## 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 Fundamental Form
 (3)

and , , and the coefficients of the second Fundamental Form given by
 (4) (5) (6)

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.