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Difference between revisions of "Isothermal net"

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An [[Orthogonal net|orthogonal net]] on a surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052900/i0529001.png" /> in Euclidean <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052900/i0529002.png" />-space in which the small quadrangles formed by two pairs of lines from distinct families are, up to infinitesimal quantities of the first order, squares. The lines of an isothermal net are level curves of two conjugate harmonic functions. In the parameters of an isothermal net the line element has the form:
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An [[Orthogonal net|orthogonal net]] on a surface $V^2$ in Euclidean $3$-space in which the small quadrangles formed by two pairs of lines from distinct families are, up to infinitesimal quantities of the first order, squares. The lines of an isothermal net are level curves of two conjugate harmonic functions. In the parameters of an isothermal net the line element has the form:
  
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<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/i/i052/i052900/i0529003.png" /></td> </tr></table>
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$$ds^2=\lambda^2(du^2+dv^2),$$
  
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052900/i0529004.png" />. An isothermal net is a particular case of a [[Rhombic net|rhombic net]]. On a surface of rotation the meridians and parallels form an isothermal net; an [[Asymptotic net|asymptotic net]] on a minimal surface is isothermal.
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where $\lambda=\lambda(u,v)$. An isothermal net is a particular case of a [[Rhombic net|rhombic net]]. On a surface of rotation the meridians and parallels form an isothermal net; an [[Asymptotic net|asymptotic net]] on a minimal surface is isothermal.
  
  

Latest revision as of 19:38, 7 July 2014

An orthogonal net on a surface $V^2$ in Euclidean $3$-space in which the small quadrangles formed by two pairs of lines from distinct families are, up to infinitesimal quantities of the first order, squares. The lines of an isothermal net are level curves of two conjugate harmonic functions. In the parameters of an isothermal net the line element has the form:

$$ds^2=\lambda^2(du^2+dv^2),$$

where $\lambda=\lambda(u,v)$. An isothermal net is a particular case of a rhombic net. On a surface of rotation the meridians and parallels form an isothermal net; an asymptotic net on a minimal surface is isothermal.


Comments

See also Isothermal coordinates and the references given there.

How to Cite This Entry:
Isothermal net. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Isothermal_net&oldid=32394
This article was adapted from an original article by V.T. Bazylev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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