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A formula for a [[Minimal surface|minimal surface]], of the form
 
A formula for a [[Minimal surface|minimal surface]], of 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/s/s083/s083520/s0835201.png" /></td> </tr></table>
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$$
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\mathbf r ( u, v)  =   \mathop{\rm Re} \left \{ \mathbf r ( t) + i \int\limits [ \mathbf n , d \mathbf r ] \right
 +
\} ,
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$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083520/s0835202.png" /> is the position vector of the minimal surface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083520/s0835203.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083520/s0835204.png" /> is the position vector of an arbitrary non-closed analytic (with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083520/s0835205.png" />) curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083520/s0835206.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083520/s0835207.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083520/s0835208.png" /> is the unit normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083520/s0835209.png" /> along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083520/s08352010.png" /> (analytically dependent on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083520/s08352011.png" />). After integration, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083520/s08352012.png" /> is replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s083/s083520/s08352013.png" />. This formula was established by H.A. Schwarz (1875); it gives an explicit solution to the [[Björling problem|Björling problem]].
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where $  \mathbf r ( u, v) $
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is the position vector of the minimal surface $  F $,  
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$  \mathop{\rm Re} \{ \mathbf r ( t) \} $
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is the position vector of an arbitrary non-closed analytic (with respect to $  t $)  
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curve $  L $
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on $  F $,  
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and $  \mathbf n ( t) $
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is the unit normal to $  F $
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along $  L $(
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analytically dependent on $  t $).  
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After integration, $  t $
 +
is replaced by $  t = u+ iv $.  
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This formula was established by H.A. Schwarz (1875); it gives an explicit solution to the [[Björling problem|Björling problem]].

Latest revision as of 08:12, 6 June 2020


A formula for a minimal surface, of the form

$$ \mathbf r ( u, v) = \mathop{\rm Re} \left \{ \mathbf r ( t) + i \int\limits [ \mathbf n , d \mathbf r ] \right \} , $$

where $ \mathbf r ( u, v) $ is the position vector of the minimal surface $ F $, $ \mathop{\rm Re} \{ \mathbf r ( t) \} $ is the position vector of an arbitrary non-closed analytic (with respect to $ t $) curve $ L $ on $ F $, and $ \mathbf n ( t) $ is the unit normal to $ F $ along $ L $( analytically dependent on $ t $). After integration, $ t $ is replaced by $ t = u+ iv $. This formula was established by H.A. Schwarz (1875); it gives an explicit solution to the Björling problem.

How to Cite This Entry:
Schwarz formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schwarz_formula&oldid=48630
This article was adapted from an original article by I.Kh. Sabitov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article