Let M be a regular surface with v_p, w_p points in the tangent space M_p of M. For M element R^3, the second fundamental form is the symmetric bilinear form on the tangent space M_p, II(v_p, w_p) = S(v_p)·w_p, where S is the shape operator. The second fundamental form satisfies II(a x_u + b x_v, a x_u + b x_v) = e a^2 + 2f a b + g b^2 for any nonzero tangent vector.
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