The Weingarten equations express the derivatives of the normal vector to a surface using derivatives of the position vector. Let x:U->R^3 be a regular patch, then the shape operator S of x is given in terms of the basis {x_u, x_v} by -S(x_u) | = | N_u = (f F - e G)/(E G - F^2) x_u + (e F - f E)/(E G - F^2) x_v -S(x_v) | = | N_v = (g F - f G)/(E G - F^2) x_u + (f F - g E)/(E G - F^2) x_v, where N is the normal vector, E, F, and G the coefficients of the first fundamental form d s^2 = E d u^2 + 2F d u d v + G d v^2, and e, f, and g the coefficients of the second fundamental form given by
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