The characteristic polynomial of an arbitrary linear operator S is p(k) = det(A-kI), where A is any matrix of S. (a) Show that the characteristic polynomial of the shape operator is k-2Hk+K. (b) Every linear operator satisfies its characteristic equation; that is, p(S) is the zero operator when S is formally substituted in p(k). Prove this in the case of the shape operator by showing that S(v) S(w)-2HS(v) w+Kvw=0 for any pair of tangent vectors to M. The real-valued functions I(v, w) w, II(v, w) = S(v) w, III(v, w) S(v) w = S(v) S(w), defined for all pairs of tangent vectors to an oriented surface, are traditionally called the first, second, and third fundamental forms of M. They are not differential forms; in fact, they are symmetric in v and w rather than alternating. The shape operator does not appear explicitly in the classical treatment of this subject; it is replaced by the second fundamental form.