Math 171, Fall 2010

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Math 171 will meet Tu-Th from 2:15 to 3:30 in room 260-113. The class will cover material from the first ten chapters Fundamental Concepts of Analysis by Johnsonbaugh and Pfaffenberger. However, we will not have time to cover all of this material. Chapters 1,4 and 5 should be mostly review for you, and will not be covered much during the lectures. After the midterm, we will settle down in Chapter VII, where there is substantial material that we will look at closely.

There will be one midterm, and one writing assignment. The writing assignment will be handed in, critiqued, and returned to you; you will then turn it in a second time.

Office Hours

My office hours, and those of the two Course Assistants are as follows (perhaps subject to change):

Notes for the class

Through the first midterm

Chapters 5 and 6 are less important than Chapter 7. Chapter 5 is less important because series (especially power series, radius of convergence) are properly studied in complex analysis. Note that we the tools at hand, we do not even have the integral test for convergence! And Chapter 6 is less important, because its subject matter (limits and continuous functions) is redone in greater generality in Chapter 7.

Therefore Chapters 5 and 6 will get one lecture each: Chapter 5 on October 7, and Chapter 6 on October 12.

The midterm will be on October 14 and will cover through Chapter 5. After this, we will slow down and take our time with Chapter 7.

About the Midterm

The midterm on October 14 will cover the book through Chapter V. The Bolzano-Weierstrass theorem is particularly important. There will be something about Chapter III. Anything that is covered in one of the homework problems could be asked about. There will be no problems involving non-integer exponents. Unfortunately I did not have time to say much about lim sup and lim inf in class, though they did come up during my lecture on the Bolzano-Weierstrass theorem, which was related to the characterization in Theorem 21.1. I would not ask you to prove Theorem 20.3 or 21.1 but at least understand the statements of these.

You will need to be able to do some proofs, but I will be limited by the fact that you have only 75 minutes as to how difficult I can make such questions.

Compactness

The notion of compactness is very important. In $\mathbb{R}^n$ a subset is compact if it is closed and bounded. For example, the closed interval $[a,b]$ in $\mathbb{R}$ is compact, and there are two relevant theorems about this: the Bolzano-Weierstrass Theorem (Section 14) and the Heine-Borel Theorem (Section 34). The topic of compactness is taken up in earnest in Sections 42, 43 and 44. So the sections that I have just mentioned are among the most important that we will cover.

Writing Assignment

A first draft of the Writing Assignment is due October 28. I would like the first draft to be good, so feel free to run an earlier draft by me before October 28 for some quick feedback. I will return the writing assignment to you on November 2 with my comments. Unless there is nothing to criticize, you will be asked to rewrite it and turn in both drafts on November 16. Both drafts will be returned to you.

I encourage you to do the assignment using LaTeX. This is not required, but would be helpful. I request that you use 12 point font and doublespace. You may accomplish this by copying the file cauchy-ds.tex and using it as a model.

Before you write the assignment, please read the document Writing Mathematics because it contains a discussion of good mathematical writing in the style that I would like you to use.

Here is the assignment. You are to write something about compactness, including some discussion of its importance. Include the three Propositions stated without proof in Point Set Topology. Your document should include proofs of these. Feel free to ask me or the CAs for help if you have trouble figuring out how to prove them. The compactness of $X\times Y$ is probably the hardest of these, and itself a good illustration of compactness in action.

Uniform Continuity

(Material that was here now has its own page)

Integration

We will cover Chapter 9 during the week of November 8. The discussion in the book is too detailed for the amount of time we have left. We will follow the book for Section 51, then the following notes contain simple proofs of the Fundamental Theorem of Calculus based on the Mean Value Theorem.

Homework

Homework is due on Wednesdays, any time. (This is not a class day.) You may turn it in by putting it in my mailbox if you can't find me. I will not accept homework after it has been given to the grader.

Final Exam

The Final will be Thursday, December 9 at 7 PM in 380Y. It is in the basement of the math building.

There will be no Stieltjes integrals on the final. That is, in your study of the Riemann integral, you may restrict yourself to the case $\alpha(x)=x$. Both mean value theorems (for derivatives and for integrals) will be covered. Chapter 7 is of course very important. The following sections are relevant.

Practice Problems

Here are a few problems that you might have a look at to give focus to your study of Chapters 9 and 10. They will not be collected.

Calendar

Grading

The data will be assigned the following weights:

Midterm25%
Writing Assignment25%
Final35%
Homework15%