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 x_{u} and x_{v} 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 x_{u} and x_{v}
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 GaussBonnet theorem.
Review problems:
 6.8
 3,8
 7.6
 3,8