Differential Geometry (3E) -Banchoff/Lovett. Prob 6.4.7, 6.4.8: Coordinate change -Fundamental Form

Differential Geometry of Curves and Surfaces (3rd Edition) - Thomas Banchoff and Stephen Lovett Chapter 6: First and Second Fundamental Forms 6.4: The Second Fundamental Form Prob 7: Suppose that an open set V of a regular oriented surface has two systems of coordinates (x1, x2) and (x'1, x'2) and suppose that V is parametrized by vect{X}(x1, x2) in terms of (x1, x2) in U and by vect{Y}(x'1, x'2) in terms of (x'1, x'2). Call Lij the terms of the second fundamental form in terms of the (x1, x2) coordinates and call L'ij the terms of the second fundamental form in terms of the (x'1, x'2) coordinates. Prove that L'kl = sum_{i, j = 1}^{2} (pxi/px'k)(pxj/px'l)Lij. Prob 8: Use the previous exercise to show that under the same conditions det(L'kl) = (p(x1, x2)/p(x'1, x'2))^2 det(Lij).