x^{i_{p}}}}{\frac {\partial x^{j_{1}}}{\partial y^{\ell _{1}}}}\cdots {\frac {\partial x^{j_{q}}}{\partial y^{\ell _{q}}}}}</annotation>

</semantics>

</math></span></span>

The system of indexed functions <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{j_{1},\ldots ,j_{q}}^{i_{1},\ldots i_{p}}(x^{1},\ldots ,x^{n})}">

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<annotation encoding="application/x-tex">{\displaystyle T_{j_{1},\ldots ,j_{q}}^{i_{1},\ldots i_{p}}(x^{1},\ldots ,x^{n})}</annotation>

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</math></span><img src="./96bcaf08e25ca48b7ec5981be917671c0379f939.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:19.314ex; height:4.343ex;" alt="{\displaystyle T_{j_{1},\ldots ,j_{q}}^{i_{1},\ldots i_{p}}(x^{1},\ldots ,x^{n})}" loading="lazy"></span> (one system for each choice of coordinate system) connected by transformations as above are the tensors in the definitions below.

</p><p><b>Remark</b>

One can, more generally, take

</mstyle>

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

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</math></span><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> to be any <a href="Vector_bundle" title="Vector bundle">vector bundle</a> on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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

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</math></span><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span>, and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V^{*}}">

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<annotation encoding="application/x-tex">{\displaystyle V^{*}}</annotation>

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</math></span><img src="./5910e6a94f4f7ee2ee85ceed9dacef3eff7a6242.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.971ex; height:2.343ex;" alt="{\displaystyle V^{*}}" loading="lazy"></span> its <a href="Dual_bundle" title="Dual bundle">dual bundle</a>. In that case can be a more general topological space. These sections are called tensors of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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

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</math></span><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> or tensors for short if no confusion is possible .

</p>

<div class="mw-heading mw-heading2"><h2 id="Geometric_introduction">Geometric introduction</h2></div>

<p>Intuitively, a vector field is best visualized as an "arrow" attached to each point of a region, with variable length and direction. One example of a vector field on a <a href="Curved_space" title="Curved space">curved space</a> is a weather map showing horizontal wind velocity at each point of the Earth's surface.

</p><p>Now consider more complicated fields. For example, if the manifold is Riemannian, then it has a metric field <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

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

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</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span>, such that given any two vectors <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v,w}">

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<annotation encoding="application/x-tex">{\displaystyle v,w}</annotation>

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</math></span><img src="./6425c6e94fa47976601cb44d7564b5d04dcfbfef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.826ex; height:2.009ex;" alt="{\displaystyle v,w}" loading="lazy"></span> at point <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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

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</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span>, their inner product is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{x}(v,w)}">

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<annotation encoding="application/x-tex">{\displaystyle g_{x}(v,w)}</annotation>

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</math></span><img src="./813ee644f6cd5cfc8eadef74e6752300a548d8bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.917ex; height:2.843ex;" alt="{\displaystyle g_{x}(v,w)}" loading="lazy"></span>. The field <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

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

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</math></span><img src="./d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"></span> could be given in matrix form, but it depends on a choice of coordinates. It could instead be given as an ellipsoid of radius 1 at each point, which is coordinate-free. Applied to the Earth's surface, this is <a href="Tissot's_indicatrix" title="Tissot's indicatrix">Tissot's indicatrix</a>.

</p><p>In general, we want to specify tensor fields in a coordinate-independent way: It should exist independently of latitude and longitude, or whatever particular "cartographic projection" we are using to introduce numerical coordinates.

</p>

<div class="mw-heading mw-heading2"><h2 id="Via_coordinate_transitions">Via coordinate transitions</h2></div>

<p>Following <a href="#CITEREFSchouten1951">Schouten (1951)</a> and <a href="#CITEREFMcConnell1957">McConnell (1957)</a>, the concept of a tensor relies on a concept of a reference frame (or <a href="Coordinate_system" title="Coordinate system">coordinate system</a>), which may be fixed (relative to some background reference frame), but in general may be allowed to vary within some class of transformations of these coordinate systems.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p><p>For example, coordinates belonging to the <i>n</i>-dimensional <a href="Real_coordinate_space" title="Real coordinate space">real coordinate space</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>

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</math></span><img src="./c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"></span> may be subjected to arbitrary <a href="Affine_transformation" title="Affine transformation">affine transformations</a>:

</p>

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<annotation encoding="application/x-tex">{\displaystyle x^{k}\mapsto A_{j}^{k}x^{j}+a^{k}}</annotation>

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</math></span><img src="./010e062eea17bbfa49a6985aacdcee34ffdc8f80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:16.263ex; height:3.676ex;" alt="{\displaystyle x^{k}\mapsto A_{j}^{k}x^{j}+a^{k}}" loading="lazy"></span></dd></dl>

<p>(with <i>n</i>-dimensional indices, <a href="Einstein_summation_convention" class="mw-redirect" title="Einstein summation convention">summation implied</a>). A covariant vector, or covector, is a system of functions <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{k}}">

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

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</math></span><img src="./d142b4083872eb72f81c1e20fd2c91d02b4a9838.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.216ex; height:2.009ex;" alt="{\displaystyle v_{k}}" loading="lazy"></span> that transforms under this affine transformation by the rule

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v_{k}\mapsto v_{i}A_{k}^{i}.}</annotation>

</semantics>

</math></span><img src="./5d49c9d0f5605f5de96ce0d4bfbb490f366eed71.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.236ex; height:3.176ex;" alt="{\displaystyle v_{k}\mapsto v_{i}A_{k}^{i}.}" loading="lazy"></span></dd></dl>

<p>The list of Cartesian coordinate basis vectors <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {e} _{k}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./afc2926ab5c390c48c2abe97d1cc7bf00e972677.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.314ex; height:2.009ex;" alt="{\displaystyle \mathbf {e} _{k}}" loading="lazy"></span> transforms as a covector, since under the affine transformation <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {e} _{k}\mapsto A_{k}^{i}\mathbf {e} _{i}}">

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./3d4d4b1a532eb33b7eb9e6fe32bd971ae0645ef1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.785ex; height:3.176ex;" alt="{\displaystyle \mathbf {e} _{k}\mapsto A_{k}^{i}\mathbf {e} _{i}}" loading="lazy"></span>. A contravariant vector is a system of functions <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v^{k}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./a5323ed8f5cc12a3bd966f11d3f4d7546b5c8268.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.216ex; height:2.676ex;" alt="{\displaystyle v^{k}}" loading="lazy"></span> of the coordinates that, under such an affine transformation undergoes a transformation

</p>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v^{k}\mapsto (A^{-1})_{j}^{k}v^{j}.}</annotation>

</semantics>

</math></span><img src="./7c0d59e9684eadd24ed80a1e8f642d57494fab4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:15.489ex; height:3.676ex;" alt="{\displaystyle v^{k}\mapsto (A^{-1})_{j}^{k}v^{j}.}" loading="lazy"></span></dd></dl>

<p>This is precisely the requirement needed to ensure that the quantity <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v^{k}\mathbf {e} _{k}}">

<msup>

</mrow>

</msup>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./1659eaba600f244b720c209a51cc0966b1116372.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.53ex; height:3.009ex;" alt="{\displaystyle v^{k}\mathbf {e} _{k}}" loading="lazy"></span> is an invariant object that does not depend on the coordinate system chosen. More generally, the coordinates of a tensor of valence (<i>p</i>,<i>q</i>) have <i>p</i> upper indices and <i>q</i> lower indices, with the transformation law being

</p>

<msub>

<msup>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</msup>

</mrow>

<msub>

</mrow>

</msub>

<msub>

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</msub>

</mrow>

</msub>

</mrow>

</msubsup>

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<msub>

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</msub>

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</msubsup>

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<msub>

</mrow>

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</msubsup>

<msub>

<msup>

</mrow>

</msubsup>

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</msubsup>

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</msup>

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</mrow>

</msubsup>

</mrow>

</msubsup>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {T^{i_{1}\cdots i_{p}}}_{j_{1}\cdots j_{q}}\mapsto A_{i'_{1}}^{i_{1}}\cdots A_{i'_{p}}^{i_{p}}{T^{i'_{1}\cdots i'_{p}}}_{j'_{1}\cdots j'_{q}}(A^{-1})_{j_{1}}^{j'_{1}}\cdots (A^{-1})_{j_{q}}^{j'_{q}}.}</annotation>

</semantics>

</math></span><img src="./8d937ee8e0f88cbd9574a8c52ccc74c37fae33f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:56.813ex; height:4.843ex;" alt="{\displaystyle {T^{i_{1}\cdots i_{p}}}_{j_{1}\cdots j_{q}}\mapsto A_{i'_{1}}^{i_{1}}\cdots A_{i'_{p}}^{i_{p}}{T^{i'_{1}\cdots i'_{p}}}_{j'_{1}\cdots j'_{q}}(A^{-1})_{j_{1}}^{j'_{1}}\cdots (A^{-1})_{j_{q}}^{j'_{q}}.}" loading="lazy"></span></dd></dl>

<p>The concept of a tensor field may be obtained by specializing the allowed coordinate transformations to be <a href="Smooth_function" class="mw-redirect" title="Smooth function">smooth</a> (or <a href="Differentiable_function" title="Differentiable function">differentiable</a>, <a href="Analytic_function" title="Analytic function">analytic</a>, etc.). A covector field is a function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./d142b4083872eb72f81c1e20fd2c91d02b4a9838.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.216ex; height:2.009ex;" alt="{\displaystyle v_{k}}" loading="lazy"></span> of the coordinates that transforms by the <a href="Jacobian_matrix" class="mw-redirect" title="Jacobian matrix">Jacobian</a> of the transition functions (in the given class). Likewise, a contravariant vector field <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v^{k}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./a5323ed8f5cc12a3bd966f11d3f4d7546b5c8268.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.216ex; height:2.676ex;" alt="{\displaystyle v^{k}}" loading="lazy"></span> transforms by the inverse Jacobian.

</p>

<div class="mw-heading mw-heading2"><h2 id="Tensor_bundles">Tensor bundles</h2></div>

<p>A tensor bundle is a <a href="Fiber_bundle" title="Fiber bundle">fiber bundle</a> where the fiber is a tensor product of any number of copies of the <a href="Tangent_space" title="Tangent space">tangent space</a> and/or <a href="Cotangent_space" title="Cotangent space">cotangent space</a> of the base space, which is a manifold. As such, the fiber is a <a href="Vector_space" title="Vector space">vector space</a> and the tensor bundle is a special kind of <a href="Vector_bundle" title="Vector bundle">vector bundle</a>.

</p><p>The vector bundle is a natural idea of "vector space depending continuously (or smoothly) on parameters" – the parameters being the points of a manifold <i>M</i>. For example, a <i>vector space of one dimension depending on an angle</i> could look like a <a href="M%C3%B6bius_strip" title="Möbius strip">Möbius strip</a> or alternatively like a <a href="Cylinder_(geometry)" class="mw-redirect" title="Cylinder (geometry)">cylinder</a>. Given a vector bundle <i>V</i> over <i>M</i>, the corresponding field concept is called a <i>section</i> of the bundle: for <i>m</i> varying over <i>M</i>, a choice of vector

</p>

<p>where <i>V<sub>m</sub></i> is the vector space "at" <i>m</i>.

</p><p>Since the <a href="Tensor_product" title="Tensor product">tensor product</a> concept is independent of any choice of basis, taking the tensor product of two vector bundles on <i>M</i> is routine. Starting with the <a href="Tangent_bundle" title="Tangent bundle">tangent bundle</a> (the bundle of <a href="Tangent_space" title="Tangent space">tangent spaces</a>) the whole apparatus explained at <a href="Component-free_treatment_of_tensors" class="mw-redirect" title="Component-free treatment of tensors">component-free treatment of tensors</a> carries over in a routine way – again independently of coordinates, as mentioned in the introduction.

</p><p>We therefore can give a definition of <b>tensor field</b>, namely as a <a href="Section_(fiber_bundle)" title="Section (fiber bundle)">section</a> of some <a href="Tensor_bundle" title="Tensor bundle">tensor bundle</a>. (There are vector bundles that are not tensor bundles: the Möbius band for instance.) This is then guaranteed geometric content, since everything has been done in an intrinsic way. More precisely, a tensor field assigns to any given point of the manifold a tensor in the space

</p>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V\otimes \cdots \otimes V\otimes V^{*}\otimes \cdots \otimes V^{*},}</annotation>

</semantics>

</math></span><img src="./9b8acf562eadd38731e92d26cd1543fdf8c9c029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.812ex; height:2.676ex;" alt="{\displaystyle V\otimes \cdots \otimes V\otimes V^{*}\otimes \cdots \otimes V^{*},}" loading="lazy"></span></dd></dl>

<p>where <i>V</i> is the <a href="Tangent_space" title="Tangent space">tangent space</a> at that point and <i>V</i><sup>∗</sup> is the <a href="Cotangent_space" title="Cotangent space">cotangent space</a>. See also <a href="Tangent_bundle" title="Tangent bundle">tangent bundle</a> and <a href="Cotangent_bundle" title="Cotangent bundle">cotangent bundle</a>.

</p><p>Given two tensor bundles <i>E</i> → <i>M</i> and <i>F</i> → <i>M</i>, a linear map <i>A</i>: Γ(<i>E</i>) → Γ(<i>F</i>) from the space of sections of <i>E</i> to sections of <i>F</i> can be considered itself as a tensor section of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle E^{*}\otimes F}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle E^{*}\otimes F}</annotation>

</semantics>

</math></span><img src="./a962578c4bb2134f1df2f3a9f17ffda172ef897c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.61ex; height:1.843ex;" alt="{\displaystyle \scriptstyle E^{*}\otimes F}" loading="lazy"></span> if and only if it satisfies <i>A</i>(<i>fs</i>) = <i>fA</i>(<i>s</i>), for each section <i>s</i> in Γ(<i>E</i>) and each smooth function <i>f</i> on <i>M</i>. Thus a tensor section is not only a linear map on the vector space of sections, but a <i>C</i><sup>∞</sup>(<i>M</i>)-linear map on the <a href="Module_(mathematics)" title="Module (mathematics)">module</a> of sections. This property is used to check, for example, that even though the <a href="Lie_derivative" title="Lie derivative">Lie derivative</a> and <a href="Covariant_derivative" title="Covariant derivative">covariant derivative</a> are not tensors, the <a href="Torsion_tensor" title="Torsion tensor">torsion</a> and <a href="Affine_connection" title="Affine connection">curvature tensors</a> built from them are.

</p>

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

<p>The notation for tensor fields can sometimes be confusingly similar to the notation for tensor spaces. Thus, the tangent bundle <i>TM</i> = <i>T</i>(<i>M</i>) might sometimes be written as

</p>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{0}^{1}(M)=T(M)=TM}</annotation>

</semantics>

</math></span><img src="./1716f0bf2e9b608de2dc1add0008e84481403144.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.189ex; height:3.176ex;" alt="{\displaystyle T_{0}^{1}(M)=T(M)=TM}" loading="lazy"></span></dd></dl>

<p>to emphasize that the tangent bundle is the range space of the (1,0) tensor fields (i.e., vector fields) on the manifold <i>M</i>. This should not be confused with the very similar looking notation

</p>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle T_{0}^{1}(V)}</annotation>

</semantics>

</math></span><img src="./e12edb0bdfa786843471270e2932b3cefdcaa195.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.371ex; height:3.176ex;" alt="{\displaystyle T_{0}^{1}(V)}" loading="lazy"></span>;</dd></dl>

<p>in the latter case, we just have one tensor space, whereas in the former, we have a tensor space defined for each point in the manifold <i>M</i>.

</p><p>Curly (script) letters are sometimes used to denote the set of <a href="Smooth_function" class="mw-redirect" title="Smooth function">infinitely-differentiable</a> tensor fields on <i>M</i>. Thus,

</p>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {T}}_{n}^{m}(M)}</annotation>

</semantics>

</math></span><img src="./63e5d0a5eb83c73d4e204fcee51b411968fbc1eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.063ex; height:2.843ex;" alt="{\displaystyle {\mathcal {T}}_{n}^{m}(M)}" loading="lazy"></span></dd></dl>

<p>are the sections of the (<i>m</i>,<i>n</i>) tensor bundle on <i>M</i> that are infinitely-differentiable. A tensor field is an element of this set.

</p>

<div class="mw-heading mw-heading2"><h2 id="Tensor_fields_as_multilinear_forms">Tensor fields as multilinear forms</h2></div>

<p>There is another more abstract (but often useful) way of characterizing tensor fields on a manifold <i>M</i>, which makes tensor fields into honest tensors (i.e. <i>single</i> multilinear mappings), though of a different type (although this is <i>not</i> usually why one often says "tensor" when one really means "tensor field"). First, we may consider the set of all smooth (<i>C</i><sup>∞</sup>) vector fields on <i>M</i>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {X}}(M):={\mathcal {T}}_{0}^{1}(M)}">

</mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {X}}(M):={\mathcal {T}}_{0}^{1}(M)}</annotation>

</semantics>

</math></span><img src="./16169969df1c40d30cd4d5e9b71285cff2dd3f0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.11ex; height:3.176ex;" alt="{\displaystyle {\mathfrak {X}}(M):={\mathcal {T}}_{0}^{1}(M)}" loading="lazy"></span> (see the section on notation above) as a single space – a <a href="Module_(mathematics)" title="Module (mathematics)">module</a> over the <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a> of smooth functions, <i>C</i><sup>∞</sup>(<i>M</i>), by pointwise scalar multiplication. The notions of multilinearity and tensor products extend easily to the case of modules over any <a href="Commutative_ring" title="Commutative ring">commutative ring</a>.

</p><p>As a motivating example, consider the space <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega ^{1}(M)={\mathcal {T}}_{1}^{0}(M)}">

<msup>

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</msup>

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</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega ^{1}(M)={\mathcal {T}}_{1}^{0}(M)}</annotation>

</semantics>

</math></span><img src="./f80270e8c59076d114acf6d291fab9b3540e101a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.525ex; height:3.176ex;" alt="{\displaystyle \Omega ^{1}(M)={\mathcal {T}}_{1}^{0}(M)}" loading="lazy"></span> of smooth covector fields (<a href="Differential_form" title="Differential form">1-forms</a>), also a module over the smooth functions. These act on smooth vector fields to yield smooth functions by pointwise evaluation, namely, given a covector field <i>ω</i> and a vector field <i>X</i>, we define

</p>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {\omega }}(X)(p):=\omega (p)(X(p)).}</annotation>

</semantics>

</math></span><img src="./5259fe4d292ddd8b072771bb438c82f60b49014d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.799ex; height:2.843ex;" alt="{\displaystyle {\tilde {\omega }}(X)(p):=\omega (p)(X(p)).}" loading="lazy"></span></dd></dl>

<p>Because of the pointwise nature of everything involved, the action of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {\omega }}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./e60cbeebb31255ddce04c7526d2631bfd49fb8bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:2.176ex;" alt="{\displaystyle {\tilde {\omega }}}" loading="lazy"></span> on <i>X</i> is a <i>C</i><sup>∞</sup>(<i>M</i>)-linear map, that is,

</p>

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {\omega }}(fX)(p)=\omega (p)((fX)(p))=\omega (p)(f(p)X(p))=f(p)\omega (p)(X(p))=(f\omega )(p)(X(p))=(f{\tilde {\omega }})(X)(p)}</annotation>

</semantics>

</math></span><img src="./c1c4e7ebc2d3ecba9d0ee4fed7482ce91427473d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:95.747ex; height:2.843ex;" alt="{\displaystyle {\tilde {\omega }}(fX)(p)=\omega (p)((fX)(p))=\omega (p)(f(p)X(p))=f(p)\omega (p)(X(p))=(f\omega )(p)(X(p))=(f{\tilde {\omega }})(X)(p)}" loading="lazy"></span></dd></dl>

<p>for any <i>p</i> in <i>M</i> and smooth function <i>f</i>. Thus we can regard covector fields not just as sections of the cotangent bundle, but also linear mappings of vector fields into functions. By the double-dual construction, vector fields can similarly be expressed as mappings of covector fields into functions (namely, we could start "natively" with covector fields and work up from there).

</p><p>In a complete parallel to the construction of ordinary single tensors (not tensor fields!) on <i>M</i> as multilinear maps on vectors and covectors, we can regard general (<i>k</i>,<i>l</i>) tensor fields on <i>M</i> as <i>C</i><sup>∞</sup>(<i>M</i>)-multilinear maps defined on <i>k</i> copies of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {X}}(M)}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./98adcdb076cb8069c2d19d06030cf7e2c2f237f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.923ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {X}}(M)}" loading="lazy"></span> and <i>l</i> copies of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega ^{1}(M)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./cb6e51da90c4ca5b118c211c7b6c89105e3b7c95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.984ex; height:3.176ex;" alt="{\displaystyle \Omega ^{1}(M)}" loading="lazy"></span> into <i>C</i><sup>∞</sup>(<i>M</i>).

</p><p>Now, given any arbitrary mapping <i>T</i> from a product of <i>k</i> copies of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {X}}(M)}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./98adcdb076cb8069c2d19d06030cf7e2c2f237f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.923ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {X}}(M)}" loading="lazy"></span> and <i>l</i> copies of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega ^{1}(M)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./cb6e51da90c4ca5b118c211c7b6c89105e3b7c95.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.984ex; height:3.176ex;" alt="{\displaystyle \Omega ^{1}(M)}" loading="lazy"></span> into <i>C</i><sup>∞</sup>(<i>M</i>), it turns out that it arises from a tensor field on <i>M</i> if and only if it is multilinear over <i>C</i><sup>∞</sup>(<i>M</i>). Namely <i>C</i><sup>∞</sup>(<i>M</i>)-module of tensor fields of type <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (k,l)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./c6596d38bbba96f6eda9c984af791cdd7790a802.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.748ex; height:2.843ex;" alt="{\displaystyle (k,l)}" loading="lazy"></span> over <i>M</i> is canonically isomorphic to <i>C</i><sup>∞</sup>(<i>M</i>)-module of <i>C</i><sup>∞</sup>(<i>M</i>)-<a href="Multilinear_form" title="Multilinear form">multilinear forms</a>

</p>

<mrow>

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<annotation encoding="application/x-tex">{\displaystyle \underbrace {\Omega ^{1}(M)\times \ldots \times \Omega ^{1}(M)} _{l\ \mathrm {times} }\times \underbrace {{\mathfrak {X}}(M)\times \ldots \times {\mathfrak {X}}(M)} _{k\ \mathrm {times} }\to C^{\infty }(M).}</annotation>

</semantics>

</math></span><img src="./24d69c0835100689d2aeb07011f2b215e3622f6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:57.647ex; height:6.676ex;" alt="{\displaystyle \underbrace {\Omega ^{1}(M)\times \ldots \times \Omega ^{1}(M)} _{l\ \mathrm {times} }\times \underbrace {{\mathfrak {X}}(M)\times \ldots \times {\mathfrak {X}}(M)} _{k\ \mathrm {times} }\to C^{\infty }(M).}" loading="lazy"></span><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></dd></dl>

<p>This kind of multilinearity implicitly expresses the fact that we're really dealing with a pointwise-defined object, i.e. a tensor field, as opposed to a function which, even when evaluated at a single point, depends on all the values of vector fields and 1-forms simultaneously.

</p><p>A frequent example application of this general rule is showing that the <a href="Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita connection</a>, which is a mapping of smooth vector fields <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X,Y)\mapsto \nabla _{X}Y}">

<msub>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (X,Y)\mapsto \nabla _{X}Y}</annotation>

</semantics>

</math></span><img src="./8c2cfcc75d0612d0510f1335ad8bb4df160f1f55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.552ex; height:2.843ex;" alt="{\displaystyle (X,Y)\mapsto \nabla _{X}Y}" loading="lazy"></span> taking a pair of vector fields to a vector field, does not define a tensor field on <i>M</i>. This is because it is only <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"></span>-linear in <i>Y</i> (in place of full <i>C</i><sup>∞</sup>(<i>M</i>)-linearity, it satisfies the <i>Leibniz rule,</i> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla _{X}(fY)=(Xf)Y+f\nabla _{X}Y}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \nabla _{X}(fY)=(Xf)Y+f\nabla _{X}Y}</annotation>

</semantics>

</math></span><img src="./fca329a81e231f8ac0cdc94a434ac0673a0cfd52.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.83ex; height:2.843ex;" alt="{\displaystyle \nabla _{X}(fY)=(Xf)Y+f\nabla _{X}Y}" loading="lazy"></span>)). Nevertheless, it must be stressed that even though it is not a tensor field, it still qualifies as a geometric object with a component-free interpretation.

</p>

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

<p>The curvature tensor is discussed in differential geometry and the <a href="Stress%E2%80%93energy_tensor" title="Stress–energy tensor">stress–energy tensor</a> is important in physics, and these two tensors are related by Einstein's theory of <a href="General_relativity" title="General relativity">general relativity</a>.

</p><p>In <a href="Electromagnetism" title="Electromagnetism">electromagnetism</a>, the electric and magnetic fields are combined into an <a href="Electromagnetic_tensor" title="Electromagnetic tensor">electromagnetic tensor field</a>.

</p><p><a href="Differential_form" title="Differential form">Differential forms</a>, used in defining integration on manifolds, are a type of tensor field.

</p>

<div class="mw-heading mw-heading2"><h2 id="Tensor_calculus">Tensor calculus</h2></div>

<p>In <a href="Theoretical_physics" title="Theoretical physics">theoretical physics</a> and other fields, <a href="Differential_equation" title="Differential equation">differential equations</a> posed in terms of tensor fields provide a very general way to express relationships that are both geometric in nature (guaranteed by the tensor nature) and conventionally linked to <a href="Differential_calculus" title="Differential calculus">differential calculus</a>. Even to formulate such equations requires a fresh notion, the <a href="Covariant_derivative" title="Covariant derivative">covariant derivative</a>. This handles the formulation of variation of a tensor field <i>along</i> a <a href="Vector_field" title="Vector field">vector field</a>. The original <i>absolute differential calculus</i> notion, which was later called <i><a href="Tensor_calculus" class="mw-redirect" title="Tensor calculus">tensor calculus</a></i>, led to the isolation of the geometric concept of <a href="Connection_(differential_geometry)" class="mw-redirect" title="Connection (differential geometry)">connection</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Twisting_by_a_line_bundle">Twisting by a line bundle</h2></div>

<p>An extension of the tensor field idea incorporates an extra <a href="Line_bundle" title="Line bundle">line bundle</a> <i>L</i> on <i>M</i>. If <i>W</i> is the tensor product bundle of <i>V</i> with <i>L</i>, then <i>W</i> is a bundle of vector spaces of just the same dimension as <i>V</i>. This allows one to define the concept of <b>tensor density</b>, a 'twisted' type of tensor field. A <i>tensor density</i> is the special case where <i>L</i> is the bundle of <i>densities on a manifold</i>, namely the <a href="Determinant_bundle" class="mw-redirect" title="Determinant bundle">determinant bundle</a> of the <a href="Cotangent_bundle" title="Cotangent bundle">cotangent bundle</a>. (To be strictly accurate, one should also apply the <a href="Absolute_value" title="Absolute value">absolute value</a> to the <a href="Topology" title="Topology">transition functions</a> – this makes little difference for an <a href="Orientable_manifold" class="mw-redirect" title="Orientable manifold">orientable manifold</a>.) For a more traditional explanation see the <a href="Tensor_density" title="Tensor density">tensor density</a> article.

</p><p>One feature of the bundle of densities (again assuming orientability) <i>L</i> is that <i>L</i><sup><i>s</i></sup> is well-defined for real number values of <i>s</i>; this can be read from the transition functions, which take strictly positive real values. This means for example that we can take a <i>half-density</i>, the case where <i>s</i> = <style data-mw-deduplicate="TemplateStyles:r1214402035">

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.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}

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</style><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">2</span></span>⁠</span>. In general we can take sections of <i>W</i>, the tensor product of <i>V</i> with <i>L</i><sup><i>s</i></sup>, and consider <b>tensor density fields</b> with weight <i>s</i>.

</p><p>Half-densities are applied in areas such as defining <a href="Integral_operator" title="Integral operator">integral operators</a> on manifolds, and <a href="Geometric_quantization" title="Geometric quantization">geometric quantization</a>.

</p>

<p>When <i>M</i> is a <a href="Euclidean_space" title="Euclidean space">Euclidean space</a> and all the fields are taken to be invariant by <a href="Translation_(geometry)" title="Translation (geometry)">translations</a> by the vectors of <i>M</i>, we get back to a situation where a tensor field is synonymous with a tensor 'sitting at the origin'. This does no great harm, and is often used in applications. As applied to tensor densities, it <i>does</i> make a difference. The bundle of densities cannot seriously be defined 'at a point'; and therefore a limitation of the contemporary mathematical treatment of tensors is that tensor densities are defined in a roundabout fashion.

</p>

<div class="mw-heading mw-heading2"><h2 id="Cocycles_and_chain_rules">Cocycles and chain rules</h2></div>

<p>As an advanced explanation of the <i>tensor</i> concept, one can interpret the <a href="Chain_rule" title="Chain rule">chain rule</a> in the multivariable case, as applied to coordinate changes, also as the requirement for self-consistent concepts of tensor giving rise to tensor fields.

</p><p>Abstractly, we can identify the chain rule as a 1-<a href="Cochain_(algebraic_topology)" class="mw-redirect" title="Cochain (algebraic topology)">cocycle</a>. It gives the consistency required to define the tangent bundle in an intrinsic way. The other vector bundles of tensors have comparable cocycles, which come from applying <a href="Functorial" class="mw-redirect" title="Functorial">functorial</a> properties of tensor constructions to the chain rule itself; this is why they also are intrinsic (read, 'natural') concepts.

</p><p>What is usually spoken of as the 'classical' approach to tensors tries to read this backwards – and is therefore a heuristic, <i>post hoc</i> approach rather than truly a foundational one. Implicit in defining tensors by how they transform under a coordinate change is the kind of self-consistency the cocycle expresses. The construction of tensor densities is a 'twisting' at the cocycle level. Geometers have not been in any doubt about the <i>geometric</i> nature of tensor <i>quantities</i>; this kind of <a href="Descent_(category_theory)" class="mw-redirect" title="Descent (category theory)">descent</a> argument justifies abstractly the whole theory.

</p>

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

<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>

<p>The concept of a tensor field can be generalized by considering objects that transform differently. An object that transforms as an ordinary tensor field under coordinate transformations, except that it is also multiplied by the determinant of the <a href="Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a> of the inverse coordinate transformation to the <i>w</i>th power, is called a tensor density with weight <i>w</i>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Invariantly, in the language of multilinear algebra, one can think of tensor densities as <a href="Multilinear_map" title="Multilinear map">multilinear maps</a> taking their values in a <a href="Density_bundle" class="mw-redirect" title="Density bundle">density bundle</a> such as the (1-dimensional) space of <i>n</i>-forms (where <i>n</i> is the dimension of the space), as opposed to taking their values in just <b>R</b>. Higher "weights" then just correspond to taking additional tensor products with this space in the range.

</p><p>A special case are the scalar densities. Scalar 1-densities are especially important because it makes sense to define their integral over a manifold. They appear, for instance, in the <a href="Einstein%E2%80%93Hilbert_action" title="Einstein–Hilbert action">Einstein–Hilbert action</a> in general relativity. The most common example of a scalar 1-density is the <a href="Volume_element" title="Volume element">volume element</a>, which in the presence of a metric tensor <i>g</i> is the square root of its <a href="Determinant" title="Determinant">determinant</a> in coordinates, denoted <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {\det g}}}">

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {\det g}}}</annotation>

</semantics>

</math></span><img src="./1d1c11a4b11cee04bca273c82817fffb72ecaa96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.056ex; height:3.509ex;" alt="{\displaystyle {\sqrt {\det g}}}" loading="lazy"></span>. The metric tensor is a covariant tensor of order 2, and so its determinant scales by the square of the coordinate transition:

</p>

<msup>

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<annotation encoding="application/x-tex">{\displaystyle \det(g')=\left(\det {\frac {\partial x}{\partial x'}}\right)^{2}\det(g),}</annotation>

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</math></span><img src="./9137cf596737b5963c21f27e2e5f2721bbdac061.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.39ex; height:6.509ex;" alt="{\displaystyle \det(g')=\left(\det {\frac {\partial x}{\partial x'}}\right)^{2}\det(g),}" loading="lazy"></span></dd></dl>

<p>which is the transformation law for a scalar density of weight +2.

</p><p>More generally, any tensor density is the product of an ordinary tensor with a scalar density of the appropriate weight. In the language of <a href="Vector_bundle" title="Vector bundle">vector bundles</a>, the determinant bundle of the <a href="Tangent_bundle" title="Tangent bundle">tangent bundle</a> is a <a href="Line_bundle" title="Line bundle">line bundle</a> that can be used to 'twist' other bundles <i>w</i> times. While locally the more general transformation law can indeed be used to recognise these tensors, there is a global question that arises, reflecting that in the transformation law one may write either the Jacobian determinant, or its absolute value. Non-integral powers of the (positive) transition functions of the bundle of densities make sense, so that the weight of a density, in that sense, is not restricted to integer values. Restricting to changes of coordinates with positive Jacobian determinant is possible on <a href="Orientable_manifold" class="mw-redirect" title="Orientable manifold">orientable manifolds</a>, because there is a consistent global way to eliminate the minus signs; but otherwise the line bundle of densities and the line bundle of <i>n</i>-forms are distinct. For more on the intrinsic meaning, see <i><a href="Density_on_a_manifold" title="Density on a manifold">Density on a manifold</a></i>.

</p>

<ul><li><a href="Bitensor" title="Bitensor">Bitensor</a> – Tensorial object depending on two points in a manifold</li>

<li><a href="Jet_bundle" title="Jet bundle">Jet bundle</a> – Construction in differential topology</li>

<li><a href="Ricci_calculus" title="Ricci calculus">Ricci calculus</a> – Tensor index notation for tensor-based calculations</li>

<li><a href="Spinor_field" class="mw-redirect" title="Spinor field">Spinor field</a> – Geometric structure<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li></ul>

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

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</style><div class="reflist">

</div>

<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">O'Neill, Barrett. <i>Semi-Riemannian Geometry With Applications to Relativity</i></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">The term "<a href="Affinor" class="mw-redirect" title="Affinor">affinor</a>" employed in the English translation of Schouten is no longer in use.</span>

</li>

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.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

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</style><cite id="CITEREFClaudio_Gorodski" class="citation web cs1">Claudio Gorodski. <a rel="nofollow" class="external text" href="https://www.ime.usp.br/~gorodski/teaching/mat5799-2015/gorodski-smooth-manifolds-2013.pdf">"Notes on Smooth Manifolds"</a> <span class="cs1-format">(PDF)</span><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-06-24</span></span>.</cite> </span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Tensor_density">"Tensor density"</a>, <i><a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite></span>

</li>

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

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

<ul><li><cite id="CITEREFO'neill1983" class="citation book cs1">O'neill, Barrett (1983). <i>Semi-Riemannian Geometry With Applications to Relativity</i>. Elsevier Science. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780080570570</bdi>.</cite></li>

<li><cite id="CITEREFFrankel2012" class="citation cs2"><a href="Theodore_Frankel" title="Theodore Frankel">Frankel, T.</a> (2012), <i>The Geometry of Physics (3rd edition)</i>, Cambridge University Press, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-107-60260-1</bdi></cite>.</li>

<li><cite id="CITEREFLambourne_[Open_University]2010" class="citation cs2">Lambourne [Open University], R.J.A. (2010), <i>Relativity, Gravitation, and Cosmology</i>, Cambridge University Press, <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/2010rgc..book.....L">2010rgc..book.....L</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-13138-4</bdi></cite>.</li>

<li><cite id="CITEREFLernerTrigg1991" class="citation cs2"><a href="Rita_G._Lerner" title="Rita G. Lerner">Lerner, R.G.</a>; Trigg, G.L. (1991), <i>Encyclopaedia of Physics (2nd Edition)</i>, VHC Publishers</cite>.</li>

<li><cite id="CITEREFMcConnell1957" class="citation cs2">McConnell, A. J. (1957), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ZCP0AwAAQBAJ"><i>Applications of Tensor Analysis</i></a>, Dover Publications, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780486145020</bdi></cite> <span class="cs1-hidden-error citation-comment"><code class="cs1-code">{{citation}}</code>: </span><span class="cs1-hidden-error citation-comment">ISBN / Date incompatibility (help)</span>.</li>

<li><cite id="CITEREFMcMahon2006" class="citation cs2">McMahon, D. (2006), <i>Relativity DeMystified</i>, McGraw Hill (USA), <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-07-145545-0</bdi></cite>.</li>

<li><cite id="CITEREFC._Misner,_K._S._Thorne,_J._A._Wheeler1973" class="citation cs2">C. Misner, K. S. Thorne, J. A. Wheeler (1973), <i><a href="Gravitation_(book)" title="Gravitation (book)">Gravitation</a></i>, W.H. Freeman & Co, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-7167-0344-0</bdi></cite><span class="cs1-maint citation-comment"><code class="cs1-code">{{citation}}</code>: CS1 maint: multiple names: authors list (link)</span>.</li>

<li><cite id="CITEREFParker1994" class="citation cs2">Parker, C.B. (1994), <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/mcgrawhillencycl1993park"><i>McGraw Hill Encyclopaedia of Physics (2nd Edition)</i></a></span>, McGraw Hill, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-07-051400-3</bdi></cite>.</li>

<li><cite id="CITEREFSchouten1951" class="citation cs2"><a href="Jan_Arnoldus_Schouten" title="Jan Arnoldus Schouten">Schouten, Jan Arnoldus</a> (1951), <i>Tensor Analysis for Physicists</i>, Oxford University Press</cite>.</li>

<li><cite id="CITEREFSteenrod1999" class="citation book cs1"><a href="Norman_Steenrod" title="Norman Steenrod">Steenrod, Norman</a> (5 April 1999). <i>The Topology of Fibre Bundles</i>. Princeton Mathematical Series. Vol. 14. Princeton, N.J.: Princeton University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-691-00548-5</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/40734875">40734875</a>.</cite></li></ul>

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</style><div id="Tensors176" style="font-size:114%;margin:0 4em"><a href="Tensor" title="Tensor">Tensors</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div><i><a href="Glossary_of_tensor_theory" title="Glossary of tensor theory">Glossary of tensor theory</a></i></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<br>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_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<br>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>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Manifolds_(Glossary,_List,_Category)274" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Manifolds_(Glossary,_List,_Category)274" style="font-size:114%;margin:0 4em"><a href="Manifold" title="Manifold">Manifolds</a> (<a href="Glossary_of_differential_geometry_and_topology" title="Glossary of differential geometry and topology">Glossary</a>, <a href="List_of_manifolds" title="List of manifolds">List</a>, Category)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</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="Topological_manifold" title="Topological manifold">Topological manifold</a>

<ul><li><a href="Atlas_(topology)" title="Atlas (topology)">Atlas</a></li></ul></li>

<li><a href="Differentiable_manifold" title="Differentiable manifold">Differentiable/Smooth manifold</a>

<ul><li><a href="Differential_structure" title="Differential structure">Differential structure</a></li>

<li><a href="Smooth_structure" title="Smooth structure">Smooth atlas</a></li></ul></li>

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

<li><a href="Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a></li>

<li><a href="Smoothness" title="Smoothness">Smooth map</a></li>

<li><a href="Submersion_(mathematics)" title="Submersion (mathematics)">Submersion</a></li>

<li><a href="Pushforward_(differential)" title="Pushforward (differential)">Pushforward</a></li>

<li><a href="Tangent_space" title="Tangent space">Tangent space</a></li>

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

<li><a href="Vector_field" title="Vector field">Vector field</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Main results <span style="font-size: 85%;">(list)</span></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="Atiyah%E2%80%93Singer_index_theorem" title="Atiyah–Singer index theorem">Atiyah–Singer index</a></li>

<li><a href="Darboux's_theorem" title="Darboux's theorem">Darboux's</a></li>

<li><a href="Frobenius_theorem_(differential_topology)" title="Frobenius theorem (differential topology)">Frobenius</a></li>

<li><a href="Generalized_Stokes_theorem" title="Generalized Stokes theorem">Generalized Stokes</a></li>

<li><a href="Hopf%E2%80%93Rinow_theorem" title="Hopf–Rinow theorem">Hopf–Rinow</a></li>

<li><a href="Noether's_theorem" title="Noether's theorem">Noether's</a></li>

<li><a href="Whitney_embedding_theorem" title="Whitney embedding theorem">Whitney embedding</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Smoothness" title="Smoothness">Maps</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="Differentiable_curve" title="Differentiable curve">Curve</a></li>

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

<ul><li><a href="Local_diffeomorphism" title="Local diffeomorphism">Local</a></li></ul></li>

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

<li><a href="Exponential_map_(Riemannian_geometry)" title="Exponential map (Riemannian geometry)">Exponential map</a>

<ul><li><a href="Exponential_map_(Lie_theory)" title="Exponential map (Lie theory)">in Lie theory</a></li></ul></li>

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

<li><a href="Immersion_(mathematics)" title="Immersion (mathematics)">Immersion</a></li>

<li><a href="Integral_curve" title="Integral curve">Integral curve</a></li>

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

<li><a href="Section_(fiber_bundle)" title="Section (fiber bundle)">Section</a></li>

<li><a href="Submersion_(mathematics)" title="Submersion (mathematics)">Submersion</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types of<br>manifolds</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="Closed_manifold" title="Closed manifold">Closed</a></li>

<li><a href="Collapsing_manifold" title="Collapsing manifold">Collapsing</a></li>

<li><a href="Complete_manifold" title="Complete manifold">Complete</a></li>

<li>(<a href="Almost_complex_manifold" title="Almost complex manifold">Almost</a>) <a href="Complex_manifold" title="Complex manifold">Complex</a></li>

<li>(<a href="Almost-contact_manifold" title="Almost-contact manifold">Almost</a>) <a href="Contact_manifold" class="mw-redirect" title="Contact manifold">Contact</a></li>

<li><a href="Fibered_manifold" title="Fibered manifold">Fibered</a></li>

<li><a href="Finsler_manifold" title="Finsler manifold">Finsler</a></li>

<li>(<a href="Almost_flat_manifold" title="Almost flat manifold">Almost</a>) <a href="Flat_manifold" title="Flat manifold">Flat</a></li>

<li><a href="G-structure_on_a_manifold" title="G-structure on a manifold">G-structure</a></li>

<li><a href="Hadamard_manifold" title="Hadamard manifold">Hadamard</a></li>

<li><a href="Hermitian_manifold" title="Hermitian manifold">Hermitian</a></li>

<li><a href="Hyperbolic_manifold" title="Hyperbolic manifold">Hyperbolic</a></li>

<li><a href="K%C3%A4hler_manifold" title="Kähler manifold">Kähler</a></li>

<li><a href="Kenmotsu_manifold" title="Kenmotsu manifold">Kenmotsu</a></li>

<li><a href="Lie_group" title="Lie group">Lie group</a>

<ul><li><a href="Lie_group%E2%80%93Lie_algebra_correspondence" title="Lie group–Lie algebra correspondence">Lie algebra</a></li></ul></li>

<li><a href="Manifold_with_boundary" class="mw-redirect" title="Manifold with boundary">Manifold with boundary</a></li>

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

<li><a href="Orientability" title="Orientability">Oriented</a></li>

<li><a href="Parallelizable_manifold" title="Parallelizable manifold">Parallelizable</a></li>

<li><a href="Poisson_manifold" title="Poisson manifold">Poisson</a></li>

<li><a href="Prime_manifold" title="Prime manifold">Prime</a></li>

<li><a href="Quaternionic_manifold" title="Quaternionic manifold">Quaternionic</a></li>

<li><a href="Hypercomplex_manifold" title="Hypercomplex manifold">Hypercomplex</a></li>

<li>(<a href="Pseudo-Riemannian_manifold" title="Pseudo-Riemannian manifold">Pseudo-</a>, <a href="Sub-Riemannian_manifold" title="Sub-Riemannian manifold">Sub-</a>) <a href="Riemannian_manifold" title="Riemannian manifold">Riemannian</a></li>

<li><a href="Rizza_manifold" title="Rizza manifold">Rizza</a></li>

<li><a href="Stein_manifold" title="Stein manifold">Stein</a></li>

<li>(<a href="Almost_symplectic_manifold" title="Almost symplectic manifold">Almost</a>) <a href="Symplectic_manifold" title="Symplectic manifold">Symplectic</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Tensor" title="Tensor">Tensors</a></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%">Vectors</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="Distribution_(differential_geometry)" title="Distribution (differential geometry)">Distribution</a></li>

<li><a href="Lie_bracket_of_vector_fields" title="Lie bracket of vector fields">Lie bracket</a></li>

<li><a href="Pushforward_(differential)" title="Pushforward (differential)">Pushforward</a></li>

<li><a href="Tangent_space" title="Tangent space">Tangent space</a>

<ul><li><a href="Tangent_bundle" title="Tangent bundle">bundle</a></li></ul></li>

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

<li><a href="Vector_field" title="Vector field">Vector field</a></li>

<li><a href="Vector_flow" title="Vector flow">Vector flow</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Covectors</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="Closed_and_exact_differential_forms" title="Closed and exact differential forms">Closed/Exact</a></li>

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

<li><a href="Cotangent_space" title="Cotangent space">Cotangent space</a>

<ul><li><a href="Cotangent_bundle" title="Cotangent bundle">bundle</a></li></ul></li>

<li><a href="De_Rham_cohomology" title="De Rham cohomology">De Rham cohomology</a></li>

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

<ul><li><a href="Vector-valued_differential_form" title="Vector-valued differential form">Vector-valued</a></li></ul></li>

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

<li><a href="Interior_product" title="Interior product">Interior product</a></li>

<li><a href="Pullback_(differential_geometry)" title="Pullback (differential geometry)">Pullback</a></li>

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

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

<li>

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

<li><a href="Volume_form" title="Volume form">Volume form</a></li>

<li><a href="Wedge_product" class="mw-redirect" title="Wedge product">Wedge product</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Fiber_bundle" title="Fiber bundle">Bundles</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="Adjoint_bundle" title="Adjoint bundle">Adjoint</a></li>

<li><a href="Affine_bundle" title="Affine bundle">Affine</a></li>

<li><a href="Associated_bundle" title="Associated bundle">Associated</a></li>

<li><a href="Cotangent_bundle" title="Cotangent bundle">Cotangent</a></li>

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

<li>(<a href="Cofibration" title="Cofibration">Co-</a>) <a href="Fibration" title="Fibration">Fibration</a></li>

<li><a href="Lie_algebra_bundle" title="Lie algebra bundle">Lie algebra</a></li>

<li>(<a href="Stable_normal_bundle" title="Stable normal bundle">Stable</a>) <a href="Normal_bundle" title="Normal bundle">Normal</a></li>

<li><a href="Principal_bundle" title="Principal bundle">Principal</a></li>

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

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

<li><a href="Tangent_bundle" title="Tangent bundle">Tangent</a></li>

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

<li><a href="Vector_bundle" title="Vector bundle">Vector</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Connection_(mathematics)" title="Connection (mathematics)">Connections</a></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</a></li>

<li><a href="Cartan_connection" title="Cartan connection">Cartan</a></li>

<li><a href="Ehresmann_connection" title="Ehresmann connection">Ehresmann</a></li>

<li><a href="Connection_(fibred_manifold)" title="Connection (fibred manifold)">Generalized</a></li>

<li><a href="Koszul_connection" class="mw-redirect" title="Koszul connection">Koszul</a></li>

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

<li><a href="Connection_(principal_bundle)" title="Connection (principal bundle)">Principal</a></li>

<li><a href="Connection_(vector_bundle)" title="Connection (vector bundle)">Vector</a></li>

<li><a href="Parallel_transport" title="Parallel transport">Parallel transport</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</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="Classification_of_manifolds" title="Classification of manifolds">Classification of manifolds</a></li>

<li><a href="Gauge_theory_(mathematics)" title="Gauge theory (mathematics)">Gauge theory</a></li>

<li><a href="History_of_manifolds_and_varieties" title="History of manifolds and varieties">History</a></li>

<li><a href="Morse_theory" title="Morse theory">Morse theory</a></li>

<li><a href="Moving_frame" title="Moving frame">Moving frame</a></li>

<li><a href="Singularity_theory" title="Singularity theory">Singularity theory</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Generalizations</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="Banach_manifold" title="Banach manifold">Banach manifold</a></li>

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

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

<li><a href="Fr%C3%A9chet_manifold" title="Fréchet manifold">Fréchet manifold</a></li>

<li><a href="K-theory" title="K-theory">K-theory</a></li>

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

<li><a href="Secondary_calculus_and_cohomological_physics" title="Secondary calculus and cohomological physics">Secondary calculus</a>

<ul><li><a href="Differential_calculus_over_commutative_algebras" title="Differential calculus over commutative algebras">over commutative algebras</a></li></ul></li>

<li><a href="Sheaf_(mathematics)" title="Sheaf (mathematics)">Sheaf</a></li>

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

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

<li><a href="Stratified_space" title="Stratified space">Stratified space</a></li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-06-18" href="https://en.wikipedia.org/wiki/?title=Tensor_field&oldid=1296250015">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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