Difference between revisions of "Blaschke-Weyl formula"
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| + | A variant of the [[Green formulas|Green formulas]] for the rotation field $ y $ | ||
| + | of an [[Infinitesimal deformation|infinitesimal deformation]] of a surface with position vector $ \mathbf x $: | ||
| + | |||
| + | $$ | ||
| + | 2 {\int\limits \int\limits } _ { G } | ||
| + | \left ( \mathbf x | ||
| + | \frac{\partial \mathbf y }{\partial u } | ||
| + | |||
| + | \frac{\partial \mathbf y }{\partial v } | ||
| + | \right ) du dv = \ | ||
| + | \cit _ {\partial G } | ||
| + | ( \mathbf x\mathbf y d \mathbf y ) . | ||
| + | $$ | ||
The proof and the idea of applying the Blaschke–Weyl formula to demonstrate the rigidity of ovaloids is due to W. Blaschke [[#References|[1]]] and to H. Weyl [[#References|[2]]]. For other applications see [[#References|[3]]]. The [[Herglotz formula|Herglotz formula]] is an analogue of the Blaschke–Weyl formula. The formula has been generalized to the case of infinitesimal deformations of surfaces in spaces of constant curvature. | The proof and the idea of applying the Blaschke–Weyl formula to demonstrate the rigidity of ovaloids is due to W. Blaschke [[#References|[1]]] and to H. Weyl [[#References|[2]]]. For other applications see [[#References|[3]]]. The [[Herglotz formula|Herglotz formula]] is an analogue of the Blaschke–Weyl formula. The formula has been generalized to the case of infinitesimal deformations of surfaces in spaces of constant curvature. | ||
Revision as of 10:59, 29 May 2020
A variant of the Green formulas for the rotation field $ y $
of an infinitesimal deformation of a surface with position vector $ \mathbf x $:
$$ 2 {\int\limits \int\limits } _ { G } \left ( \mathbf x \frac{\partial \mathbf y }{\partial u } \frac{\partial \mathbf y }{\partial v } \right ) du dv = \ \cit _ {\partial G } ( \mathbf x\mathbf y d \mathbf y ) . $$
The proof and the idea of applying the Blaschke–Weyl formula to demonstrate the rigidity of ovaloids is due to W. Blaschke [1] and to H. Weyl [2]. For other applications see [3]. The Herglotz formula is an analogue of the Blaschke–Weyl formula. The formula has been generalized to the case of infinitesimal deformations of surfaces in spaces of constant curvature.
References
| [1] | W. Blaschke, "Ein Beweis für die Unverbiegbarkeit geschlossener konvexer Flächen" Gött. Nachr. (1912) pp. 607–610 |
| [2] | H. Weyl, "Ueber die Starrheit der Eiflächen und konvexer Polyeder" Sitzungsber. Akad. Wiss. Berlin (1917) pp. 250–266 (Also: Gesammelte Abh., Vol. 1, Springer, 1968, pp. 646–662) |
| [3] | N.V. Efimov, "Qualitative problems in the theory of deformations of surfaces" Uspekhi Mat. Nauk , 3 : 2 (24) (1948) pp. 47–158 (In Russian) (Translated into German as book) |
| [4] | W. Blaschke, "Einführung in die Differentialgeometrie" , Springer (1950) |
How to Cite This Entry:
Blaschke-Weyl formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Blaschke-Weyl_formula&oldid=46082
Blaschke-Weyl formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Blaschke-Weyl_formula&oldid=46082
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article