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Difference between revisions of "Potential net"

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''Egorov net''
 
''Egorov net''
  
 
An [[Orthogonal net|orthogonal net]] on a two-dimensional surface in Euclidean space that is mapped to itself by the potential motion of a fluid on this surface. In parameters of the potential net the line element of this surface has the form
 
An [[Orthogonal net|orthogonal net]] on a two-dimensional surface in Euclidean space that is mapped to itself by the potential motion of a fluid on this surface. In parameters of the potential net the line element of this surface has 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/p/p074/p074110/p0741101.png" /></td> </tr></table>
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$$
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d s  ^ {2}  =
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\frac{\partial  \Phi }{\partial  u }
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\
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d u  ^ {2} +
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\frac{\partial  \Phi }{\partial  v }
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  d v  ^ {2} ,
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$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074110/p0741102.png" /> is the potential of the velocity field of the fluid. Each orthogonal semi-geodesic net is potential. A particular case of a potential net is a [[Liouville net|Liouville net]]. D.F. Egorov was the first (1901) to consider potential nets.
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where $  \Phi = \Phi ( u , v ) $
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is the potential of the velocity field of the fluid. Each orthogonal semi-geodesic net is potential. A particular case of a potential net is a [[Liouville net|Liouville net]]. D.F. Egorov was the first (1901) to consider potential nets.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D.F. Egorov,  "Papers in differential geometry" , Moscow  (1970)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.I. Shulikovskii,  "Classical differential geometry in a tensor setting" , Moscow  (1963)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D.F. Egorov,  "Papers in differential geometry" , Moscow  (1970)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.I. Shulikovskii,  "Classical differential geometry in a tensor setting" , Moscow  (1963)  (In Russian)</TD></TR></table>

Latest revision as of 08:07, 6 June 2020


Egorov net

An orthogonal net on a two-dimensional surface in Euclidean space that is mapped to itself by the potential motion of a fluid on this surface. In parameters of the potential net the line element of this surface has the form

$$ d s ^ {2} = \frac{\partial \Phi }{\partial u } \ d u ^ {2} + \frac{\partial \Phi }{\partial v } d v ^ {2} , $$

where $ \Phi = \Phi ( u , v ) $ is the potential of the velocity field of the fluid. Each orthogonal semi-geodesic net is potential. A particular case of a potential net is a Liouville net. D.F. Egorov was the first (1901) to consider potential nets.

References

[1] D.F. Egorov, "Papers in differential geometry" , Moscow (1970) (In Russian)
[2] V.I. Shulikovskii, "Classical differential geometry in a tensor setting" , Moscow (1963) (In Russian)
How to Cite This Entry:
Potential net. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Potential_net&oldid=48263
This article was adapted from an original article by V.T. Bazylev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article