Math 323, Differential Geometry, Spring 1999

Homework Assignments

Homework assignments will be due each Wednesday.
Due Jan 26:
2.2:
1, 2, 10
Due Feb 2:
2.3:
2, 5, 9, 10
(For #10, sigma = 1/tau, and gamma = alpha + rho N + rho' sigma B.)
Due Feb 9:
2.4:
3, 5, 11(ab), 18
(For #18, assume the curve is simple, i.e. does not intersect itself.)
Due Feb 16:
You do not need to turn in solutions to the practice problems for the first midterm.
Due Mar 1:
4.2:
3, 8
4.3:
2, 5, 8, 9
Due Mar 8:
5.1:
2, 5
5.3:
1, 2, 3
Due Mar 22:
5.3:
4, 5
5.4:
2, 3, 6
Due Mar 29:
5.4:
13, 16, 17
5.6:
2
Due Apr 5:
5.6:
4, 8, 15
5.7:
2
(For 5.6#15, prove also in part (b) that this parameterization is conformal. In part (d), the asymptotic curves should be x(u,c+-u) for any constant c.)
Due Apr 21:
5.6:
11
5.7:
3
We gave intrinsic formulas for K in two cases: first for an orthogonal parameterization, and second for normal coordinates around a point p. Suppose x(u,v) is a conformal parameterization (with xu and xv of common length lambda(u,v)), and suppose at point p, lambda=1 and its derivatives (in u and v) vanish. Then both our formulas for K(p) are valid. Show that in this case, they give the same result, and express this in terms of lambda and its derivatives.
Due Apr 28:
1.
We gave in class an intrinsic formula for the Gauss curvature K, in terms of the lengths of xu and xv in an arbitrary orthogonal parameterization. Use this to rederive the formula for the Gauss curvature K of a round torus of revolution (p. 139).
2.
Integrate the formula you got in the first question, to show explicitly that the total Gauss curvature of the round torus is zero, as predicted by the Gauss-Bonnet theorem.
Review problems:
6.8
3,8
7.6
3,8