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.Let S be a surface in three-dimensional Euclidean space that is parametrized by position vector r of the surface. Let P = P be a fixed point on this surface. Then
are two tangent vectors at point P.
Let n be the unit normal vector and let and 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 ru and rv:
This can be expressed compactly in index notation as
where Kab are the components of the surface's curvature tensor.
