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Volume element
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<h2 id="volume-element-in-euclidean-space">Volume element in Euclidean space</h2> <p>In <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a>, the volume element is given by the product of the differentials of the Cartesian coordinates d V = d x d y d z . {\displaystyle \mathrm {d} V=\mathrm {d} x\,\mathrm {d} y\,\mathrm {d} z.} In different coordinate systems of the form x = x ( u 1 , u 2 , u 3 ) {\displaystyle x=x(u_{1},u_{2},u_{3})} , y = y ( u 1 , u 2 , u 3 ) {\displaystyle y=y(u_{1},u_{2},u_{3})} , z = z ( u 1 , u 2 , u 3 ) {\displaystyle z=z(u_{1},u_{2},u_{3})} , the volume element <a href="/facts/Jacobian_matrix_and_determinant/0tNv6o8j">changes by the Jacobian</a> (determinant) of the coordinate change: d V = | ∂ ( x , y , z ) ∂ ( u 1 , u 2 , u 3 ) | d u 1 d u 2 d u 3 . {\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}.} For example, in spherical coordinates (mathematical convention) x = ρ cos θ sin ϕ y = ρ sin θ sin ϕ z = ρ cos ϕ {\displaystyle {\begin{aligned}x&=\rho \cos \theta \sin \phi \\y&=\rho \sin \theta \sin \phi \\z&=\rho \cos \phi \end{aligned}}} the Jacobian determinant is | ∂ ( x , y , z ) ∂ ( ρ , ϕ , θ ) | = ρ 2 sin ϕ {\displaystyle \left|{\frac {\partial (x,y,z)}{\partial (\rho ,\phi ,\theta )}}\right|=\rho ^{2}\sin \phi } so that d V = ρ 2 sin ϕ d ρ d θ d ϕ . {\displaystyle \mathrm {d} V=\rho ^{2}\sin \phi \,\mathrm {d} \rho \,\mathrm {d} \theta \,\mathrm {d} \phi .} This can be seen as a special case of the fact that differential forms transform through a pullback F ∗ {\displaystyle F^{*}} as F ∗ ( u d y 1 ∧ ⋯ ∧ d y n ) = ( u ∘ F ) det ( ∂ F j ∂ x i ) d x 1 ∧ ⋯ ∧ d x n {\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}} </p> <h2 id="volume-element-of-a-linear-subspace">Volume element of a linear subspace</h2> <p>Consider the <a href="/facts/Linear_subspace/2wtKTgHL">linear subspace</a> of the <i>n</i>-dimensional <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a> R<i>n</i> that is spanned by a collection of <a href="/facts/Linearly_independent/8JrHfIIa">linearly independent</a> vectors X 1 , … , X k . {\displaystyle X_{1},\dots ,X_{k}.} To find the volume element of the subspace, it is useful to know the fact from linear algebra that the volume of the parallelepiped spanned by the X i {\displaystyle X_{i}} is the square root of the <a href="/facts/Determinant/eGYqJ2TF">determinant</a> of the <a href="/facts/Gramian_matrix/SGb2VGTU">Gramian matrix</a> of the X i {\displaystyle X_{i}} : det ( X i ⋅ X j ) i , j = 1 … k . {\displaystyle {\sqrt {\det(X_{i}\cdot X_{j})_{i,j=1\dots k}}}.} </p><p>Any point <i>p</i> in the subspace can be given coordinates ( u 1 , u 2 , … , u k ) {\displaystyle (u_{1},u_{2},\dots ,u_{k})} such that p = u 1 X 1 + ⋯ + u k X k . {\displaystyle p=u_{1}X_{1}+\cdots +u_{k}X_{k}.} At a point <i>p</i>, if we form a small parallelepiped with sides d u i {\displaystyle \mathrm {d} u_{i}} , then the volume of that parallelepiped is the square root of the determinant of the Grammian matrix det ( ( d u i X i ) ⋅ ( d u j X j ) ) i , j = 1 … k = det ( X i ⋅ X j ) i , j = 1 … k d u 1 d u 2 ⋯ d u k . {\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}.} This therefore defines the volume form in the linear subspace. </p> <h2 id="volume-element-of-manifolds">Volume element of manifolds</h2> <p class="note">See also: <a href="/facts/Riemannian_volume_form/2x37pDLg">Riemannian volume form</a></p> <p>On an <i>oriented</i> <a href="/facts/Riemannian_manifold/5KYnmEgF">Riemannian manifold</a> of dimension <i>n</i>, the volume element is a volume form equal to the <a href="/facts/Hodge_dual/AH6tmHhU">Hodge dual</a> of the unit constant function, f ( x ) = 1 {\displaystyle f(x)=1} : ω = ⋆ 1. {\displaystyle \omega =\star 1.} Equivalently, the volume element is precisely the <a href="/facts/Levi-Civita_tensor/WBKUA3Tw">Levi-Civita tensor</a> ϵ {\displaystyle \epsilon } .<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> In coordinates, ω = ϵ = | det g | d x 1 ∧ ⋯ ∧ d x n {\displaystyle \omega =\epsilon ={\sqrt {\left|\det g\right|}}\,\mathrm {d} x^{1}\wedge \cdots \wedge \mathrm {d} x^{n}} where det g {\displaystyle \det g} is the <a href="/facts/Determinant/eGYqJ2TF">determinant</a> of the <a href="/facts/Metric_tensor/tbqek1gJ">metric tensor</a> <i>g</i> written in the coordinate system. </p> <h3>Area element of a surface</h3> <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="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a>. Such a volume element is sometimes called an <i>area element</i>. Consider a subset U ⊂ R 2 {\displaystyle U\subset \mathbb {R} ^{2}} and a mapping function φ : U → R n {\displaystyle \varphi :U\to \mathbb {R} ^{n}} thus defining a surface embedded in R n {\displaystyle \mathbb {R} ^{n}} . 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 f ( u 1 , u 2 ) d u 1 d u 2 {\displaystyle f(u_{1},u_{2})\,\mathrm {d} u_{1}\,\mathrm {d} u_{2}} that allows one to compute the area of a set <i>B</i> lying on the surface by computing the integral Area ( B ) = ∫ B f ( u 1 , u 2 ) d u 1 d u 2 . {\displaystyle \operatorname {Area} (B)=\int _{B}f(u_{1},u_{2})\,\mathrm {d} u_{1}\,\mathrm {d} u_{2}.} </p><p>Here we will find the volume element on the surface that defines area in the usual sense. The <a href="/facts/Jacobian_matrix/0tNv6o8j">Jacobian matrix</a> of the mapping is J i j = ∂ φ i ∂ u j {\displaystyle J_{ij}={\frac {\partial \varphi _{i}}{\partial u_{j}}}} 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="/facts/Metric_(mathematics)/RkUptSo8">metric</a> in the <i>n</i>-dimensional space induces a metric g = J T J {\displaystyle g=J^{T}J} on the set <i>U</i>, with matrix elements g i j = ∑ k = 1 n J k i J k j = ∑ k = 1 n ∂ φ k ∂ u i ∂ φ k ∂ u j . {\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}}}.} </p><p>The <a href="/facts/Determinant/eGYqJ2TF">determinant</a> of the metric is given by det g = | ∂ φ ∂ u 1 ∧ ∂ φ ∂ u 2 | 2 = det ( J T J ) {\displaystyle \det g=\left|{\frac {\partial \varphi }{\partial u_{1}}}\wedge {\frac {\partial \varphi }{\partial u_{2}}}\right|^{2}=\det(J^{T}J)} </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="/facts/Diffeomorphism/855N5YYj">diffeomorphism</a> f : U → U , {\displaystyle f\colon U\to U,} so that the coordinates ( u 1 , u 2 ) {\displaystyle (u_{1},u_{2})} are given in terms of ( v 1 , v 2 ) {\displaystyle (v_{1},v_{2})} by ( u 1 , u 2 ) = f ( v 1 , v 2 ) {\displaystyle (u_{1},u_{2})=f(v_{1},v_{2})} . The Jacobian matrix of this transformation is given by F i j = ∂ f i ∂ v j . {\displaystyle F_{ij}={\frac {\partial f_{i}}{\partial v_{j}}}.} </p><p>In the new coordinates, we have ∂ φ i ∂ v j = ∑ k = 1 2 ∂ φ i ∂ u k ∂ f k ∂ v j {\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}}}} and so the metric transforms as g ~ = F T g F {\displaystyle {\tilde {g}}=F^{T}gF} where g ~ {\displaystyle {\tilde {g}}} is the pullback metric in the <i>v</i> coordinate system. The determinant is det g ~ = det g ( det F ) 2 . {\displaystyle \det {\tilde {g}}=\det g\left(\det F\right)^{2}.} </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 B ⊂ U {\displaystyle B\subset U} is given by the integral Area ( B ) = ∬ B det g d u 1 d u 2 = ∬ B det g | det F | d v 1 d v 2 = ∬ B det g ~ d v 1 d v 2 . {\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}}} </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> <h3>Example: Sphere</h3> <p>For example, consider the sphere with radius <i>r</i> centered at the origin in R3. This can be parametrized using <a href="/facts/Spherical_coordinates/5cVnNof3">spherical coordinates</a> with the map ϕ ( u 1 , u 2 ) = ( r cos u 1 sin u 2 , r sin u 1 sin u 2 , r cos u 2 ) . {\displaystyle \phi (u_{1},u_{2})=(r\cos u_{1}\sin u_{2},r\sin u_{1}\sin u_{2},r\cos u_{2}).} Then g = ( r 2 sin 2 u 2 0 0 r 2 ) , {\displaystyle g={\begin{pmatrix}r^{2}\sin ^{2}u_{2}&0\\0&r^{2}\end{pmatrix}},} and the area element is ω = det g d u 1 d u 2 = r 2 sin u 2 d u 1 d u 2 . {\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}.} </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Cylindrical_coordinate_system/1Vc6lhxD">Cylindrical coordinate system § Line and volume elements</a></li> <li><a href="/facts/Spherical_coordinate_system/5cVnNof3">Spherical coordinate system § Integration and differentiation in spherical coordinates</a></li> <li><a href="/facts/Volume_integral/0fHWrQe7">Volume integral</a></li> <li><a href="/facts/Surface_integral/oPk5IRI1">Surface integral</a></li> <li><a href="/facts/Line_integral/10s4B3Nn">Line integral</a></li> <li><a href="/facts/Line_element/asquI22v">Line element</a></li></ul> <ul><li>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="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, pp. xii+510, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-15279-8</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Carroll, Sean. Spacetime and Geometry. Addison Wesley, 2004, p. 90 <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> </ol>