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Difference between revisions of "Flux of a vector field"

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A concept in the theory of vector fields. The flux of a vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040670/f0406701.png" /> through the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040670/f0406702.png" /> is expressed, up to sign, by the surface integral
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{{TEX|done}}
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{{MSC|26B20}}
  
<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/f/f040/f040670/f0406703.png" /></td> </tr></table>
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A concept in the [[Integral calculus|integral calculus]] of functions in several variables. Let $\Omega\subset \mathbb R^n$ be an open set and $v$ a (continuous) [[Vector field|vector field]] on $\Omega$, namely a map $v: \Omega\to \mathbb R^n$. If $\Sigma\subset\Omega$ is a $C^1$ $n-1$-dimensional surface oriented by a (continuous) unit normal $\nu$, the flux of the vector field $v$ through the surface $\Sigma$ is given by the integral
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\begin{equation}\label{e:flux}
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\int_\Sigma v\cdot \nu\, .
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\end{equation}
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The integral in \eqref{e:flux} is a surface integral, which is computed using the [[Area formula]]. If $\Sigma$ is given by the graph of a function $f: \mathbb R^{n-1} \supset V \to \mathbb R$ with its natural orientation, namely with
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\[
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\nu (x_1, \ldots, x_{n-1}, f(x_1, \ldots, x_{n-1})) = \frac{(-\nabla f, 1)}{\sqrt{1+|\nabla f|^2}} (x_1, \ldots, x_{n-1})\, ,
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\]
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then we have the useful formula
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\[
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\int_{\partial U} v\cdot \nu = \int \left(v_n (x', f (x'))
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- \frac{\partial f}{\partial x_1} (x') v_1 (x', f(x'))
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- \ldots - \frac{\partial f}{\partial x_{n-1}} (x') v_{n-1} (x', f (x'))\right)\, dx'\, ,
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\]
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where $x' = (x_1, \ldots , x_{n-1})$. The latter formula can be used to define the flux of a vector field over a general $C^1$ surface using a [[Partition of unity|partition of unity]].  
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040670/f0406704.png" /> is the unit normal vector to the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040670/f0406705.png" /> (it is assumed that the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040670/f0406706.png" /> changes continuously over the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040670/f0406707.png" />). The flux of the velocity field of a fluid is equal to the volume of fluid passing through the surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040670/f0406708.png" /> per unit time.
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An alternative powerful way to define the surface integral in \eqref{e:flux} is to resort to [[Differential form|differential forms]] and their [[Integration on manifolds|integration of manifolds]], see {{Cite|Sp}}. More precisely, if $v_1, \ldots, v_n$ are the components of the vector function $v$, it is convenient to introduce the $n-1$-form
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\[
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\omega = \sum_{i=1}^n (-1)^{i-1} v_i dx_1 \wedge \ldots \wedge dx_{i-1}\wedge dx_{i+1} \wedge \ldots \wedge dx_n\, .
 +
\]
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Then it turns out that the integral in \eqref{e:flux} is in fact
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\[
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\int_{\Sigma} \omega\, .
 +
\]
  
 
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The [[Divergence theorem|divergence theorem]] relates the flux of a differentiable vector field of $v$ through the boundary of a regular open set $U$ to the integral over $U$ of the divergence of $v$. This important theorem (which goes also under the name Green formula, Gauss-Green formula, Gauss formula, Ostrogradski formula, Gauss-Ostrogradski formula or Gauss-Green-Ostrogradski formula) is a generalization of the [[Fundamental theorem of calculus]] and it is a particular case of the more general [[Stokes theorem|Stokes theorem]] on integral of differential forms.
 
 
====Comments====
 
The flux of a differential vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040670/f0406709.png" /> (defined by the formula above) is related to the [[Divergence|divergence]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040670/f04067010.png" />:
 
 
 
<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/f/f040/f040670/f04067011.png" /></td> </tr></table>
 
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040670/f04067012.png" /> is the volume element in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040670/f04067013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040670/f04067014.png" /> is the [[Hamilton operator|Hamilton operator]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040670/f04067015.png" />. This equation is called the divergence theorem or also Green's theorem in space, cf. [[#References|[a1]]] and [[Stokes theorem|Stokes theorem]].
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> M. Spivak,  "Calculus on manifolds" , Benjamin (1965)</TD></TR></table>
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{|
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|valign="top"|{{Ref|CH}}|| R. Courant,  D. Hilbert,    "Methods of mathematical physics. Partial  differential equations" ,  '''2''' , Interscience  (1965)  (Translated  from German)  {{MR|0195654}} {{ZBL|}}
 +
|-
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|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}}
 +
|-
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|valign="top"|{{Ref|Kr}}|| A.M. Krall,  "Applied analysis" , Reidel  (1986)  pp. 380
 +
|-
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|valign="top"|{{Ref|Os1}}||  M.V. Ostrogradski,  ''Mém. Acad. Sci. St. Petersbourg. Sér. 6. Sci.  Math. Phys. et Naturelles'' , '''1'''  (1831)  pp. 117–122
 +
|-
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|valign="top"|{{Ref|Os2}}||  M.V. Ostrogradski,  ''Mém. Acad. Sci. St. Petersbourg. Sér. 6. Sci.  Math. Phys. et Naturelles'' , '''1'''  (1838)  pp. 35–58
 +
|-
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|valign="top"|{{Ref|Tr}}|| H. Triebel,  "Analysis and mathematical physics" , Reidel  (1986)  pp. Sect. 9.3.1
 +
|-
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|valign="top"|{{Ref|Wi}}||  A.P. Wills,  "Vector analysis with an introduction to tensor analysis"  , Dover, reprint  (1958)  pp. 97ff
 +
|-
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|valign="top"|{{Ref|vW }}||C. von Westenholz,  "Differential forms in mathematical physics" , North-Holland (1981) pp. 286ff
 +
|-
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|}

Revision as of 15:20, 4 February 2014

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

A concept in the integral calculus of functions in several variables. Let $\Omega\subset \mathbb R^n$ be an open set and $v$ a (continuous) vector field on $\Omega$, namely a map $v: \Omega\to \mathbb R^n$. If $\Sigma\subset\Omega$ is a $C^1$ $n-1$-dimensional surface oriented by a (continuous) unit normal $\nu$, the flux of the vector field $v$ through the surface $\Sigma$ is given by the integral \begin{equation}\label{e:flux} \int_\Sigma v\cdot \nu\, . \end{equation} The integral in \eqref{e:flux} is a surface integral, which is computed using the Area formula. If $\Sigma$ is given by the graph of a function $f: \mathbb R^{n-1} \supset V \to \mathbb R$ with its natural orientation, namely with \[ \nu (x_1, \ldots, x_{n-1}, f(x_1, \ldots, x_{n-1})) = \frac{(-\nabla f, 1)}{\sqrt{1+|\nabla f|^2}} (x_1, \ldots, x_{n-1})\, , \] then we have the useful formula \[ \int_{\partial U} v\cdot \nu = \int \left(v_n (x', f (x')) - \frac{\partial f}{\partial x_1} (x') v_1 (x', f(x')) - \ldots - \frac{\partial f}{\partial x_{n-1}} (x') v_{n-1} (x', f (x'))\right)\, dx'\, , \] where $x' = (x_1, \ldots , x_{n-1})$. The latter formula can be used to define the flux of a vector field over a general $C^1$ surface using a partition of unity.

An alternative powerful way to define the surface integral in \eqref{e:flux} is to resort to differential forms and their integration of manifolds, see [Sp]. More precisely, if $v_1, \ldots, v_n$ are the components of the vector function $v$, it is convenient to introduce the $n-1$-form \[ \omega = \sum_{i=1}^n (-1)^{i-1} v_i dx_1 \wedge \ldots \wedge dx_{i-1}\wedge dx_{i+1} \wedge \ldots \wedge dx_n\, . \] Then it turns out that the integral in \eqref{e:flux} is in fact \[ \int_{\Sigma} \omega\, . \]

The divergence theorem relates the flux of a differentiable vector field of $v$ through the boundary of a regular open set $U$ to the integral over $U$ of the divergence of $v$. This important theorem (which goes also under the name Green formula, Gauss-Green formula, Gauss formula, Ostrogradski formula, Gauss-Ostrogradski formula or Gauss-Green-Ostrogradski formula) is a generalization of the Fundamental theorem of calculus and it is a particular case of the more general Stokes theorem on integral of differential forms.

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:
Flux of a vector field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Flux_of_a_vector_field&oldid=31304
This article was adapted from an original article by BSE-3 (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article