Difference between revisions of "Weierstrass criterion"
From Encyclopedia of Mathematics
(Importing text file) |
(TeX) |
||
| Line 1: | Line 1: | ||
| + | {{TEX|done}} | ||
''for a minimal surface'' | ''for a minimal surface'' | ||
| − | For a two-dimensional surface in | + | For a two-dimensional surface in $n$-dimensional Euclidean space $E^n$, $n\geq3$, with [[Isothermal coordinates|isothermal coordinates]] $u$ and $v$ of class $C^2$, to be minimal (cf. [[Minimal surface|Minimal surface]]), it is necessary and sufficient that the components of its position vector be harmonic functions of $(u,v)$. |
Revision as of 11:50, 5 July 2014
for a minimal surface
For a two-dimensional surface in $n$-dimensional Euclidean space $E^n$, $n\geq3$, with isothermal coordinates $u$ and $v$ of class $C^2$, to be minimal (cf. Minimal surface), it is necessary and sufficient that the components of its position vector be harmonic functions of $(u,v)$.
Comments
References
| [a1] | J.C.C. Nitsche, "Vorlesungen über Minimalflächen" , Springer (1975) pp. §455 |
| [a2] | K. Weierstrass, "Math. Werke" , 3 , G. Olms, reprint (1967) |
How to Cite This Entry:
Weierstrass criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weierstrass_criterion&oldid=14513
Weierstrass criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Weierstrass_criterion&oldid=14513
This article was adapted from an original article by I.Kh. Sabitov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article