Homework Assignments for

M365G, Curves and Surfaces

This page is always under construction, so
you
should check it regularly. Assignments with the **due date**
in **bold face** are set in stone.
Other assignments are still tentative.

The homework is due **every Tuesday** in lecture (except the first
day of class). Most of the homework is from the book, and those problems
problems all have solutions in the back of the book. As the preface recommends,
"these should be consulted only after the reader has obtained his or her
own solution, or in case of desperation." The remaining problems are from
the online supplement that you can download at
http://www.springer.com/mathematics/geometry/book/978-1-84882-890-2?changeHeader
(Look under "Additional Information" and download the "Additional Exercises".)
This file includes all of the book exercises, and some others.
I will assign occasional problems from this set. (Marked with stars).
If you get a problem number
and it's not in the book, look in the downloaded file!

Feb 7:** I mistakenly returned the first two homeworks before the grades were
recorded. Please bring these papers to class so I can correctly record
your grades. I apologize for the snafu.
**

**Tuesday, January 17** First week. No homework due.

**Tuesday, January 24** 1.1: Problems 1,2,6,9,15*; 1.2: Problems 1,3,4,5*,6*
[Problem 1.2.6 looks ugly, but simplifies using some trig identities. The
final answer is quite simple.]

**Tuesday, January 31** Section 1.3: 1, 4, 7*; Section 1.4: 2, 4;
Section 1.5: 2, 4; Section 2.1: 1, 2

**Tuesday, February 7** Problems 2.2.1, 2.2.2, 2.2.6, 2.2.8;
2.3.1, 2.3.2, 2.3.6; 3.3.1; 3.2.2

**Tuesday, February 14** The cylinder x^{2} + y^{2}
= 1 is a surface, and the book (Ex. 4.1.3) shows how to find an atlas with
two surface patches (aka charts).
Find an atlas with *one* chart. It *can* be done!

Problems 3.1.1, 4.1.2, 4.1.3, 4.2.2, 4.2.4, 4.2.5, 4.2.7, and 4.3.1
from the book.

Problem 4.1.3 says you can use a single chart, but this requires the result
of the cylinder problem. If you can't solve the cylinder problem, just
write a solution involving two charts for problem 4.1.3.

**Tuesday, February 21** A short set, thanks to the exam.
Problems 4.4.1, 4.4.3 (this is the reasoning
behind Lagrange multipliers), 4.4.4, 4.5.1, 4.5.2, 4.5.3.

**Tuesday, February 28** Problems 5.1.1, 5.1.2, 5.1.3, 5.2.2, 5.2.3,
5.2.4, 5.3.1, 5.3.2 and 5.3.3.

**Tuesday, March 6** Problems 5.6.3, 5.6.4, 6.1.2, 6.1.3, 6.1.4, 6.2.1, 6.2.3

**Tuesday, March 13** Spring break. No homework due.

**Tuesday, March 20** No homework. Study for the exam instead!

**Tuesday, March 27** Problems 6.3.4, 6.3.5, 6.4.3. Also, write up clean
solutions to problems 2 and 3 from the midterm, including the "extra credit"
part of problem 3. (I'll try to post the exam online from afar, but there's
a chance that it won't appear until Sunday evening.)

**Tuesday, April 3** Problems 7.1.1, 7.1.2, 7.1.3, 7.1.5, 7.1.6, 7.2.1,
7.2.2, and 7.2.3.

**Tuesday, April 10** Work out the equations for parallel transport
on the unit sphere in two ways: (1) with latitude/longitude coordinates
and (2) with stereographic projection. In each case compute the Christoffel
symbols. (We'll do much of (1) in class on Thursday).

Now consider great circles going through the point (1,0,0), and imagine
a satelite orbiting at constant speed along this path. Compute the coordinates
as a function of time and show that they satisfy the geodesic equations. (You
only need to do this for latitude/longitude.)

Now work out parallel transport along the following path, using
latitude/longitude coordinates. Start at the
north pole, head along the prime meridian until the latitude is THETA, then
head due east until the longitude is PHI, and then head north to the pole.
If you start with the vector (1,0,0) and move at around that path, what vector
do you get at the end? What about if you start with the vector (0,1,0)?
[Warning: strictly speaking,
latitude and longitude don't make sense AT the north pole, so to do this
properly you'll
need to start and end a distance EPSILON away from the pole and then take a
limit. Also, you may with to assume that THETA and PHI aren't multiples of
pi/2, so that all of their trigonometric functions are well-defined.]

Problems 7.3.1, 7.3.6, 8.1.1, 8.1.7.

**Tuesday, April 17** 8.2.1, 8.2.2, 8.2.3, 8.3.1. Also, write up clean
solutions to all of the problems on the exam. (I'll try to post the exam
as soon after 12:30PM Thursday as possible. Solutions to the exam will
also be posted, but as with solutions in the back of the book, look at them
only after you've done your best on the problems.)

**Tuesday, April 24** First review problem 8.3.1 (in case you didn't do
it for the last set). Then write down the Geodesic Equations (as in thm 9.2.1)
for the first two parametrizations. In model (i), show that geodesics
correspond
either to vertical lines or to arcs of circles whose centers are on the
horizontal (v) axis.
In model (ii), geodesics are either lines through the center
or arcs of circles that intersect the unit circle at right angles. You can
prove this either by (a) analyzing the geodesic equations
directly, or (b) showing that the
transformation from the upper half plane to the Poincare disk is conformal and
sends circles
and lines to circles and lines. If you think in terms of a complex variable
z = v+iw, then V + i W = (z-i)/(z+i).
Then analyze the image of the geodesics you
found in the first half of this problem. [I recommend the second approach,
but either way can work.]

In addition, do problems 9.1.1, 9.1.2, 9.1.3, 9.2.1, 9.3.2 and 9.3.3

**Tuesday, May 1** The final problem set can be found
here. I made some corrections on April 25, so be sure
you're using an up-to-date copy. In the spirit of the book, I am placing
solutions here. But as with the book problems,
please work as much as possible yourself before peeking. (I updated the
solutions on 4/30 to fix a typo. G is 1 + (stuff), not (stuff). Hat tip
to James Staff for catching the mistake.)