Weingarten equations

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

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

are two tangent vectors at point P.

Let n(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 r_u and r_v:

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

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

This can be expressed compactly in index notation as

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

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

• 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.