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A formula in the [[Integral calculus|integral calculus]] of functions in several variables that establishes a link between an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070600/o0706001.png" />-fold integral over a domain and an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070600/o0706002.png" />-fold integral over its boundary. Let the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070600/o0706003.png" /> and their partial derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070600/o0706004.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070600/o0706005.png" />, be Lebesgue integrable in a bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070600/o0706006.png" /> whose boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070600/o0706007.png" /> is the union of a finite set of piecewise-smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070600/o0706008.png" />-dimensional hypersurfaces oriented using the exterior normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070600/o0706009.png" />. The Ostrogradski formula then takes the form
 
  
<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/o/o070/o070600/o07060010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
 
  
<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/o/o070/o070600/o07060011.png" /></td> </tr></table>
+
{{TEX|done}}
 +
{{MSC|26B20}}
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070600/o07060012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070600/o07060013.png" />, are the direction cosines of the exterior normals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070600/o07060014.png" /> of the hypersurfaces forming the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070600/o07060015.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070600/o07060016.png" />, then formula (1) can be expressed in the form
+
The divergence theorem gives a formula in the [[Integral calculus|integral calculus]] of functions in several variables that establishes a link between an $n$-fold integral over a domain and an $n-1$-fold integral over its boundary. The formula, which can be regarded as a direct generalization of the [[Fundamental theorem of calculus]], is often referred to as:
 +
Green formula, Gauss-Green formula, Gauss formula, Ostrogradski formula, Gauss-Ostrogradski formula or Gauss-Green-Ostrogradski formula.
  
<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/o/o070/o070600/o07060017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
Let us recall that, given an open set $U\subset \mathbb R^n$, a [[Vector field|vector field]] on $U$ is a map $v: U \to \mathbb R^n$. If $v$ is differentiable and the components of the vector field are denoted by $v_1, \ldots, v_n$, then the [[Divergence|divergence]] of $v$ is given by the function
 +
\[
 +
{\rm div}\, v := \sum_{i=1}^n \frac{\partial v_i}{\partial x_i}\, .
 +
\]
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070600/o07060018.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070600/o07060019.png" />-dimensional volume element in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070600/o07060020.png" /> while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070600/o07060021.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070600/o07060022.png" />-dimensional volume element on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070600/o07060023.png" />.
+
The divergence theorem asserts that
  
In terms of the vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070600/o07060024.png" />, the formulas (1) and (2) signify the equality of the integral of the [[Divergence|divergence]] of this field over the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070600/o07060025.png" /> to its flux (see [[Flux of a vector field|Flux of a vector field]]) over the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070600/o07060026.png" />:
+
'''Theorem 1'''
 +
If $v$ is a $C^1$ vector field, $\partial U$ is regular (i.e. can be described locally as the graph of a $C^1$ function) and $U$ is bounded, then
 +
\begin{equation}\label{e:divergence_thm}
 +
\int_U {\rm div}\, v = \int_{\partial U} v\cdot \nu\, ,
 +
\end{equation}
 +
where $\nu$ denotes the unit normal to $\partial U$ pointing towards the "exterior" (namely $\mathbb R^n \setminus \overline{U}$).
  
<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/o/o070/o070600/o07060027.png" /></td> </tr></table>
+
When the dimension $n$ is $1$ and $U$ is an interval $I =[a,b]$, the left hand side of \eqref{e:divergence_thm} is given by
 +
\[
 +
\int_a^b f' (x)\, dx
 +
\]
 +
and the right hand side is given by $f(b)-f(a)$: the theorem is therefore a generalization of the [[Fundamental theorem of calculus]]. For larger $n$ the integral on the right hand side of \eqref{e:divergence_thm} is a surface integral, which is computed using the [[Area formula]], and is called [[Flux of a vector field|flux of the vector field]] $v$ through $\partial U$. If $v$ is compactly supported in a region $V$ where $U\cap V$ is the subgraph of a function $f$, then the flux of $v$ takes a simple form. More precisely, assume that $U\cap V = \{(x_1, \ldots, x_n)\in V : x_n < f(x_1, \ldots, x_{n-1})\}$. Then, under the assumption that $v$ vanishes outside $V$, we have
 +
\[
 +
\int_{\partial U} v\cdot \nu = \int \left(v_n (x', f (x'))
 +
- \frac{\partial f}{\partial x_1} (x') v_1 (x', f(x'))
 +
- \ldots - \frac{\partial f}{\partial x_{n-1}} (x') v_{n-1} (x', f (x'))\right)\, dx'\, ,
 +
\]
 +
where $x'= (x_1, \ldots, x_{n-1})$. This formula can be used, together with a [[Partition of unity|partition of unity]], to compute (or define) the flux of a general vector field $v$.
  
For smooth functions, the formula was first obtained for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070600/o07060028.png" />-dimensional case by M.V. Ostrogradski in 1828 (published in 1831, see [[#References|[1]]]). He later extended it (1834) to cover <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070600/o07060029.png" />-fold integrals for an arbitrary natural <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070600/o07060030.png" /> (published in 1838, see [[#References|[2]]]). Using this formula, Ostrogradski found an expression for the derivative with respect to a parameter of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070600/o07060031.png" />-fold integral with variable limits, and obtained a formula for the variation of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070600/o07060032.png" />-fold integral; in one particular case, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070600/o07060033.png" />, the formula was obtained by C.F. Gauss in 1813, for this reason it is also sometimes called the Ostrogradski–Gauss formula. A generalization of this formula is the [[Stokes formula|Stokes formula]] for manifolds with boundary.
+
'''Remark 2''' The three key assumptions of Theorem 1 can be all heavily relaxed:
 +
* The boundedness of $U$ might be dropped if we assume that the vector field $v$ has suitable decay properties for $|x|\to \infty$.
 +
* The regularity of $\partial U$ might be considerably weakened. For instance the theorem holds when $\partial U$ is piecewise $C^1$ and the singularities are "corner-like". More generaly, it still holds if $\partial U$ is Lipschitz. An important generalization holds for [[Set of finite perimeter|sets of finite perimeter]]: in this case the flux of the vector field through $\partial U$ must be suitably defined in a measure theoretic sense.
 +
* The formula still holds when $v$ belongs to the [[Sobolev space]] $W^{1,p}$: in this case the right hand side of \eqref{e:divergence_thm} must be suitably interpreted, since $v$ is not necessarily continuous. Note that the almost everywhere differentiability of $v$ is not sufficient to guarantee \eqref{e:divergence_thm}, even when $v$ is continuous: see [[Absolute continuity]] for a counterexample.
 +
Simultaneous weakenings of more than one assumption need to be handled with care.
  
====References====
+
'''Remark 3''' The formula has also important generalizations of geometrical flavour. In particular, it holds on regular open subsets of [[Riemannian manifold|Riemannian manifolds]]. A far-reaching generalization is given by the [[Stokes formula]], using the language of [[Differential form|differential forms]].
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.V. Ostrogradski,  ''Mém. Acad. Sci. St. Petersbourg. Sér. 6. Sci. Math. Phys. et Naturelles'' , '''1'''  (1831)  pp. 117–122</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.V. Ostrogradski,  ''Mém. Acad. Sci. St. Petersbourg. Sér. 6. Sci. Math. Phys. et Naturelles'' , '''1'''  (1838)  pp. 35–58</TD></TR></table>
 
 
 
 
 
 
 
====Comments====
 
The result embodied in formula (2) above is most often known as the divergence theorem. It is (finally) equivalent to the Gauss formula (Gauss integral formula)
 
 
 
<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/o/o070/o070600/o07060034.png" /></td> </tr></table>
 
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070600/o07060035.png" /> is the surface element for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070600/o07060036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070600/o07060037.png" /> is the outward pointing unit normal at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070600/o07060038.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070600/o07060039.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070600/o07060040.png" />-th standard unit vector.
+
'''Remark 4''' Theorem 1 is attributed to different people. The $2$-dimensional case is credited often to Green, see {{Cite|Gr}}. The $3$-dimensional formula is attributed to Gauss, who proved a particular case in 1813, and to Ostrogradski (see {{Cite|Os1}}), who later generalized it to general dimension, {{Cite|Os2}}. Sometimes also Riemann is credited. However, it must be noted that the formula is already present in the works of Euler and other mathematician of the 18th century.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Triebel,  "Analysis and mathematical physics" , Reidel (1986)  pp. Sect. 9.3.1</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.M. Krall,  "Applied analysis" , Reidel  (1986)  pp. 380</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A.P. Wills,  "Vector analysis with an introduction to tensor analysis" , Dover, reprint  (1958)  pp. 97ff</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  C. von Westenholz,  "Differential forms in mathematical physics" , North-Holland  (1981)  pp. 286ff</TD></TR></table>
+
{|
 +
|valign="top"|{{Ref|CH}}|| R. Courant,  D. Hilbert,  "Methods of mathematical physics. Partial  differential equations" , '''2''' , Interscience  (1965)  (Translated  from German)  {{MR|0195654}} {{ZBL|}}
 +
|-
 +
|valign="top"|{{Ref|Gr}}|| G. Green,  "An essay on the application of mathematical analysis to the  theories of electricity and magnetism" , Nottingham (1828(Reprint:  Mathematical papers, Chelsea, reprint, 1970, pp. 1–82)  {{MR|}}  {{ZBL|21.0014.03}}
 +
|-
 +
|valign="top"|{{Ref|Kr}}|| A.M. Krall,  "Applied analysis" , Reidel  (1986)  pp. 380
 +
|-
 +
|valign="top"|{{Ref|Os1}}|| M.V. Ostrogradski,  ''Mém. Acad. Sci. St. Petersbourg. Sér. 6. Sci. Math. Phys. et Naturelles'' , '''1'''  (1831)  pp. 117–122
 +
|-
 +
|valign="top"|{{Ref|Os2}}|| M.V. Ostrogradski,  ''Mém. Acad. Sci. St. Petersbourg. Sér. 6. Sci. Math. Phys. et Naturelles'' , '''1'''  (1838) pp. 35–58
 +
|-
 +
|valign="top"|{{Ref|Tr}}|| H. Triebel,  "Analysis and mathematical physics" , Reidel  (1986)  pp. Sect. 9.3.1
 +
|-
 +
|valign="top"|{{Ref|Wi}}|| A.P. Wills,  "Vector analysis with an introduction to tensor analysis" , Dover, reprint  (1958)  pp. 97ff
 +
|-
 +
|valign="top"|{{Ref|vW }}||C. von Westenholz,  "Differential forms in mathematical physics" , North-Holland  (1981)  pp. 286ff
 +
|-
 +
|}

Revision as of 10:07, 3 February 2014

2020 Mathematics Subject Classification: Primary: 26B20 [MSN][ZBL]

The divergence theorem gives a formula in the integral calculus of functions in several variables that establishes a link between an $n$-fold integral over a domain and an $n-1$-fold integral over its boundary. The formula, which can be regarded as a direct generalization of the Fundamental theorem of calculus, is often referred to as: Green formula, Gauss-Green formula, Gauss formula, Ostrogradski formula, Gauss-Ostrogradski formula or Gauss-Green-Ostrogradski formula.

Let us recall that, given an open set $U\subset \mathbb R^n$, a vector field on $U$ is a map $v: U \to \mathbb R^n$. If $v$ is differentiable and the components of the vector field are denoted by $v_1, \ldots, v_n$, then the divergence of $v$ is given by the function \[ {\rm div}\, v := \sum_{i=1}^n \frac{\partial v_i}{\partial x_i}\, . \]

The divergence theorem asserts that

Theorem 1 If $v$ is a $C^1$ vector field, $\partial U$ is regular (i.e. can be described locally as the graph of a $C^1$ function) and $U$ is bounded, then \begin{equation}\label{e:divergence_thm} \int_U {\rm div}\, v = \int_{\partial U} v\cdot \nu\, , \end{equation} where $\nu$ denotes the unit normal to $\partial U$ pointing towards the "exterior" (namely $\mathbb R^n \setminus \overline{U}$).

When the dimension $n$ is $1$ and $U$ is an interval $I =[a,b]$, the left hand side of \eqref{e:divergence_thm} is given by \[ \int_a^b f' (x)\, dx \] and the right hand side is given by $f(b)-f(a)$: the theorem is therefore a generalization of the Fundamental theorem of calculus. For larger $n$ the integral on the right hand side of \eqref{e:divergence_thm} is a surface integral, which is computed using the Area formula, and is called flux of the vector field $v$ through $\partial U$. If $v$ is compactly supported in a region $V$ where $U\cap V$ is the subgraph of a function $f$, then the flux of $v$ takes a simple form. More precisely, assume that $U\cap V = \{(x_1, \ldots, x_n)\in V : x_n < f(x_1, \ldots, x_{n-1})\}$. Then, under the assumption that $v$ vanishes outside $V$, we have \[ \int_{\partial U} v\cdot \nu = \int \left(v_n (x', f (x')) - \frac{\partial f}{\partial x_1} (x') v_1 (x', f(x')) - \ldots - \frac{\partial f}{\partial x_{n-1}} (x') v_{n-1} (x', f (x'))\right)\, dx'\, , \] where $x'= (x_1, \ldots, x_{n-1})$. This formula can be used, together with a partition of unity, to compute (or define) the flux of a general vector field $v$.

Remark 2 The three key assumptions of Theorem 1 can be all heavily relaxed:

  • The boundedness of $U$ might be dropped if we assume that the vector field $v$ has suitable decay properties for $|x|\to \infty$.
  • The regularity of $\partial U$ might be considerably weakened. For instance the theorem holds when $\partial U$ is piecewise $C^1$ and the singularities are "corner-like". More generaly, it still holds if $\partial U$ is Lipschitz. An important generalization holds for sets of finite perimeter: in this case the flux of the vector field through $\partial U$ must be suitably defined in a measure theoretic sense.
  • The formula still holds when $v$ belongs to the Sobolev space $W^{1,p}$: in this case the right hand side of \eqref{e:divergence_thm} must be suitably interpreted, since $v$ is not necessarily continuous. Note that the almost everywhere differentiability of $v$ is not sufficient to guarantee \eqref{e:divergence_thm}, even when $v$ is continuous: see Absolute continuity for a counterexample.

Simultaneous weakenings of more than one assumption need to be handled with care.

Remark 3 The formula has also important generalizations of geometrical flavour. In particular, it holds on regular open subsets of Riemannian manifolds. A far-reaching generalization is given by the Stokes formula, using the language of differential forms.

Remark 4 Theorem 1 is attributed to different people. The $2$-dimensional case is credited often to Green, see [Gr]. The $3$-dimensional formula is attributed to Gauss, who proved a particular case in 1813, and to Ostrogradski (see [Os1]), who later generalized it to general dimension, [Os2]. Sometimes also Riemann is credited. However, it must be noted that the formula is already present in the works of Euler and other mathematician of the 18th century.

References

[CH] R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 2 , Interscience (1965) (Translated from German) MR0195654
[Gr] G. Green, "An essay on the application of mathematical analysis to the theories of electricity and magnetism" , Nottingham (1828) (Reprint: Mathematical papers, Chelsea, reprint, 1970, pp. 1–82) Zbl 21.0014.03
[Kr] A.M. Krall, "Applied analysis" , Reidel (1986) pp. 380
[Os1] M.V. Ostrogradski, Mém. Acad. Sci. St. Petersbourg. Sér. 6. Sci. Math. Phys. et Naturelles , 1 (1831) pp. 117–122
[Os2] M.V. Ostrogradski, Mém. Acad. Sci. St. Petersbourg. Sér. 6. Sci. Math. Phys. et Naturelles , 1 (1838) pp. 35–58
[Tr] H. Triebel, "Analysis and mathematical physics" , Reidel (1986) pp. Sect. 9.3.1
[Wi] A.P. Wills, "Vector analysis with an introduction to tensor analysis" , Dover, reprint (1958) pp. 97ff
[vW ] C. von Westenholz, "Differential forms in mathematical physics" , North-Holland (1981) pp. 286ff
How to Cite This Entry:
Divergence theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Divergence_theorem&oldid=31299
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article