Homework 1, due Friday September 30: 1.1.16, 1.1.17,
1.1.25, 1.1.27, 1.2.7, 1.2.8
In problem 1.2.7, the "some
point" where the end of the string is attached is the same as
the "some fixed point" from which arclength is measured, take
t = 0 in both cases.
Homework 2, due Friday October 7: 1.3.2, 1.3.22, 1.3.27, 1.4.6,
1.4.8, 1.4.10, 1.4.14, 1.5.13, 1.5.15 and the following:
Suppose that all the normal lines to a unit speed curve pass
through a common point P. Show that the curve is part of a
circle of center P (similar to 1.3.22).
Homework 3, due Friday October 14: 2.1.11, 2.1.12, 2.1.13,
2.1.16, 2.2.5, 2.2.14, 2.2.15, 2.2.16.
In problem 2.1.11 change the question "Why is" to "When is". In
problem 2.1.13, R is the distance from the z axis to the center
of the revolving circle and R >r. In problem 2.2.5 delete the
word ``orientable''. In problem 2.2.15 remove the minus signs in
the two formulas for the shape operator S.
Additional problems:
(1) Deduce from the isoperimetric inequality
the solution to Dido's problem: find the maximum area that can
be enclosed between a rope of length L and a straight
fence. This maximum area is L2/2π and occurs when
the rope forms a semicircle (don't compute, rather try to do
some clever arguments).
(2) Does the equation xyz=12 define a surface? How about xyz=0?
Find the equation of the tangent plane to xyz=12 at the point (2,2,3).
Homework 4, due Wednesday November 2: (the Gauss map is the map
sending a point of the surface to its unit normal vector U: we
think of U as
a point on the unit sphere in ℝ3, computing the
derivative of the Gauss map
means computing (minus) the shape operator S, also, if the image
of the Gauss map is a curve, then the area it covers is zero)
2.3.9, 2.3.10, 2.3.11,
2.4.4, 3.1.6, 3.1.7
Additional problems:
(1) Verify the following useful trick for finding eigenvectors of
any 2 × 2 matrix A. If λ is one of the eigenvalues, then
any nonzero column of the matrix A - λ I is an eigenvector
of A for the other eigenvalue
(2) If A is a 2 × 2 symmetric matrix, show that its
eigenvalues are always real, and that when the two eigenvalues
are distinct, their eigenvectors are perpendicular.
Homework 5, due Wednesday November 9: 3.1.5, 3.1.10, 3.2.5,
3.2.7, 3.2.13, 3.2.16, 3.2.19
Additional problem:
Let α be an (arclength parametrized) asymptotic curve on a
surface M, so that S(α')·α'=0. Show that B = ± U
along the curve. By choosing the opposite normal vector if
necessary, we can make the sign here +. Then show that at each
point of the curve, we have S(T) = τ N, and K =
-τ2. Here S is the shape operator of M, K is its
Gaussian curvature, T and N are the curve's unit tangent and
normal vectors, B is its unit binormal vector, and τ is its
torsion. Hint: if we use T and N as a basis for the tangent
plane, what will the matrix of S look like?
Homework 6, due Wednesday November 16: 1.3.8, 3.3.4, 3.3.5, 3.3.6, 3.3.7, 3.4.2, 3.4.5, 3.4.6 (in the hint for problem 3.4.5 (back of the book), the last term in the second displayed formula should read EvGvG / 2(EG)3/2)
Homework 7, due Friday December 2: 5.1.2, 5.1.3, 5.1.9,
5.1.10, 5.1.11 (in all problems assume the curve α has
constant speed, in problem 5.1.3 only prove the assertions at
the point 0)
Additional Problems:
(1) On the paraboloid φ(u,v) =
(u,v,u2+v2), find the area of the region
u2+v2 ≤ R2. Compare to the
area πR2 of the same region in the uv plane.
(2) Show that the normal curvature of a surface M in the direction
of a unit vector aφu+bφv is given
by the formula la2+2mab+nb2. (This is a
quadratic form in a,b which accounts for the terminology
''second fundamental form''.)
(3) Recall the Möbius strip whose parametrization is given by
the first equation in Example 2.2.4. Use the formula from
Gauss' Theorema Egregium (Theorem 3.4.1) to show that its
Gaussian curvature is K=-[(v2/4) + (2-v sin
(u/2))2]-2. In particular, K is negative
at all points. How do you make sense of this? Shouldn't K be
zero for a strip made from paper?