Weingarten equations

The Weingarten equations give the expansion of the derivative of the unit normal vector to a surface in terms of the first derivatives of the position vector of a point on the surface. These formulas were established in 1861 by the German mathematician Julius Weingarten.[1]

## Statement in classical differential geometry

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

𝐫u=∂𝐫∂u,𝐫v=∂𝐫∂v

are two tangent vectors at point P.

Let n(u, v) 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 the tangent vectors ru and rv:

𝐧u=FM−GLEG−F2𝐫u+FL−EMEG−F2𝐫v

𝐧v=FN−GMEG−F2𝐫u+FM−ENEG−F2𝐫v

This can be expressed compactly in index notation as

∂a𝐧=Kab𝐫b,

where Kab are the components of the surface's second fundamental form (shape tensor).

## Notes

1. ↑ J. Weingarten (1861). "Ueber eine Klasse auf einander abwickelbarer Flächen". Journal für die Reine und Angewandte Mathematik 59: 382–393.

## References

* Weisstein, Eric W.. "Weingarten Equations". http://mathworld.wolfram.com/WeingartenEquations.html.
* Springer Encyclopedia of Mathematics, Weingarten derivational formulas
* Struik, Dirk J. (1988), Lectures on Classical Differential Geometry, Dover Publications, p. 108, ISBN 0-486-65609-8
* Erwin Kreyszig, Differential Geometry, Dover Publications, 1991, ISBN 0-486-66721-9, section 45.

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