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A closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096530/v0965301.png" /> of points of a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096530/v0965302.png" /> in which there has been specified a [[Vector field|vector field]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096530/v0965303.png" /> such that the normal vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096530/v0965304.png" /> is orthogonal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096530/v0965305.png" /> everywhere on its boundary surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096530/v0965306.png" />. The vector tube <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096530/v0965307.png" /> consists of vector lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096530/v0965308.png" /> of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096530/v0965309.png" />, i.e. curves in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096530/v09653010.png" /> at each point of which the tangent direction coincides with the direction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096530/v09653011.png" />. A line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096530/v09653012.png" /> is completely contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096530/v09653013.png" /> if one point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096530/v09653014.png" /> is contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096530/v09653015.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096530/v09653016.png" /> is the field of velocities of a stationary liquid flow, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096530/v09653017.png" /> is the trajectory of the liquid particles, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096530/v09653018.png" /> is the part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096530/v09653019.png" /> "swept along"  by the motion of a given amount of liquid particles.
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The intensity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096530/v09653020.png" /> of the tube <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096530/v09653021.png" /> in the cross-section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096530/v09653022.png" /> is the flux (cf. [[Vector analysis|Vector analysis]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096530/v09653023.png" /> across <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096530/v09653024.png" />:
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{{TEX|auto}}
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{{TEX|done}}
  
<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/v096530/v09653025.png" /></td> </tr></table>
+
A closed set  $  \Phi $
 +
of points of a space  $  \Omega $
 +
in which there has been specified a [[Vector field|vector field]]  $  \mathbf a ( M) $
 +
such that the normal vector  $  \mathbf n $
 +
is orthogonal to  $  \mathbf a $
 +
everywhere on its boundary surface  $  S $.
 +
The vector tube  $  \Phi $
 +
consists of vector lines  $  \Gamma $
 +
of the field  $  \mathbf a $,
 +
i.e. curves in  $  \Omega $
 +
at each point of which the tangent direction coincides with the direction of  $  \mathbf a $.  
 +
A line  $  \Gamma $
 +
is completely contained in  $  \Phi $
 +
if one point of  $  \Gamma $
 +
is contained in  $  \Phi $.  
 +
If  $  \mathbf a $
 +
is the field of velocities of a stationary liquid flow, then  $  \Gamma $
 +
is the trajectory of the liquid particles, while  $  \Phi $
 +
is the part of  $  \Omega $"
 +
swept along" by the motion of a given amount of liquid particles.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096530/v09653026.png" /> is the unit normal vector to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096530/v09653027.png" />. If the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096530/v09653028.png" /> is solenoidal (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096530/v09653029.png" />), the law of preservation of the intensity of the vector tube holds:
+
The intensity  $  I $
 +
of the tube  $  \Phi $
 +
in the cross-section  $  S  ^  \prime  $
 +
is the flux (cf. [[Vector analysis|Vector analysis]]) of $  \mathbf a $
 +
across  $  S  ^  \prime  $:
  
<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/v096530/v09653030.png" /></td> </tr></table>
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$$
 +
I( S  ^  \prime  )  = \int\limits \int\limits S  ^  \prime  ( \mathbf a , \mathbf n )  d \sigma ,
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096530/v09653031.png" /> be the orthogonal Cartesian coordinates of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096530/v09653032.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096530/v09653033.png" /> be the coordinates of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096530/v09653034.png" />. Then the boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096530/v09653035.png" /> is locally defined by an equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096530/v09653036.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096530/v09653037.png" /> satisfies the partial differential equation
+
where  $  \mathbf n $
 +
is the unit normal vector to  $  S  ^  \prime  $.  
 +
If the field  $  \mathbf a $
 +
is solenoidal ( $  \mathop{\rm div}  \mathbf a = 0 $),
 +
the law of preservation of the intensity of the vector tube holds:
  
<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/v096530/v09653038.png" /></td> </tr></table>
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$$
 +
I( S  ^  \prime  )  = I( S  ^ {\prime\prime} ) .
 +
$$
  
<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/v096530/v09653039.png" /></td> </tr></table>
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Let  $  a _ {1} ( x, y, z), a _ {2} ( x, y, z), a _ {3} ( x, y, z) $
 +
be the orthogonal Cartesian coordinates of the vector  $  \mathbf a = \mathbf a ( M), $
 +
and let  $  x, y, z $
 +
be the coordinates of the point  $  M $.
 +
Then the boundary of  $  \Phi $
 +
is locally defined by an equation  $  F( x, y, z) = \textrm{ const } $,
 +
where  $  F( x, y, z) $
 +
satisfies the partial differential equation
  
 +
$$
 +
( \mathbf a , \nabla F  ) =
 +
$$
  
 +
$$
 +
= \
 +
a _ {1} ( x, y, z)
 +
\frac{\partial  F }{\partial  x }
 +
+
 +
a _ {2} ( x, y, z)
 +
\frac{\partial  F }{\partial  y }
 +
+ a _ {3} ( x, y, z)
 +
\frac{\partial  F }{\partial  z }
 +
  =  0.
 +
$$
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.P. Wills,  "Vector analysis with an introduction to tensor analysis" , Dover, reprint  (1958)  pp. Sect. 45</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.P. Wills,  "Vector analysis with an introduction to tensor analysis" , Dover, reprint  (1958)  pp. Sect. 45</TD></TR></table>

Latest revision as of 08:28, 6 June 2020


A closed set $ \Phi $ of points of a space $ \Omega $ in which there has been specified a vector field $ \mathbf a ( M) $ such that the normal vector $ \mathbf n $ is orthogonal to $ \mathbf a $ everywhere on its boundary surface $ S $. The vector tube $ \Phi $ consists of vector lines $ \Gamma $ of the field $ \mathbf a $, i.e. curves in $ \Omega $ at each point of which the tangent direction coincides with the direction of $ \mathbf a $. A line $ \Gamma $ is completely contained in $ \Phi $ if one point of $ \Gamma $ is contained in $ \Phi $. If $ \mathbf a $ is the field of velocities of a stationary liquid flow, then $ \Gamma $ is the trajectory of the liquid particles, while $ \Phi $ is the part of $ \Omega $" swept along" by the motion of a given amount of liquid particles.

The intensity $ I $ of the tube $ \Phi $ in the cross-section $ S ^ \prime $ is the flux (cf. Vector analysis) of $ \mathbf a $ across $ S ^ \prime $:

$$ I( S ^ \prime ) = \int\limits \int\limits S ^ \prime ( \mathbf a , \mathbf n ) d \sigma , $$

where $ \mathbf n $ is the unit normal vector to $ S ^ \prime $. If the field $ \mathbf a $ is solenoidal ( $ \mathop{\rm div} \mathbf a = 0 $), the law of preservation of the intensity of the vector tube holds:

$$ I( S ^ \prime ) = I( S ^ {\prime\prime} ) . $$

Let $ a _ {1} ( x, y, z), a _ {2} ( x, y, z), a _ {3} ( x, y, z) $ be the orthogonal Cartesian coordinates of the vector $ \mathbf a = \mathbf a ( M), $ and let $ x, y, z $ be the coordinates of the point $ M $. Then the boundary of $ \Phi $ is locally defined by an equation $ F( x, y, z) = \textrm{ const } $, where $ F( x, y, z) $ satisfies the partial differential equation

$$ ( \mathbf a , \nabla F ) = $$

$$ = \ a _ {1} ( x, y, z) \frac{\partial F }{\partial x } + a _ {2} ( x, y, z) \frac{\partial F }{\partial y } + a _ {3} ( x, y, z) \frac{\partial F }{\partial z } = 0. $$

Comments

References

[a1] A.P. Wills, "Vector analysis with an introduction to tensor analysis" , Dover, reprint (1958) pp. Sect. 45
How to Cite This Entry:
Vector tube. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vector_tube&oldid=18570
This article was adapted from an original article by Yu.P. Pyt'ev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article