Math 545/6, Winter/Spring 2008 - Old Homework Assignments

• Week 1 (W). Reading Report due by noon, Tuesday, January 8. Read read pp. 302-314 of Chapter 12 in ITM (fall text book). (We will not include the details on coverings of the torus or pp. 315-332, the construction of the universal cover for compact surfaces with negative genus, in class discussion or homework.)
  Problems from ITM to work out for yourself: 12-2, 12-8, 12-9, and 12-19. Also read Problem 12-16. This characterization of a proper action will be very useful later in the year. It may be helpful to give it a preliminary pondering now.
  Written homework, due Friday, 1/11: ITM Problems 11-6, 12-7, 12-10, and then use 12-10 and Theorem 12.15 to redo 11-3.


• Week 2 (W). All reading and problems from ISM from here on.
  Reading Report due by noon, Tuesday, January 15, on Chapter 1. Recommended: read pp. iii-v of the preface.
  Also read the appendices B and C, but you do not have to send a report. (If you have questions about something in the appendices, however, please write or come to office hours about it.)
  Problems to work out for yourself: Problem 1-5ab. Remember that you should always work all the exercises for yourself. In the Appendices, in particular be sure you understand all exercises related to the ones to be handed in.
  Written homework, due Friday, 1/18: Exercises B.4abc, B.8, B.20cdef, C.2ace, and C.9, and Problem 1-5cd. (You may assume 1-5ab.)


• Week 3 (W). Reading Report due by noon, Tuesday, January 22 on Chapter 2 and Chapter 3 through p. 67. Note: You may skip pp. 53-54.
  Problems to work out for yourself: Problems 1-6, 2-1, 2-9, and 2-10. Note that 2-9 and 2-10 are the extension to the smooth category of the final exam problem on topological groups, and the same proofs work.
  TA Office hour change this week only: Sean's going to the colloquium tomorrow, so on 1/24, his office hours will be 2-3:50.
  Written homework, due Friday, 1/25: Problems 1-4, 2-2b, 2-5, 2-6, 2-12, and the following:
  Suppose you know that f is a continuous, real-valued, not necessarily differentiable function on an open subset of Rⁿ and it has a directional derivative in every direction at point a. (Use the definition in equation (C.6), even though we do not know that our function f is smooth.) Can you conclude that f is differentiable at a? Justify your answer; that is, if your answer is "yes", give a proof, and if it is "no", give a counterexample.


• Week 4 (W). Reading Report due by noon, Tuesday, January 29 on the rest of Chapter 3 and Chapter 4.
  Problem to work out for yourself: Problem 3-2.
  Written homework, due Friday, 2/1: Problems 3-1, 3-3, 3-4*, 3-5, 3-7, and 4-2.
  *In 3-4, you may use stereographic projection with any intersection point with a coordinate axis as the pole.
  Another TA Office hour change this week: Sean will be out of town for a funeral Thursday, so will have office hours Tu 1-3 (as usual) and W 4:10-6 (instead of Thursday).
  Prof Arms will have extra office hours Thursday 2-4 as well as the usual after class and F 10-11.


• Week 5 (W). Reading Report due by noon, Tuesday, February 5 on Chapter 5 at least through p. 134.
  Problems to work out for yourself: Problems 4-8 and 4-13.
  Written homework, due Friday, 2/8: Problems 4-3, 4-6, 4-7, 4-10, and 4-11.
  JMA Office hour change this Friday: 10-11 as usual, then 12:45-1:20 instead of at 2:30.


• Week 6 (W). Reading Report due by noon, Tuesday, February 12 on Chapter 6. (Also finish Chapter 5 if you didn't before, report optional.)
  Problems to work out for yourself: Problems 5-2, 5-9, 5-12, 5-15, 5-16, and 5-18. (Some will be discussed in class.)
  Written homework, due Friday, 2/15: Problems 5-3*, 5-6, 5-7, 5-8, 5-11, and 5-19.
  *The caution in Problem 5-2 applies in Problm 5-3, also.


• Week 7 (W). No reading report this week. Between the time spent on Chapter 5 and the holiday on 2/18, we'll be in Chapter 6 all week. (But if we're near the end of Chapter 6 on Friday, it might be good to read Chapter 7 before Monday 2/25.)
  Problems to work out for yourself: Problems 6-3, 6-4, 6-6, and 6-15.
  Written homework, due Friday, 2/22: Problems 6-2, 6-5, 6-9, 6-16, and CHANGE! 6-18. (If you started on 6-19, just save it for next week.)


• Week 8 (W). Reading Report due by noon, Tuesday, February 26 on Chapter 7. (The last topic, "Time-Dependent Vector Fields," is optional.)
  Problems to work out for yourself: Problems 6-20, 6-21, and 6-24.
  Written homework, due Friday, 2/22: Problems 6-19, 6-25, 6-26 - SL(n,R), SO(n), and U(n) only, *7-2(ab), and *7-3(ab).
  *Problems 7-2 and 7-3 may be presented together as a single problem. For 7-2, you may simply state the flow and verify it has the vector field as a generator; that is, you need not show the process of solving the ODE's.
  In 7-3, be sure to give the coordinate domain.


• Week 9 (W). Reading Report due by noon, Tuesday, March 4 on Chapter 8.
  Problems to work out for yourself: Problems 7-10 and 7-21. Note we found the flow of V in 7-21 in class. Also, at least read 7-20 after doing 7-8.
  Written homework, due Friday, 3/7: Problems 7-1, 7-5, 7-8, 7-18, and 7-22.


• Week 10 (W). Reading Report due by noon, Tuesday, March 11 on pp. 225-240 of Chapter 9.
  Problem to work out for yourself: Problems 8-9.
  Written homework, due Friday, 3/14: Problems *8-7 + last line of 8-3; 8-8, 9-1, and 9-9.
  *You may assume the claims in Example 8.3 (but it is of course recommended that you think through their proofs).


• Winter Quarter Final Exam.


• Week 1 (Sp). Reading: Review/Read all of Chapter 9. Report optional, but if you report by Wednesday noon, it may be counted in place of a skipped report later in the quarter.
  Problems to work out for yourself: Problem 9-5 and as much of Problems 9-2 and 9-6 as needed to be comfortable with pullbacks. Also read Problem 9-8. The categorically inclined are encouraged to read and ponder any or all of Problems 8-2, 8-15, 9-4, and 9-18.
  Written homework, due Friday, 4/4: Problems 6-10, 7-4, 9-3 (don't miss part (c)!), and 9-7(ab). Do 9-7(b) two ways: convert f to polar coordinates and compute directly, and also compute using the result of part (a) and the pullback, as in Example 9.23.


• Week 2 (Sp). Reading report due by noon, Tuesday, 4/8 on Chapter 10.
  Problem to work out for yourself: Problem 9-10. Also read Problem 9-15.
  Written homework, due Friday, 4/11: Problems 9-11, 9-12, 9-13, 9-14, and 9-16.


• Week 3 (Sp). Reading report due by noon, Tuesday, 4/15 on Chapter 11.
  Problems to work out for yourself: Problems 10-2, 10-10, and 10-11.
  CORRECTED written homework, due Friday, 4/18: Problems 10-1, 10-3, 10-6, 10-9, 11-1, and the following: Prove the last line before Prop. 10.6 on p. 263 by finding a counterexample when the dimensions of both spaces are at least two.


• Week 4 (Sp). Reading report due by noon, Tuesday, 4/22 on Chapter 12. As a preview of some ideas in Math 547, include in your reading Problems 11-4, 11-5, 11-13, and 11-14.
  Written homework, due Friday, 4/25: Problems 11-7, 11-8, 11-9, 11-10, and 11-11.


• Week 5 (Sp). Reading report due by noon, Tuesday, 4/23 on Chapter 14. (Chapter 13 optional.) Include in your reading Problem 12-3, especially the remark at the end, and if categorically inclined, Problem 12-9.
  Written homework, due Friday, 5/2: Problems 12-1, 12-5, 12-6, 12-7, 12-8, and the following.
  Do Exercise 12.27, then compute the Lie derivative of a local coframe element "epsilonⁱ" with respect to a local frame element E_j as a linear combination of coframe elements by two methods, using Prop. 10.29(e) and using Cartan's Magic Formula (Prop. 12.32).


• Week 6 (Sp). Reading report on Chapter 15. Because we won't start on this chapter in class until the end of the week, you may postpone your report until Thursday if you wish.
  Problems to work out for yourself: Problems 14-3 & 14-6.
  Written homework, due Friday, 5/9: Problems 14-4, 14-7, 14-9, & 14-11.


• Week 7 (Sp). Reading report on Chapter 16. I'm going to try to start this chapter on Wednesday, but I'm flexible on getting reports anytime through Thursday.
  Problems to work out for yourself: Problems 14-13 & 15-3.
  Written homework, due Friday, 5/9: Problems 14-2, 14-5, 14-15, 15-4 & 15-5.


• Week 8 (Sp). No reading report this week. If you haven't reported on Chapter 16 and want to, do so soon. We will skip Chapter 17. Reading Report on Chapter 18 will be due 5/27, the day after the Memorial Day Holiday.
  Problems to work out for yourself: Problems 15-15, 16-1, and 16-6. Recommended: read 15-10 and 15-11.
  Written homework, due Friday, 5/23: Problems 15-6, 15-12, 15-14, 16-3, & 16-7.
• Week 9 (Sp). Reading report due by noon, Tuesday, 5/27 on Chapter 18. You may omit the sections on the orientation covering, pp. 451-455, and densities, pp. 470-475. I will discuss the former briefly in class. If you are in interested in analysis on manifolds, you should probably read the section on densities sometime.
  Problems to work out for yourself: Problems 6-12, 18-1, 18-2, 18-3, and 18-4.
  Written homework, due Friday, 5/30: The problems below and Problems 16-11, 16-13, & 18-10.
  1) We skipped problems earlier about complex projective space because now we can construct its smooth manifold structure more easily. Consider the action of C \ {0}, the nonzero complex numbers, on Cⁿ⁺¹ \ {0} by multiplication. Show that the orbit space is a topological manifold with a unique smooth structure such that the quotient map is a smooth submersion. We will define the complex projective space CPⁿ to be the orbit space with this smooth structure.
  2) (A special case of Problem 16-10.) Let X be the set of type (1,3) flags in R⁴. Show that Proposition 16.19 may be used to give X the structure of a homogeneous space. What is its dimension?


• Week 10 (Sp). Reading report due by noon, Tuesday, 6/3 on Chapter 19. You may omit the section on the Riemannian density, pp. 492-494. Optional; continue on to read part or all of Chapter 20.
  Problems to work out for yourself: Problems 18-9 (notice it follows easily from 18-10(b)), 18-12, & 18-13.
  If you are interested in analysis on manifolds (PDE's, math physics, etc.), you should read all the problems 19-6 through 19-16 sometime. (Perhaps include them in the Riemannian reading course next year?)
  Written homework, due Friday, 6/6: Problems 18-17, 19-1, 19-2 with addition below, 19-4, & 19-5.
  Addition to 19-2: (c) In R³, let E be the ellipsoid given by (x-1)²/4 + (y-2)²/9 + z² = 1, oriented by the outward normal vector and the standard orientation on R³. Evaluate the integral of the form from 19-2 over E.


• Spring Quarter Final Exam.

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