Homework Assignments for Math 563H

• Assignment 1   Due Monday, February 4.
  • Section 1.2 page 5:   2, 5
  • Section 1.3 page 7:   1, 4, 6, 8 (recommended, not to be handed in, depends on Math 523), 10
  • Section 1.4 page 14:   1, 3, 5, 9
• Assignment 2   Due Friday, February 8.
  • Section 1.5 page page 22:   1, 2, 4, 6, 9, 12, 13, 17
  • Note for 6b: Denote the dot product of u and v by (u,v). Recall that an orthogonal transformation is a linear transformation r:R^3->R^3, satisfying (r(u),r(v))=(u,v). Every orthogonal transformation is given by multiplication by an orthogonal matrix A, i.e., a matrix whose transpose is equal to its inverse A^t=A^{-1}. Hence,
    det(A)=det(A^t)=det(A^{-1})=det(A)^{-1},
    so such a matrix must have determinant 1 or -1. If the determinant is positive, then it is 1. An orthogonal 3 by 3 matrix with determinant one is the matrix of a rotation of R^3 about some line through the origin.
    Denote the vector product of u and v by u^v. In part 6b you are asked to show that r(u)^r(v)=r(u^v). Do it by showing that r(u^v) satisfies the defining equation of r(u)^r(v), i.e., for every vector w, the dot product of w with r(u^v) is equal to det(r(u),r(v),w).
  • Note for 9:   First calculate the curvature of the given curve. Then use the Fundamental Theorem Of The Local Theory of Curves in order to conclude that any other curve with the same curvature function is obtained from the one given by rigid motion.
  • Note for 13:   Write a(s) instead of alpha(s). Assume a(I) is on a sphere of radius r^2. We may assume that the sphere is centered at the origin (why?). Show that a(s) is orthogonal to t(s):=a'(s). Hence, we can express a(s) as a linear combination of the normal and binormal: a(s)=nu(s)n(s)+betta(s)b(s). Differentiate both sides and equate coefficients, in order to relate nu and beta to the curvature and torsion. Then use the fact that nu^2+beta^2=r^2 to obtain the desired differential equation. Next you need to show that the differential equation implies the curve is on a sphere. This direction is challenging. See the back of the text for a solution.
• Assignment 3   Due Friday, February 22.
  • Section 2.2 page 65:   1, 3, 7, 11, 16
  • Section 2.3 page 80:   1, 2, 3, 4 (carefully prove that the map you constructed is differentiable, invertible, and that the inverse is differentiable as well),
    5, 10
• Assignment 4   Due Friday, March 7.
  • Section 2.4 page 88:   1, 2, 3, 4, 6, 9, 13, 17, 18 (use problem 17), 22, 23
  • 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, 6 (derive and solve a differential equation along the lines of the example of loxodromes on page 96),
    
• Assignment 5   Due Monday, March 31.
  • Sec 2.5 page 99:   5, 12, 14, 15
  • Sec 3.2:   Work out in detail Example 5 page 140.
  • 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),
    8a
• Assignment 6   Due Friday, April 11.
  • Sec 3.2 page 151: 5, 6, 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. See the appendix to Ch. 3 on quadratic forms. The term non-degenerate means that B(v,w)=0, for all w, if and only if v is the zero vector. Let 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
  • Sec 3.3 page 168: 4, 5
• Assignment 7   Due Monday, April 21.
  • Sec 3.3 page 168: 6, 7 (note that we did in class part a in case K=-1),
    8 (parts a to e, read first Example 5 page 162),
    16 (use the hint), 20, 21
• Assignment 8   Due Friday, April 25.
  • Sec 4.2 page 227: 1, 2, 3, 4, 5, 6, 8, 9, 12, Highly recommended: 15
• Assignment 9   Due Monday, May 5.
  • Sec 4.3 page 237: 1, 2, 3, 6, 8, 9
  • Sec 4.4 page 260: 3, 4, 5
• Assignment 10   Due Friday, May 9.
  • Sec 4.4 page 260: 2, 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))
• Assignment 11   Due Friday, May 16 at 10AM (Day and time of the final, Room LGRT 1334!).
  • 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)
  • Additional problems: May be announced.