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 x2 + y2 = 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.)