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)
