Weingarten equations

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

$\mathbf {r} _{u}={\frac {\partial \mathbf {r} }{\partial u}},\quad \mathbf {r} _{v}={\frac {\partial \mathbf {r} }{\partial v}}$

are two 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:

$\mathbf {n} _{u}={\frac {FM-GL}{EG-F^{2}}}\mathbf {r} _{u}+{\frac {FL-EM}{EG-F^{2}}}\mathbf {r} _{v}$

$\mathbf {n} _{v}={\frac {FN-GM}{EG-F^{2}}}\mathbf {r} _{u}+{\frac {FM-EN}{EG-F^{2}}}\mathbf {r} _{v}$

This can be expressed compactly in index notation as

$\partial _{a}\mathbf {n} =K_{a}^{~b}\mathbf {r} _{b}$,

where K_ab are the components of the surface's curvature tensor.

References

• Weisstein, Eric W. "Weingarten Equations". MathWorld.
• 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.