Homework Assignments for Math 563

• Assignment 1   Due Thursday, Feb 4.
  • Sec 1.2 page 5: 2, 5
  • Sec 1.3 page 7: 1, 4, 6, 10 (the curve in Problem 4 will appear later when we discuss the pseudosphere, Figure 3-22 page 169)
  • Sec 1.4 page 14: 1, 3, 5, 9
• Assignment 2   Due Thursday, Feb 11.
  • Sec 1.5 page 22: 1, 2, 4, 6, 9, 12, 13, 17
  • Sec 2.2 page 65: 1, 3, 7, 11, 16
• Assignment 3   Due Thursday, Feb 25.
  • Sec 2.3 page 80: 1, 2, 3, 4, 5, 10
• Assignment 4   Due Thursday, March 4.
  • Sec 2.4 page 88: 1, 2, 3, 4, 6, 9, 13, 17, 18 (use problem 17), 22, 23
• Assignment 5   Due Thursday, March 11.
  • Sec 2.5 page 99: 1b, 2 (translate the equation x = z cotan(alpha) to an equation in the (phi,theta) plane and use the first fundamental form in spherecal coordinates),
    4, 5, 6 (derive and solve a differential equation along the lines of the example of loxodromes on page 96),
    12, 14, 15
• Assignment 6   Due Thursday, March 25.
  • Sec 3.2 page 151: 2, 3 (start with the inequality:
        k > absolute value of the normal curvature.
    Then express the normal curvature in terms of the principal curvatures and a unit tangent vector to the curve),
    5, 6, 8a, 9 (parametrize C by arclength. Its image on the sphere is not parametrized by arclength, but you found a formula for its curvature in Exercise 12 in section 1.5),
    12 (Prove and use the following
    Lemma: Let B(v,w) be a non-degenerate symmetric bilinear form on R^2,   Q(w)=B(w,w) the corresponding quadratic function, and D the curve defined by the equation Q(w)=1. Then the tangent line to a point v on D is parallel to the line {w : B(v,w)=0}.
    Prove this lemma by calculating the differential of Q at the point v),
    13
• Assignment 7   Due Thursday, April 15.
  • Sec 3.3 page 168: 5, 6, 7, 16 (use the hint), 20, 21
• Assignment 8   Due Tuesday, April 27.
  • Sec 4.2 page 227: 1, 3, 4, 5*, 6*, 10, 11*, 14*, 15*
• Assignment 9   Due Monday, May 17
  • Sec 4.3 page 237: 1, 2, 8, 9
  • Sec 4.4 page 260: 2, 5, 6, 9 (Choose an orientation on the sphere and be very careful with the orientation of the angles),
    15 (without using Gauss-Bonet Theorem. Choose the normal to point inside, be careful with the orientation of the angle, and relate your answer to all three exterior angles)
    Note: The answer to 15(b) is: 2pi-theta(1-cos(phi))
  • Sec 4.5 page 282: 2 (both with and without the global Gauss-Bonet. The calculation without Gauss-Bonet can be done explicitly using the formula for the curvature on page 157. A more elegant proof in obtained by considering separately the two regions on the torus separated by the top and bottom parallels and using the chain rule to calculate the integral on each),
    5 (there is a misprint in the text. The point p should be on C but the limit should be taken as the region R shrinks to the north pole),
    and the following two problems:
    A) Solve problem 15 in section 4.4 using the Gauss-Bonet Theorem.
    B) Solve problem 13 in section 4.4 using the interpretation of the Gaussian curvature in terms of parallelism (formula (2) page 271 in Section 4.5)