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.
- Euler's Phi Function
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- Eventually Periodic