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Difference between revisions of "Blaschke-Weyl formula"

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A variant of the [[Green formulas|Green formulas]] for the rotation field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016650/b0166501.png" /> of an [[Infinitesimal deformation|infinitesimal deformation]] of a surface with position vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016650/b0166502.png" />:
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A variant of the [[Green formulas|Green formulas]] for the rotation field  $  y $
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of an [[Infinitesimal deformation|infinitesimal deformation]] of a surface with position vector  $  \mathbf x $:
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$$
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2 {\int\limits \int\limits } _ { G }
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\left ( \mathbf x
 +
\frac{\partial  \mathbf y }{\partial  u }
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\frac{\partial  \mathbf y }{\partial  v }
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\right )  du  dv  = \
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\cit _ {\partial  G }
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( \mathbf x\mathbf y  d \mathbf y ) .
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$$
  
 
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=22141
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article