ebab7d0da53c51438720d60c6e2e467c - /page/wikibook/entry.mu

Micron Document

<!DOCTYPE html>

<title>Volume element</title>

</head>

<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Volume element</span></span>

</h1>

</header>

<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, a <b>volume element</b> provides a means for <a href="Integral" title="Integral">integrating</a> a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> with respect to <a href="Volume" title="Volume">volume</a> in various coordinate systems such as <a href="Spherical_coordinate_system" title="Spherical coordinate system">spherical coordinates</a> and <a href="Cylindrical_coordinate_system" title="Cylindrical coordinate system">cylindrical coordinates</a>. Thus a volume element is an expression of the form

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} V=\rho (u_{1},u_{2},u_{3})\,\mathrm {d} u_{1}\,\mathrm {d} u_{2}\,\mathrm {d} u_{3}}</annotation>

</semantics>

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

where the <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_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./14f13cb025ff2e136dcbd2fc81ddf965b728e3d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle u_{i}}" loading="lazy"></span> are the coordinates, so that the volume of any set <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 B}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"></span> can be computed by

<mi>Volume</mi>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Volume} (B)=\int _{B}\rho (u_{1},u_{2},u_{3})\,\mathrm {d} u_{1}\,\mathrm {d} u_{2}\,\mathrm {d} u_{3}.}</annotation>

</semantics>

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

For example, in spherical 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 \mathrm {d} V=u_{1}^{2}\sin u_{2}\,\mathrm {d} u_{1}\,\mathrm {d} u_{2}\,\mathrm {d} u_{3}}">

</mrow>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} V=u_{1}^{2}\sin u_{2}\,\mathrm {d} u_{1}\,\mathrm {d} u_{2}\,\mathrm {d} u_{3}}</annotation>

</semantics>

</math></span><img src="./c5dc32ad557bbd48d23e9ce97de80cc65b4cb657.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.766ex; height:3.176ex;" alt="{\displaystyle \mathrm {d} V=u_{1}^{2}\sin u_{2}\,\mathrm {d} u_{1}\,\mathrm {d} u_{2}\,\mathrm {d} u_{3}}" loading="lazy"></span>, and so <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 \rho =u_{1}^{2}\sin u_{2}}">

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho =u_{1}^{2}\sin u_{2}}</annotation>

</semantics>

</math></span><img src="./6b1c7e12373c4cbd552b900cd2b9176be011630a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.698ex; height:3.176ex;" alt="{\displaystyle \rho =u_{1}^{2}\sin u_{2}}" loading="lazy"></span>.

</p><p>The notion of a volume element is not limited to three dimensions: in two dimensions it is often known as the <b>area element</b>, and in this setting it is useful for doing <a href="Surface_integral" title="Surface integral">surface integrals</a>. Under changes of coordinates, the volume element changes by the absolute value of the <a href="Jacobian_determinant" class="mw-redirect" title="Jacobian determinant">Jacobian determinant</a> of the coordinate transformation (by the <a href="Integration_by_substitution#Substitution_for_multiple_variables" title="Integration by substitution">change of variables formula</a>). This fact allows volume elements to be defined as a kind of <a href="Measure_(mathematics)" title="Measure (mathematics)">measure</a> on a <a href="Manifold" title="Manifold">manifold</a>. On an <a href="Orientability" title="Orientability">orientable</a> <a href="Differentiable_manifold" title="Differentiable manifold">differentiable manifold</a>, a volume element typically arises from a <a href="Volume_form" title="Volume form">volume form</a>: a top degree <a href="Differential_form" title="Differential form">differential form</a>. On a non-orientable manifold, the volume element is typically the <a href="Absolute_value" title="Absolute value">absolute value</a> of a (locally defined) volume form: it defines a <a href="Density_on_a_manifold" title="Density on a manifold">1-density</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Volume_element_in_Euclidean_space">Volume element in Euclidean space</h2></div>

<p>In <a href="Euclidean_space" title="Euclidean space">Euclidean space</a>, the volume element is given by the product of the differentials of the Cartesian coordinates

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} V=\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z.}</annotation>

</semantics>

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

In different coordinate systems of the form <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=x(u_{1},u_{2},u_{3})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=x(u_{1},u_{2},u_{3})}</annotation>

</semantics>

</math></span><img src="./a0a5dbc76763c96edc3314f77323f2199a0ac80c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.787ex; height:2.843ex;" alt="{\displaystyle x=x(u_{1},u_{2},u_{3})}" loading="lazy"></span>, <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=y(u_{1},u_{2},u_{3})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=y(u_{1},u_{2},u_{3})}</annotation>

</semantics>

</math></span><img src="./a0bb0e09f8a3c5c344e42263a93975fc0e7da454.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.438ex; height:2.843ex;" alt="{\displaystyle y=y(u_{1},u_{2},u_{3})}" loading="lazy"></span>, <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 z=z(u_{1},u_{2},u_{3})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z=z(u_{1},u_{2},u_{3})}</annotation>

</semantics>

</math></span><img src="./3667c1ac2f5a1cbe58b5b896b5a184714cb7c21e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.304ex; height:2.843ex;" alt="{\displaystyle z=z(u_{1},u_{2},u_{3})}" loading="lazy"></span>, the volume element <a href="Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">changes by the Jacobian</a> (determinant) of the coordinate change:

</mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} V=\left|{\frac {\partial (x,y,z)}{\partial (u_{1},u_{2},u_{3})}}\right|\,\mathrm {d} u_{1}\,\mathrm {d} u_{2}\,\mathrm {d} u_{3}.}</annotation>

</semantics>

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

For example, in spherical coordinates (mathematical convention)

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x&=\rho \cos \theta \sin \phi \\y&=\rho \sin \theta \sin \phi \\z&=\rho \cos \phi \end{aligned}}}</annotation>

</semantics>

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

the Jacobian determinant is

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left|{\frac {\partial (x,y,z)}{\partial (\rho ,\phi ,\theta )}}\right|=\rho ^{2}\sin \phi }</annotation>

</semantics>

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

so that

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} V=\rho ^{2}\sin \phi \,\mathrm {d} \rho \,\mathrm {d} \theta \,\mathrm {d} \phi .}</annotation>

</semantics>

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

This can be seen as a special case of the fact that differential forms transform through a pullback <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 F^{*}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./c8ab6d9b2b48fcfd2f5c741de789391d4cdd0f13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.869ex; height:2.343ex;" alt="{\displaystyle F^{*}}" loading="lazy"></span> as

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F^{*}(u\;dy^{1}\wedge \cdots \wedge dy^{n})=(u\circ F)\det \left({\frac {\partial F^{j}}{\partial x^{i}}}\right)\mathrm {d} x^{1}\wedge \cdots \wedge \mathrm {d} x^{n}}</annotation>

</semantics>

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

</p>

<div class="mw-heading mw-heading2"><h2 id="Volume_element_of_a_linear_subspace">Volume element of a linear subspace</h2></div>

<p>Consider the <a href="Linear_subspace" title="Linear subspace">linear subspace</a> of the <i>n</i>-dimensional <a href="Euclidean_space" title="Euclidean space">Euclidean space</a> <b>R</b><sup><i>n</i></sup> that is spanned by a collection of <a href="Linearly_independent" class="mw-redirect" title="Linearly independent">linearly independent</a> vectors

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X_{1},\dots ,X_{k}.}</annotation>

</semantics>

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

To find the volume element of the subspace, it is useful to know the fact from <a href="Linear_algebra" title="Linear algebra">linear algebra</a> that the volume of the <a href="Parallelepiped" title="Parallelepiped">parallelepiped</a> spanned by the <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_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./af4a0955af42beb5f85aa05fb8c07abedc13990d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.724ex; height:2.509ex;" alt="{\displaystyle X_{i}}" loading="lazy"></span> is the square root of the <a href="Determinant" title="Determinant">determinant</a> of the <a href="Gramian_matrix" class="mw-redirect" title="Gramian matrix">Gramian matrix</a> of the <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_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

<msqrt>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {\det(X_{i}\cdot X_{j})_{i,j=1\dots k}}}.}</annotation>

</semantics>

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

</p><p>Any point <i>p</i> in the subspace can be given 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 (u_{1},u_{2},\dots ,u_{k})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (u_{1},u_{2},\dots ,u_{k})}</annotation>

</semantics>

</math></span><img src="./cdf4b28d7b17cab7f5e0149c7cfd3955f57c4f09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.208ex; height:2.843ex;" alt="{\displaystyle (u_{1},u_{2},\dots ,u_{k})}" loading="lazy"></span> such that

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p=u_{1}X_{1}+\cdots +u_{k}X_{k}.}</annotation>

</semantics>

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

At a point <i>p</i>, if we form a small parallelepiped with sides <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 \mathrm {d} u_{i}}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {d} u_{i}}</annotation>

</semantics>

</math></span><img src="./cd2e793d6d6b247449777500fec90a085fb78fe1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.422ex; height:2.509ex;" alt="{\displaystyle \mathrm {d} u_{i}}" loading="lazy"></span>, then the volume of that parallelepiped is the square root of the determinant of the Grammian matrix

<msqrt>

<msub>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mrow>

</msub>

</msqrt>

</mrow>

<msqrt>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</msqrt>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {\det \left((du_{i}X_{i})\cdot (du_{j}X_{j})\right)_{i,j=1\dots k}}}={\sqrt {\det(X_{i}\cdot X_{j})_{i,j=1\dots k}}}\;\mathrm {d} u_{1}\,\mathrm {d} u_{2}\,\cdots \,\mathrm {d} u_{k}.}</annotation>

</semantics>

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

This therefore defines the volume form in the linear subspace.

</p>

<div class="mw-heading mw-heading2"><h2 id="Volume_element_of_manifolds">Volume element of manifolds</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}

/* end https://en.wikipedia.org/ */

</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="Riemannian_volume_form" class="mw-redirect" title="Riemannian volume form">Riemannian volume form</a></div>

<p>On an <i>oriented</i> <a href="Riemannian_manifold" title="Riemannian manifold">Riemannian manifold</a> of dimension <i>n</i>, the volume element is a volume form equal to the <a href="Hodge_dual" class="mw-redirect" title="Hodge dual">Hodge dual</a> of the unit constant 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 f(x)=1}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./ea78f54e69b72f398cf6077e61c50a05b532d4c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.678ex; height:2.843ex;" alt="{\displaystyle f(x)=1}" loading="lazy"></span>:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega =\star 1.}</annotation>

</semantics>

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

Equivalently, the volume element is precisely the <a href="Levi-Civita_tensor" class="mw-redirect" title="Levi-Civita tensor">Levi-Civita tensor</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 \epsilon }">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./c3837cad72483d97bcdde49c85d3b7b859fb3fd2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.944ex; height:1.676ex;" alt="{\displaystyle \epsilon }" loading="lazy"></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> In coordinates,

<msqrt>

<mrow>

<mrow>

</mrow>

</mrow>

</msqrt>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega =\epsilon ={\sqrt {\left|\det g\right|}}\,\mathrm {d} x^{1}\wedge \cdots \wedge \mathrm {d} x^{n}}</annotation>

</semantics>

</math></span></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 \det g}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./b3a689ff5058ad7c943a6ac7c6d262c7fc77287c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.733ex; height:2.509ex;" alt="{\displaystyle \det g}" loading="lazy"></span> is the <a href="Determinant" title="Determinant">determinant</a> of the <a href="Metric_tensor" title="Metric tensor">metric tensor</a> <i>g</i> written in the coordinate system.

</p>

<div class="mw-heading mw-heading3"><h3 id="Area_element_of_a_surface">Area element of a surface</h3></div>

<p>A simple example of a volume element can be explored by considering a two-dimensional surface embedded in <i>n</i>-dimensional <a href="Euclidean_space" title="Euclidean space">Euclidean space</a>. Such a volume element is sometimes called an <i>area element</i>. Consider a subset <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\subset \mathbb {R} ^{2}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U\subset \mathbb {R} ^{2}}</annotation>

</semantics>

</math></span><img src="./88bf916836435eaac9f115120ab48ec6c558b4e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.613ex; height:2.676ex;" alt="{\displaystyle U\subset \mathbb {R} ^{2}}" loading="lazy"></span> and a mapping function

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

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

thus defining a surface embedded in <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}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</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>. In two dimensions, volume is just area, and a volume element gives a way to determine the area of parts of the surface. Thus a volume element is an expression of the form

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(u_{1},u_{2})\,\mathrm {d} u_{1}\,\mathrm {d} u_{2}}</annotation>

</semantics>

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

that allows one to compute the area of a set <i>B</i> lying on the surface by computing the integral

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Area} (B)=\int _{B}f(u_{1},u_{2})\,\mathrm {d} u_{1}\,\mathrm {d} u_{2}.}</annotation>

</semantics>

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

</p><p>Here we will find the volume element on the surface that defines area in the usual sense. The <a href="Jacobian_matrix" class="mw-redirect" title="Jacobian matrix">Jacobian matrix</a> of the mapping is

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle J_{ij}={\frac {\partial \varphi _{i}}{\partial u_{j}}}}</annotation>

</semantics>

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

with index <i>i</i> running from 1 to <i>n</i>, and <i>j</i> running from 1 to 2. The Euclidean <a href="Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metric</a> in the <i>n</i>-dimensional space induces a metric <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=J^{T}J}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./28fe097305f712f3d16d198eb3c6b954e85e0f21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.601ex; height:3.009ex;" alt="{\displaystyle g=J^{T}J}" loading="lazy"></span> on the set <i>U</i>, with matrix elements

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{ij}=\sum _{k=1}^{n}J_{ki}J_{kj}=\sum _{k=1}^{n}{\frac {\partial \varphi _{k}}{\partial u_{i}}}{\frac {\partial \varphi _{k}}{\partial u_{j}}}.}</annotation>

</semantics>

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

</p><p>The <a href="Determinant" title="Determinant">determinant</a> of the metric is given by

<msup>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \det g=\left|{\frac {\partial \varphi }{\partial u_{1}}}\wedge {\frac {\partial \varphi }{\partial u_{2}}}\right|^{2}=\det(J^{T}J)}</annotation>

</semantics>

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

</p><p>For a regular surface, this determinant is non-vanishing; equivalently, the Jacobian matrix has rank 2.

</p><p>Now consider a change of coordinates on <i>U</i>, given by a <a href="Diffeomorphism" title="Diffeomorphism">diffeomorphism</a>

</mstyle>

</mrow>

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

</semantics>

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

so that 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 (u_{1},u_{2})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (u_{1},u_{2})}</annotation>

</semantics>

</math></span><img src="./bb6764e9870c65f0289890bdb7edf20be77b802a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.611ex; height:2.843ex;" alt="{\displaystyle (u_{1},u_{2})}" loading="lazy"></span> are given in terms 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_{1},v_{2})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (v_{1},v_{2})}</annotation>

</semantics>

</math></span><img src="./8e8ab5272bbaa8b9527fc9a9dde1296d18ce696c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.207ex; height:2.843ex;" alt="{\displaystyle (v_{1},v_{2})}" loading="lazy"></span> by <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_{1},u_{2})=f(v_{1},v_{2})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (u_{1},u_{2})=f(v_{1},v_{2})}</annotation>

</semantics>

</math></span><img src="./3b5d321e111dd5ff706a7431beefac3426c8aa33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.195ex; height:2.843ex;" alt="{\displaystyle (u_{1},u_{2})=f(v_{1},v_{2})}" loading="lazy"></span>. The Jacobian matrix of this transformation is given by

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{ij}={\frac {\partial f_{i}}{\partial v_{j}}}.}</annotation>

</semantics>

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

</p><p>In the new coordinates, we have

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \varphi _{i}}{\partial v_{j}}}=\sum _{k=1}^{2}{\frac {\partial \varphi _{i}}{\partial u_{k}}}{\frac {\partial f_{k}}{\partial v_{j}}}}</annotation>

</semantics>

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

and so the metric transforms as

<mover>

</mover>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tilde {g}}=F^{T}gF}</annotation>

</semantics>

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

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./bcf709e979316ee494a3f076f7e1d97be44a3f8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.232ex; height:2.509ex;" alt="{\displaystyle {\tilde {g}}}" loading="lazy"></span> is the pullback metric in the <i>v</i> coordinate system. The determinant is

<mover>

</mover>

</mrow>

<msup>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \det {\tilde {g}}=\det g\left(\det F\right)^{2}.}</annotation>

</semantics>

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

</p><p>Given the above construction, it should now be straightforward to understand how the volume element is invariant under an orientation-preserving change of coordinates.

</p><p>In two dimensions, the volume is just the area. The area of a subset <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 B\subset U}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./f98fcc1dda3a96950248395514073a118036e215.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.645ex; height:2.176ex;" alt="{\displaystyle B\subset U}" loading="lazy"></span> is given by the integral

<mtr>

<mtd>

</mstyle>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msqrt>

</msqrt>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msqrt>

</msqrt>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msqrt>

<mover>

</mover>

</mrow>

</msqrt>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\mbox{Area}}(B)&=\iint _{B}{\sqrt {\det g}}\;\mathrm {d} u_{1}\;\mathrm {d} u_{2}\\[1.6ex]&=\iint _{B}{\sqrt {\det g}}\left|\det F\right|\;\mathrm {d} v_{1}\;\mathrm {d} v_{2}\\[1.6ex]&=\iint _{B}{\sqrt {\det {\tilde {g}}}}\;\mathrm {d} v_{1}\;\mathrm {d} v_{2}.\end{aligned}}}</annotation>

</semantics>

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

</p><p>Thus, in either coordinate system, the volume element takes the same expression: the expression of the volume element is invariant under a change of coordinates.

</p><p>Note that there was nothing particular to two dimensions in the above presentation; the above trivially generalizes to arbitrary dimensions.

</p>

<div class="mw-heading mw-heading3"><h3 id="Example:_Sphere">Example: Sphere</h3></div>

<p>For example, consider the sphere with radius <i>r</i> centered at the origin in <b>R</b><sup>3</sup>. This can be parametrized using <a href="Spherical_coordinates" class="mw-redirect" title="Spherical coordinates">spherical coordinates</a> with the map

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi (u_{1},u_{2})=(r\cos u_{1}\sin u_{2},r\sin u_{1}\sin u_{2},r\cos u_{2}).}</annotation>

</semantics>

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

Then

<mrow>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g={\begin{pmatrix}r^{2}\sin ^{2}u_{2}&0\\0&r^{2}\end{pmatrix}},}</annotation>

</semantics>

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

and the area element is

<msqrt>

</msqrt>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega ={\sqrt {\det g}}\;\mathrm {d} u_{1}\mathrm {d} u_{2}=r^{2}\sin u_{2}\,\mathrm {d} u_{1}\mathrm {d} u_{2}.}</annotation>

</semantics>

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

</p>

<ul><li><a href="Cylindrical_coordinate_system#Line_and_volume_elements" title="Cylindrical coordinate system">Cylindrical coordinate system § Line and volume elements</a></li>

<li><a href="Spherical_coordinate_system#Integration_and_differentiation_in_spherical_coordinates" title="Spherical coordinate system">Spherical coordinate system § Integration and differentiation in spherical coordinates</a></li>

<li><a href="Volume_integral" title="Volume integral">Volume integral</a></li>

<li><a href="Surface_integral" title="Surface integral">Surface integral</a></li>

<li><a href="Line_integral" title="Line integral">Line integral</a></li>

<li><a href="Line_element" title="Line element">Line element</a></li></ul>

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

/* start https://en.wikipedia.org/ */

.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}}

/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFBesse1987" class="citation cs2">Besse, Arthur L. (1987), <i>Einstein manifolds</i>, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, pp. xii+510, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-15279-8</bdi></cite></li></ul>

/* start https://en.wikipedia.org/ */

.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}

/* end https://en.wikipedia.org/ */

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

<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">Carroll, Sean. <i>Spacetime and Geometry</i>. Addison Wesley, 2004, p. 90</span>

</li>

</ol></div></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2024-10-04" href="https://en.wikipedia.org/wiki/?title=Volume_element&oldid=1249322983">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.

</div>

</div></div>

</div>

</main>

</div>

</body></html>