12/3 - Here is the remaining proof we were unable to cover in lecture yesterday. Let me know if you have any questions about it.
12/2 - Possible notes on Lie groups for future study:
Math 261A, Lie Groups and Lie Algebras Spring 2006 notes,
(Geraschenko has several other sets of notes here. Gallier also has other texts on his site, should you like what you see in the one.)
11/16 - For anyone interested, Rosenlicht has a fairly short and clear exposition of the result of Liouville mentioned in lecture today: Liouville's theorem on functions with elementary integrals.
11/15 - Homework Assignment #6:
2, 3, 4, 8, 9.
Due December 2, 5pm.
10/07 - The re-lecture will be Monday at 10:30am in Evans 805.
10/05 - I was out of the loop and mistakenly assumed that Thursday morning's activities were simply an earmark to the 11:30 walkout. For anyone planning to not be in lecture Thursday, I'd like to try to organize a re-lecture. Ideally, this would be sometime Friday or Monday morning. Send me an email and I'll try to find a time of maximal convenience for all.
09/28 - The next assignment is: PDF
09/27 - For 3-26, use the definitions and results of 3-23.
09/23 - Bates's sharp version of Sard's Theorem is available here from the AMS. Whitney's classic counterexample is here, but it must be accessed from campus.
09/21 - The discussion at the end of class should have been about C^k(M)
modules, not vector spaces, as C^k(M) is only a ring. It appears no
one caught the intentional motivating mistake (perhaps due to the
honest mistake with the vector space vs module). Since too much time
will pass before Thursday, don't try to make the partition of unity
argument from the very end of class work. It won't, motivating our
discussion of orientation.
The following problems will have solutions written up: 1-19(b), 1-24(d), 2-7(b). Let me know if any other problems should be on the list. Send me an email if you would like solutions.
09/16 - As stated in class, feel free to use Sard's Theorem on the
Regarding the generalized normal bundle:
The classical way to handle this is via Riemannian geometry, where an inner product on the tangent bundle is assumed in place of a metric. This is one example of additional structure one could add to a vector bundle. In Riemannian geometry, the inner product is then used to construct a metric on the manifold. There is also a metric geometry approach, where a metric is used to define an "inner product", at least formally. If there is enough interest in either concept, we could use the last week of class for discussion. I'd also be happy to discuss these topics in office hours if anyone's curiosity is particularly piqued.
09/09 - The next assignment is: PDF
09/02 - For 1-19b, assume X is a connected manifold. Feel free to use the Axiom of Choice if needed in your proof. (If you are not familiar with the Axiom of Choice, you might end up using it anyway. Don't worry about this unless you want to.)
08/31 again - There is a typo in 1-24b. You may either replace \R^2 by S^2 in the first sentence, or show that the space of ends is homeomorphic to A with your favorite point outside of \R^2 adjoined. (Declare the adjoined point's singleton to be an open set in the topology.)
The two-weekers won, so the first homework is due September 16 at 5pm. You may turn it in during class or slide it under my door (Evans 805).
08/31 - Here are the rotatable renderings for today's lecture:
08/26 - The following problems will be assigned for Chapter 1:
3, 10, 15, 18, 19, 24, 25.
We will cover the missing terminology for 15 and 25 on Tuesday.
In the local spirit, I would like to decide democratically on Tuesday the time interval between assignments. We will choose between 1 week and 2 weeks, avoiding a possible preference paradox (PPP). A longer interval will give you more scheduling freedom, but at the cost of a longer delay on feedback.
If we choose 1 week intervals, the Chapter 1 problems will be due Thursday, September 9. If we choose 2 week intervals, the due date will be Thursday, September 16, and the following Chapter 2 problems will also be due:
1, 4, 7, 14, 23, 26.
(Naturally, the Chapter 2 problems will be due on the 16th in either case.)
I neglected to mention below that you should not be searching for solutions to the homework online, in other texts, or anywhere else. Should you innocently stumble upon a solution, however, please tell me where you found it and do not spoil the problem for your classmates.