Advanced Calculus fi..

21 4 Advanced Calculus, Fifth Edition5.{+ 2M6q + ~ q',critical points of f (6, q) = LS~where Ec2 + 2Feq + Gq2 = 1.Use Lagrange multipliers as in Problems 14, 15 following Section 2.21 to obtain thequadratic equationL-IEM-IFM-AFN-IGwith real roots I1 5 1 2 and corresponding eigenvectors vl = (61, ql), v2 = (e2, q2).Conclude that f has an absolute minimum II and absolute maximum A2 under the sidecondition. One terms H = ;(I1 + h2) the mean curvature and K = AlA2 the Gaussiancurvature of the surface at Po. By Problem 14(b) following Section 2.21, if I1 # h2,thenIIand hence the corresponding tangent vectors are orthogonal (Problem 10(e) followingSection 3.8). The corresponding directions, or their opposites, are called directions ofprincipal curvature. If hl = hz, f (6, q) has a constant value Il on the ellipse Et2 +2F5q + Gq2 = 1 and hence f = Al(Et2 + 2FCq + Gq2) for all (6, q); if furtherAl # 0, then the point Po is called an umbilical point of S-here all plane sections of Sthrough Po have the same curvature IAl I; if I1 = A2 = 0, then Po is called aparabolicumbilic of S.]Suggested ReferencesBrand, Louis, Vector and Tensor Analysis. New York: John Wiley and Sons, Inc., 1947.Coburn, N., Vector and Tensor Analysis. New York: Macmillan, 1955.Gibbs, J. Willard, Vector Analysis. New Haven: Yale University Press, 1913.Hay, G. E., Vector and Tensor Analysis. New York: Dover, 1954.Levi-Civita, T., The Absolute Differential Calculus, transl. by M. Long. London: Blackie andSon, 1927.Phillips, H. B., Vector Analysis. New York: John Wiley and Sons, Inc., 1933.Rainich, G. Y., Mathematics of Relativity. New York: John Wiley and Sons, Inc., 1950.Especially Chapters 1,2, and 4.Struik, Dirk J., Lectures on Classical Differential Geometry, 2nd ed., Reading, Mass.:Addison-Wesley, 1961.Weyl, Hermann, Space, Time, Matter, transl. by H. L. Brose. New York: Dover, 1952.I