Weingarten Equations
The Weingarten equations express the derivatives of the Normal using derivatives of the position vector. Let 
be a Regular Patch, then the Shape Operator 
of x is given in terms of the basis
by
 |  |  | (1) |
 | |  | (2) |
where
is the Normal Vector, 
, 
, and 
the coefficients of the first FundamentalForm | (3) |
, 
, and 
the coefficients of the second Fundamental Form given by | |  | (4) |
 | |  | |
|  | (5) | |
 | |  | (6) |
See also Fundamental Forms, Shape Operator
References
Gray, A. ``Calculation of the Shape Operator.'' §14.3 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 274-277, 1993.
- Lyapunov Characteristic Exponent
- Lyapunov Characteristic Number
- Lyapunov Condition
- Lyapunov Dimension
- Lyapunov Function
- Lyapunov's First Theorem
- Lyapunov's Second Theorem
- Lyndon Word
- Lyons Group
- Léon Anne's Theorem
- Lévy Constant
- Lévy Distribution
- Lévy Flight
- Lévy Fractal
- Lévy Function
- Lévy Tapestry
- Löwenheim-Skolem Theorem
- Lüroth's Theorem
- Machine
- Machin-Like Formulas
- Machin's Formula
- Mackey's Theorem
- Mac Lane's Theorem
- Maclaurin-Bezout Theorem
- Maclaurin-Cauchy Theorem