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Supplementary Problems THEORY OF SURFACES 10.30. Obtain the Christoffel symbols T's for the cylinder x = y(u) + vg, g = constant, [g) = 1. Ans. TH = y' . y"/ly' x82, Til = (9 . ')(v' .y")/ly'xg/2, otherwise I; = 0. 10.31. Verify that the functions E = 1+ 4u2, F = -4uv, G = 1+4v2, L = 2(4u2 + 4v2 + 1)-1/2, M = 0, N = -2(4u2 + 4v2+1)-1/2 satisfy the compatibility conditions, equations (10.7) and (10.8), page 203. 10.32. Using the Weingarten equations, prove that Nu X N. = (EG -F2)KN. 10.33. Solve the Gauss-Weingarten equations for the surface whose fundamental coefficients are E = 1, F = 0, G=1, L= -1, M = 0, N =0. Ans. Circular cylinder of radius 1. 10.34. Derive Rodrigues' formula from the Weingarten equations. 10.35. If the parameter curves on a patch are orthogonal, prove that K = 1 a 1 avG VEG Lau VE au + av (LOVE ( VGav 10.36. Prove that f(P) = x . x is a continuous function of a point P on a surface. 10.37. Prove that the principal curvatures x1(P) and *2(P) are continuous functions of a point P on an oriented surface. 10.38. Prove Theorem 10.6: The only connected and closed surfaces of class = 2 of which all points are planar points are planes. 10.39. Prove that the spheres are the only connected compact surfaces with positive Gaussian curvature and constant mean curvature. TENSORS 10.40. If A; and B, are the components of two covariant vectors, show that the outer product Cy = AB, are the components of a covariant tensor of rank 2. 10.41. Show that det = elika ajax where evk is defined in Example 10.6, page 210. 10.42. Show that Aag = Ajj - Ag