Shear velocity: Difference between revisions

Revision as of 12:41, 19 March 2013

Weingarten equations give expansion of the derivative of the unit normal vector to a surface in terms of the first derivatives of the position vector of this surface. These formulas were established in 1861 by German mathematician Julius Weingarten.

Statement in classical differential geometry

Let S be a surface in three-dimensional Euclidean space that is parametrized by position vector r(u, v) of the surface. Let P = P(u, v) be a fixed point on this surface. Then

r_{u} = \frac{\partial r}{\partial u}, r_{v} = \frac{\partial r}{\partial v}

are the tangent vectors at point P.

Let n be the unit normal vector and let (E, F, G) and (L, M, N) be the coefficients of the first and second fundamental forms of this surface, respectively. The Weingarten equation gives the first derivative of the unit normal vector n at point P in terms of tangent vectors r_u and r_v:

n_{u} = \frac{F M - G L}{E G - F^{2}} r_{u} + \frac{F L - E M}{E G - F^{2}} r_{v}

n_{v} = \frac{F N - G M}{E G - F^{2}} r_{u} + \frac{F M - E N}{E G - F^{2}} r_{v}

Sources

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Springer Encyclopedia of Mathematics, Weingarten derivational formulas
Erwin Kreyszig, Differential Geometry, Dover Publications, 1991, ISBN 0-486-66721-9, section 45.

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+'''Weingarten equations''' give expansion of the derivative of the unit normal vector to a surface in terms of the first derivatives of the position vector of this surface. These formulas were established in 1861 by German mathematician [[Julius Weingarten]].
+==Statement in classical differential geometry==
+Let ''S'' be a surface in three-dimensional [[Euclidean space]] that is parametrized by position vector '''r'''(''u'', ''v'') of the surface. Let ''P'' = ''P''(''u'', ''v'') be a fixed point on this surface. Then
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+:<math> \bold{r}_{u} = \frac {\partial \bold{r}} {\partial u}, \quad \bold{r}_{v} = \frac {\partial \bold{r}} {\partial v}</math>
+are the tangent vectors at point ''P''.
+Let '''n''' be the unit [[Surface normal|normal vector]] and let (''E'', ''F'', ''G'') and (''L'', ''M'', ''N'') be the coefficients of the [[first fundamental form|first]] and [[second fundamental form]]s of this surface, respectively. The Weingarten equation gives the first derivative of the unit normal vector '''n''' at point ''P'' in terms of tangent vectors '''r'''<sub>u</sub> and '''r'''<sub>v</sub>:
+:<math>\bold{n}_u = \frac {FM-GL} {EG-F^2} \bold{r}_u + \frac {FL-EM} {EG-F^2} \bold{r}_v </math>
+:<math>\bold{n}_v = \frac {FN-GM} {EG-F^2} \bold{r}_u + \frac {FM-EN} {EG-F^2} \bold{r}_v </math>
+==Sources==
+* {{MathWorld|urlname=WeingartenEquations|title=Weingarten Equations}}
+* Springer ''Encyclopedia of Mathematics'', [http://www.encyclopediaofmath.org/index.php/Weingarten_derivational_formulas Weingarten derivational formulas]
+* [[Erwin Kreyszig]], ''Differential Geometry'', Dover Publications, 1991, ISBN 0-486-66721-9, section 45.
+[[Category:Differential geometry of surfaces]]

Shear velocity: Difference between revisions

Revision as of 12:41, 19 March 2013

Statement in classical differential geometry

Sources

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