Bruce B. Peckham - Fall 2008 Real Variables (Math 5201)

Math 5201 Real Variables

Text: Real Mathematical Analysis, by Charles C. Pugh, Springer, 2002.

Set 1 Due Monday Sept. 8:
- Read: Sections 1.1-1.2, Skim the rest of Chapter 1.
- Written HW: Ch 1: 2,3,7d,8,9
Set 2 Due Monday Sept. 15:
- Read Sections 1.3, 1.4. Look through 1.5 - 1.6 again.
- Written HW: Ch 1: 11, 17, 19, 30, 35a, 36a
Set 3 Due Wednesday Sept. 24:
- Reread Ch 1.
- Written HW: Ch 1: 39, 40, 42. Extra credit: 36b, 41 (You may restrict your answer to defining how the bijection acts on the integers in [-20, 20]).
Set 4 Due Monday Oct. 6:
- Read Sections 2.1, 2.2, 2.3.
- Ch 2: 2,3,6,7,8,9,12
(mini) Set 5 Due Wednesday Oct. 15:
- Reread Sections 2.1, 2.2, 2.3.
- Ch 2: 30a, 39
Midterm 1: Monday, Oct. 20, 3-4:30 in classroom: LSBE 129. See Topic list below. "Basic theorems" and proofs. Definitions an short problems. Examples.
Test 1 topic list .
Sample Test 1 . Note: This is a test from a previous 5201 class. The topics don't match ours exactly. For example, we have not yet covered connected sets. But the style of the test will be similar to the sample test.

Set 6 Due Friday Oct. 31:
- Read Sections 2.4, 2.5.
- Ch 2: 43, 47, 48, 49, 55. Extra Credit: 46.
Set 7 Due Friday Nov. 14:
- Read Chapters 2, 3.
- Ch 2: 68, 73, 82, 84, 124a
- Ch 3: 3,5,8,9
Set 8 Due Monday Dec. 1:
- Read Chapter 3.
- Chapter 3: 14ab (For 14a, show only the first derivative at zero is zero), 17, 28, 31, 56, 59, 64
- Extra Credit: 14a. Show all derivatives at zero are zero.
Midterm 2: Wednesday, Dec. 3, 3-4:30 in classroom: LSBE 129. See Topic list below. "Basic theorems" and proofs. Definitions an short problems. Examples. More info soon.
Test 2 topic list .

Set 9 Due Monday Dec. 15 (Friday Dec. 12 would be better.):
- Read Chapter 4, Sections 1 and 5. Skim section 2.
- A. Ch 4, #5
  B. Denote successive approximations via the Picard iteration method by y_n(t). For the initial value problem y'=y², y(0) = 1, compute y₂ and y₃ by hand, and y₄ and y₅ using Mathematica (or any other computational aid). Also compute the exact solution using separation of variables. Rewrite the exact solution as a power series. Compare your approximations y₂, ..., y₅ with the series solution.
Final Exam Wed. Dec. 17, 12-1:55PM in LSBE 129. Cumulative. Optional - will only be counted if it raises your course average. See midterm 1 and 2 topics lists above. Add to them:
1. Prove that if a sequence of continuous functions converges uniformly, then the limit function is also continuous.
2. Prove that a contraction mapping on a metric space has a unique fixed point.
3. Compute one or more Picard iterates for approximate solutions to an ODE.
4. Determine the C^0 and/or L^1 distance between given functions.
5. Determine whether a sequence of functions converges pointwise, in C^0, or in L^1.
This page is maintained by Bruce Peckham (bpeckham@d.umn.edu) and was last modified on Tuesday, 16-Dec-2008 10:09:07 CST.