Weingarten Equations

The Weingarten equations express the derivatives of the normal vector to a surface using derivatives of the position vector. Let be a regular patch, then the shape operator of 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)
			(7)
			(8)
			(9)
			(10)
			(11)