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Question 1. This question concerns surfaces of revolution. (See Homeworks 5 and 6 for the definition and some earlier computations) (i) Compute the coefficients L's for a surface of revolution, and write your answer in matrix format. (ii) Compute the mean curvature H and the Gaussian curvature K (as functions on U). (iii) Draw an example of a surface of revolution that has some areas where K is positive, and others where K is negative, and roughly indicate which areas are which. (iv) NO NEED TO TURN THIS ONE IN: Compute all 16 coefficient functions Rijk fo the Riemann curvature tensor for a surface of revolution (HINT: Gauss's equations are probably an easier approach rather than going straight by definition). Question 2. Prove that IIp is symmetric (or equivalently, prove that L is self-adjoint). Question 3. Prove Proposition 5.15 (ii) in the notes relating Kin and II. Question 4. Prove that n 1 X n 2 = (K,g)n. Question 5. Suppose that 911 = 1 and 912 = 0 everywhere in U. Prove that a2 ( V 922 ) (ou¹) 2 +K1922=0. = Question 6. Viewing Kn as a function Kn : S¹ R as in class, prove that 27 H = 27 1 0 Kin Y ) d0 where 0 == / ( X (1) , Y ) as in Euler's theorem (Thm 5.21 in the course notes, which we did not cover in class, but is easy to read and understand). This shows that H is really an arithmetic average of Kin over the entire unit circle, not just an average over the two principal directions.

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