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