Difference between revisions of "Schwarz formula"
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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 | ||
| − | + | $$ | |
| + | \mathbf r ( u, v) = \mathop{\rm Re} \left \{ \mathbf r ( t) + i \int\limits [ \mathbf n , d \mathbf r ] \right | ||
| + | \} , | ||
| + | $$ | ||
| − | where | + | 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|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=17542
Schwarz formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Schwarz_formula&oldid=17542
This article was adapted from an original article by I.Kh. Sabitov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article