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mean curvature at surface point
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(Theorem)
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Let $P$ be a point on the surface $F(x,\,y,\,z) = 0$ where the function $F$ is twice continuously differentiable on a neighbourhood of $P$ . Then the normal curvature $\varkappa_\theta$ at $P$ is, by Euler's theorem, expressible via the principal curvatures $\varkappa_1$ and $\varkappa_2$ as
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(1) |
where $\theta$ is the angle between the normal section plane corresponding $\varkappa_1$ and the normal section plane corresponding $\varkappa_\theta$ . When we apply (1) by taking instead $\theta$ the angle $\theta\!+\!\frac{\pi}{2}$ , we may write $$\varkappa_{\theta+\frac{\pi}{2}} = \varkappa_1\sin^2\theta+\varkappa_2\cos^2\theta.$$ Adding this equation to (1) then yields $$\frac{\varkappa_\theta+\varkappa_{\theta+\frac{\pi}{2}}}{2} = \frac{\varkappa_1+\varkappa_2}{2}.$$
The contents of this result is the
Theorem. The arithmetic mean of the curvatures of two perpendicular normal sections has a constant value, which is equal to the arithmetic mean of the principal curvatures. This mean is called the mean curvature at the point in question.
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- ERNST LINDELÖF: Differentiali- ja integralilasku ja sen sovellutukset II. Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1932).
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"mean curvature at surface point" is owned by pahio.
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Cross-references: perpendicular, arithmetic mean, theorem, equation, angle, plane, normal section, principal curvatures, Euler's theorem, normal curvature, neighbourhood, continuously differentiable, function, surface, point
There are 4 references to this entry.
This is version 5 of mean curvature at surface point, born on 2007-08-03, modified 2007-08-05.
Object id is 9830, canonical name is MeanCurvatureAtSurfacePoint.
Accessed 1819 times total.
Classification:
| AMS MSC: | 26A24 (Real functions :: Functions of one variable :: Differentiation : general theory, generalized derivatives, mean-value theorems) | | | 26B05 (Real functions :: Functions of several variables :: Continuity and differentiation questions) | | | 53A05 (Differential geometry :: Classical differential geometry :: Surfaces in Euclidean space) |
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Pending Errata and Addenda
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