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''volume element.''
 
''volume element.''
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v0968901.png" /> be a vector space of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v0968902.png" /> with a given [[Orientation|orientation]] and an [[Inner product|inner product]]. The corresponding volume form, or volume element, is the unique element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v0968903.png" />, the space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v0968904.png" />-forms on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v0968905.png" /> (cf. [[Exterior form|Exterior form]]), such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v0968906.png" /> for each orthonormal (with respect to the given inner product) basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v0968907.png" />. Recall that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v0968908.png" /> is one-dimensional. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v0968909.png" /> with the standard inner product and orientation, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v09689010.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v09689011.png" />-tuple of vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v09689012.png" /> (written out in the standard basis for calculating the determinant) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v09689013.png" /> is the volume of the parallelopipedon spanned by the line segments from zero to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v09689014.png" />.
+
{{TEX|done}}
 +
{{MSC|53B20}}
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v09689015.png" /> is an oriented Riemannian manifold, then the volume form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v09689016.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v09689017.png" /> is defined by requiring that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v09689018.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v09689019.png" /> is the unique volume element on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v09689020.png" /> defined by inner product and orientation on each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v09689021.png" />. One often writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v09689022.png" /> for the volume form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v09689023.png" />, even though there may not be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v09689024.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v09689025.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v09689026.png" /> of which it is the exterior derivative.
+
The volume form is a special [[Differential form|differential form]] defined on oriented [[Riemannian manifold|Riemannian manifolds]] and which introduces a natural concept of [[Measure|measure]] on the manifold.
  
In given local coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v09689027.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v09689028.png" /> be the two-form (matrix) determining the inner product on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v09689029.png" /> (with respect to the basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v09689030.png" />, cf. [[Tangent vector|Tangent vector]]). Then in local coordinates,
+
===On vector spaces===
 +
First consider a real vector space $V$ of dimension $n$ with a given [[Orientation|orientation]] and an [[Inner product|inner product]]. The corresponding volume form, or volume element, is the unique element $\omega$ of the space $\Lambda^n V$ of $n$-forms (namely alternating multilinear maps $L : V^n \to \mathbb R$; cf. [[Exterior form|Exterior form]]), such that $\omega (v_1, \ldots, v_n) =1$ for each orthonormal (with respect to the given inner product) positively oriented basis of $V$. Recall that $\Lambda^n V$ is one-dimensional. If $V = \mathbb R^n$ with the standard inner product and orientation, then, for each $n$-tuple of vectors $w_1, \ldots, w_n \in \mathbb R^n$, $\omega (w_1, \ldots, w_n)$ is the [[Determinant|determinant]] of the $n\times n$ matrix whose $ij$ entry is the $i$-th corrdinate (in the standared basis) of $w_j$. Moreover, $|\omega (v_1, \ldots, v_n)|$ is the [[Lebesgue measure]] (i.e. the $n$-dimensional volume) of the [[Parallelopipedon|parallelopipedon]] $P$ spanned by the vectors $v_j$, namely
 +
\[
 +
P = \left\{\sum_j \lambda_j v_j : \lambda_j\in [0,1]\right\}\, .
 +
\]
 +
The latter property gives an intuitive justification for the terms ''volume form'' and ''volume element'': in particular, if $U$ is an open bounded subset of $\mathbb R^n$, then
 +
\begin{equation}\label{e:integra}
 +
\int_U \omega
 +
\end{equation}
 +
is the Lebesgue measure of $U$ (for the definition of the integral in \eqref{e:integra} see [[Integration on manifolds]]).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v09689031.png" /></td> </tr></table>
+
===On Riemannian manifolds===
 +
If $(\Sigma, g)$ is an oriented Riemannian manifold, then the volume form $\omega$ on $\Sigma$ is the unique element of $\Lambda^n M$ such that, for each $x\in M$, $\omega|_x$ is the volume form on the vector space $T_x M$ relative to the inner product $g_x$ and the orientation chosen. One often writes ${\rm dV}$ or ${\rm dvol}$, for the volume form on $\Sigma$, even though the volume form might not be exact (indeed the volume form is never exact if $M$ is compact and without boundary).  
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v09689032.png" /> depending on whether the orientation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v09689033.png" /> corresponds to the standard one on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v09689034.png" /> or not (under the given coordinate chart).
+
====Formula in local coordinates====
 +
Consider a local coordinate patch $U$ on $\Sigma$. Let $x_1, \ldots, x_n$ be the corresponding coordinates and denote by $g_{ij} (x)$ the inner product (at the point $x$) of the vector fields $\frac{\partial}{\partial x_i}$ and $\frac{\partial}{\partial x_j}$(see [[Tangent vector]]), namely
 +
\[
 +
g_{ij} (x) = g_x \left(\frac{\partial}{\partial x_i}, \frac{\partial}{\partial x_j}\right)\, .
 +
\]
 +
With a slight abuse of notation, let $\det g (x)$ be the determinant of the matrix $(g_{ij} (x))_{ij}$.
 +
We then have the following formula for the volume form ${\rm dvol}$ on $U$:
 +
\[
 +
{\rm dvol} = \varepsilon\, \sqrt{\det g (x)}\, dx_1\wedge \ldots \wedge dx_n
 +
\]
 +
where $\varepsilon =1 $ if $\frac{\partial}{\partial x_1}, \ldots, \frac{\partial}{\partial x_n}$ is positively oriented, and $-1$ otherwise.
  
On a Riemannian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v09689035.png" /> a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v09689036.png" /> is integrated by intergrating the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v09689037.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v09689038.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v09689039.png" /> in the sense of [[Integration on manifolds|integration on manifolds]].
+
====Volume measure====
 +
The volume form defines naturally a [[Measure|measure]] $\mu$ on any Riemannian manifold. Indeed, given a smooth continuous compactly supported function $f$ we can define
 +
\[
 +
\int_\Sigma f d\mu
 +
\]
 +
as the integral of the form $f {\rm dvol}$, in the sense of [[Integration on manifolds|integration on manifolds]]. It can be checked that such integral coincides with the [[Lebesgue integral]] with respect to a $\sigma$-additive [[Measure|measure]] on an appropriate [[Algebra of sets|$\sigma$-algebra]] (assuming that the manifold is [[Second axiom of countability|second countable]]).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v09689040.png" /> denote the Hodge star operator (cf. [[Laplace operator|Laplace operator]]). The divergence of a vector field, locally given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v09689041.png" />, is defined as the function
+
===Divergence and Hodge star===
 +
Assume $(\Sigma, g)$ is an oriented Riemannian manifold. If $\nabla$ is the [[Levi-Civita connection]] on $\Sigma$, the [[Divergence|divergence]] of a [[Vector field on a manifold|vector field]] $X$ is then given, at the point $x$, by
 +
\[
 +
{\rm div}\, X = \sum_j \nabla_{e_j} X\, ,
 +
\]
 +
where $e_1, \ldots, e_n$ is any orthonormal [[Frame|frame]] defined in a neighborhood of
 +
$x$. In local coordinates, if $X = \sum \psi^j \frac{\partial}{\partial x_j}$, then
 +
\[
 +
{\rm div}\, X = (\det g)^{-1/2} \sum_j \frac{\partial}{\partial x_j} \left( (\det g)^{1/2} \psi^j\right)\, .
 +
\]
 +
Let $*$ denote the [[Hodge star]] operator on the oriented Riemannian manifold $\Sigma$ (cf. [[Laplace operator|Laplace operator]]).It then turns out that
 +
\begin{equation}\label{e:star}
 +
d (*X) = {\rm div}\, X\, {\rm dvol}\, .
 +
\end{equation}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v09689042.png" /></td> </tr></table>
+
====Stokes and divergence theorems====
 +
The latter identity gives a link between [[Stokes theorem]] on differential forms on the divergence theorem (also called Gauss-Green theorem or [[Ostrogradski formula]]). More precisely, assume that $(\Sigma,g)$ is a compact oriented Riemannian manifold with boundary $\partial \Sigma$. Denote by:
 +
* ${\rm d Vol}$ the volume form on $\Sigma$;
 +
* ${\rm dvol}$ the volume form on $\partial \Sigma$;
 +
* $\nu$ the exterior unit normal to $\partial \Sigma$ (namely, if $e_1, \ldots, e_{n-1}$ is a positively oriented ortonormal frame on $U\cap \partial \Sigma$, then $e_1, \ldots, e_{n-1}, \nu$ is a positively oriented orthonormal frame in $U\cap \Sigma$).
 +
Stokes theorem together with \eqref{e:star} implies that
 +
\[
 +
\int_\Sigma {\rm div}\, X\, {\rm dVol} = \int_{\partial \Sigma} *X\, .
 +
\]
 +
On the other it can be shown that
 +
\[
 +
\int_{\partial \Sigma} *X = \int_{\partial \Sigma} g (X, \nu)\, {\rm dvol}\, ,
 +
\]
 +
thereby concluding
 +
\[
 +
\int_\Sigma {\rm div}\, X\, {\rm dVol} = \int_{\partial \Sigma} g (X, \nu)\, {\rm dvol}\, .
 +
\]
  
One then has
+
===Area formula===
 
+
For $C^1$ submanifolds of the Euclidean space $\mathbb R^N$, the volume form can be related to a measure-theoretic concept of volume. More precisely, assume $\Sigma$ is a (compact) $m$-dimensional $C^1$ submanifold of $\mathbb R^N$. Let ${\rm dvol}$ be the volume form on $\Sigma$ related to the Riemannian structure given by the restriction of the standard Euclidean inner product on the tangent space to $\Sigma$. The [[Area formula|area formula]] implies that, if we denote by $\mathcal{H}^m$ the $m$-dimensional [[Hausdorff measure]] on $\mathbb R^N$, then
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v09689043.png" /></td> </tr></table>
+
\[
 
+
\int_\Sigma {\rm dvol} = \mathcal{H}^m (\Sigma)\, .
and, on integration over an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v09689044.png" />-chain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v09689045.png" />, by the Stokes formula there results the higher-dimensional divergence theorem, which specializes to the usual one for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v09689046.png" /> a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v09689047.png" />-dimensional submanifold with boundary in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096890/v09689048.png" />.
+
\]
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Spivak,   "Calculus on manifolds" , Benjamin (1965)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Hazewinkel,  "A tutorial introduction to differentiable manifolds and calculus on manifolds"  W. Schiehlen (ed.)  W. Wedig (ed.) , ''Analysis and estimation of stochastic mechanical systems'' , Springer (Wien)  (1988)  pp. 316–340</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> Y. Choquet-Bruhat,  C. DeWitt-Morette,  M. Dillard-Bleick,  "Analysis, manifolds, and physics" , North-Holland (1977) (Translated from French)</TD></TR></table>
+
{|
 +
|-
 +
|valign="top"|{{Ref|Be}}|| Y. Choquet-Bruhat,  C. DeWitt-Morette,  M. Dillard-Bleick,   "Analysis, manifolds, and physics" , North-Holland  (1977) (Translated  from French)
 +
|-
 +
|valign="top"|{{Ref|Ha}}|| M. Hazewinkel,  "A tutorial introduction to differentiable manifolds and calculus on manifolds"  W. Schiehlen (ed.)  W. Wedig (ed.) , ''Analysis and estimation of stochastic mechanical systems'' , Springer (Wien)  (1988)  pp. 316–340
 +
|-
 +
|valign="top"|{{Ref|Sp}}|| M. Spivak,  "Calculus on manifolds" , Benjamin (1965)
 +
|-
 +
|}

Latest revision as of 06:58, 29 June 2014

volume element. 2020 Mathematics Subject Classification: Primary: 53B20 [MSN][ZBL]

The volume form is a special differential form defined on oriented Riemannian manifolds and which introduces a natural concept of measure on the manifold.

On vector spaces

First consider a real vector space $V$ of dimension $n$ with a given orientation and an inner product. The corresponding volume form, or volume element, is the unique element $\omega$ of the space $\Lambda^n V$ of $n$-forms (namely alternating multilinear maps $L : V^n \to \mathbb R$; cf. Exterior form), such that $\omega (v_1, \ldots, v_n) =1$ for each orthonormal (with respect to the given inner product) positively oriented basis of $V$. Recall that $\Lambda^n V$ is one-dimensional. If $V = \mathbb R^n$ with the standard inner product and orientation, then, for each $n$-tuple of vectors $w_1, \ldots, w_n \in \mathbb R^n$, $\omega (w_1, \ldots, w_n)$ is the determinant of the $n\times n$ matrix whose $ij$ entry is the $i$-th corrdinate (in the standared basis) of $w_j$. Moreover, $|\omega (v_1, \ldots, v_n)|$ is the Lebesgue measure (i.e. the $n$-dimensional volume) of the parallelopipedon $P$ spanned by the vectors $v_j$, namely \[ P = \left\{\sum_j \lambda_j v_j : \lambda_j\in [0,1]\right\}\, . \] The latter property gives an intuitive justification for the terms volume form and volume element: in particular, if $U$ is an open bounded subset of $\mathbb R^n$, then \begin{equation}\label{e:integra} \int_U \omega \end{equation} is the Lebesgue measure of $U$ (for the definition of the integral in \eqref{e:integra} see Integration on manifolds).

On Riemannian manifolds

If $(\Sigma, g)$ is an oriented Riemannian manifold, then the volume form $\omega$ on $\Sigma$ is the unique element of $\Lambda^n M$ such that, for each $x\in M$, $\omega|_x$ is the volume form on the vector space $T_x M$ relative to the inner product $g_x$ and the orientation chosen. One often writes ${\rm dV}$ or ${\rm dvol}$, for the volume form on $\Sigma$, even though the volume form might not be exact (indeed the volume form is never exact if $M$ is compact and without boundary).

Formula in local coordinates

Consider a local coordinate patch $U$ on $\Sigma$. Let $x_1, \ldots, x_n$ be the corresponding coordinates and denote by $g_{ij} (x)$ the inner product (at the point $x$) of the vector fields $\frac{\partial}{\partial x_i}$ and $\frac{\partial}{\partial x_j}$(see Tangent vector), namely \[ g_{ij} (x) = g_x \left(\frac{\partial}{\partial x_i}, \frac{\partial}{\partial x_j}\right)\, . \] With a slight abuse of notation, let $\det g (x)$ be the determinant of the matrix $(g_{ij} (x))_{ij}$. We then have the following formula for the volume form ${\rm dvol}$ on $U$: \[ {\rm dvol} = \varepsilon\, \sqrt{\det g (x)}\, dx_1\wedge \ldots \wedge dx_n \] where $\varepsilon =1 $ if $\frac{\partial}{\partial x_1}, \ldots, \frac{\partial}{\partial x_n}$ is positively oriented, and $-1$ otherwise.

Volume measure

The volume form defines naturally a measure $\mu$ on any Riemannian manifold. Indeed, given a smooth continuous compactly supported function $f$ we can define \[ \int_\Sigma f d\mu \] as the integral of the form $f {\rm dvol}$, in the sense of integration on manifolds. It can be checked that such integral coincides with the Lebesgue integral with respect to a $\sigma$-additive measure on an appropriate $\sigma$-algebra (assuming that the manifold is second countable).

Divergence and Hodge star

Assume $(\Sigma, g)$ is an oriented Riemannian manifold. If $\nabla$ is the Levi-Civita connection on $\Sigma$, the divergence of a vector field $X$ is then given, at the point $x$, by \[ {\rm div}\, X = \sum_j \nabla_{e_j} X\, , \] where $e_1, \ldots, e_n$ is any orthonormal frame defined in a neighborhood of $x$. In local coordinates, if $X = \sum \psi^j \frac{\partial}{\partial x_j}$, then \[ {\rm div}\, X = (\det g)^{-1/2} \sum_j \frac{\partial}{\partial x_j} \left( (\det g)^{1/2} \psi^j\right)\, . \] Let $*$ denote the Hodge star operator on the oriented Riemannian manifold $\Sigma$ (cf. Laplace operator).It then turns out that \begin{equation}\label{e:star} d (*X) = {\rm div}\, X\, {\rm dvol}\, . \end{equation}

Stokes and divergence theorems

The latter identity gives a link between Stokes theorem on differential forms on the divergence theorem (also called Gauss-Green theorem or Ostrogradski formula). More precisely, assume that $(\Sigma,g)$ is a compact oriented Riemannian manifold with boundary $\partial \Sigma$. Denote by:

  • ${\rm d Vol}$ the volume form on $\Sigma$;
  • ${\rm dvol}$ the volume form on $\partial \Sigma$;
  • $\nu$ the exterior unit normal to $\partial \Sigma$ (namely, if $e_1, \ldots, e_{n-1}$ is a positively oriented ortonormal frame on $U\cap \partial \Sigma$, then $e_1, \ldots, e_{n-1}, \nu$ is a positively oriented orthonormal frame in $U\cap \Sigma$).

Stokes theorem together with \eqref{e:star} implies that \[ \int_\Sigma {\rm div}\, X\, {\rm dVol} = \int_{\partial \Sigma} *X\, . \] On the other it can be shown that \[ \int_{\partial \Sigma} *X = \int_{\partial \Sigma} g (X, \nu)\, {\rm dvol}\, , \] thereby concluding \[ \int_\Sigma {\rm div}\, X\, {\rm dVol} = \int_{\partial \Sigma} g (X, \nu)\, {\rm dvol}\, . \]

Area formula

For $C^1$ submanifolds of the Euclidean space $\mathbb R^N$, the volume form can be related to a measure-theoretic concept of volume. More precisely, assume $\Sigma$ is a (compact) $m$-dimensional $C^1$ submanifold of $\mathbb R^N$. Let ${\rm dvol}$ be the volume form on $\Sigma$ related to the Riemannian structure given by the restriction of the standard Euclidean inner product on the tangent space to $\Sigma$. The area formula implies that, if we denote by $\mathcal{H}^m$ the $m$-dimensional Hausdorff measure on $\mathbb R^N$, then \[ \int_\Sigma {\rm dvol} = \mathcal{H}^m (\Sigma)\, . \]

References

[Be] Y. Choquet-Bruhat, C. DeWitt-Morette, M. Dillard-Bleick, "Analysis, manifolds, and physics" , North-Holland (1977) (Translated from French)
[Ha] M. Hazewinkel, "A tutorial introduction to differentiable manifolds and calculus on manifolds" W. Schiehlen (ed.) W. Wedig (ed.) , Analysis and estimation of stochastic mechanical systems , Springer (Wien) (1988) pp. 316–340
[Sp] M. Spivak, "Calculus on manifolds" , Benjamin (1965)
How to Cite This Entry:
Volume form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Volume_form&oldid=14716