Real Analysis I syllabus

Math 446 Real Analysis I Fall 2011, Prof. Mark Janeba

Meetings	Text
TTh 12:50 - 2:20 Ford 204	Introduction to Real Analysis, 2nd edition, by Manfred Stoll ISBN 978-0-321046-25-3, published by Addison Wesley

Course content:

This course concerns itself with the concepts of limit and continuity, which are the basis of mathematical analysis and the calculus from which it evolved. We focus on limits of sequences and of functions, and continuity of functions. While the main context is the real numbers, i.e. limits of sequences of real numbers, continuity of functions from the real numbers to the real numbers, and so on, we will introduce a more general context, that of metric spaces, in which to study the same concepts. This will allow us to extend our results (often with no added effort) to functions on

ⁿ, sequences in

, functions on spaces of functions, etc. A highlight of the course is the interaction of continuity with the concepts of completeness, compactness, and connectedness, which give us rigorous proofs of the seemingly obvious (but definitely not obvious) existence of global extrema and Intermediate Value Theorem. As time permits, we may begin the concept of differentiability before semester break.
Note: In Real Analysis II, we will see that the "proofs" about derivatives that we saw in Calculus I were mostly quite rigorous, if we have a rigorous understanding of limits. Therefore, we will quickly review what we already know about derivatives in a rigorous setting. Integrals, however, are far more complicated than a Calculus I/II course would lead us to believe, especially if they involve discontinous functions. Thus much of Real Analysis II will be concerned with properties of the Riemann integral. Other material will include some introductory material on measure theory and/or sequences of functions, e.g. uniform convergence.

Course Goals:
At the completion of this course, the successful student will have demonstrated these abilities:

The ability to understand both abstract and concrete mathematical reasoning.
The ability to differentiate between sound mathematical reasoning, flawed reasoning, and non-rigorous reasoning.
The ability to use the basic tools and methods of proof seen in analysis, in particular set theory and epsilon-delta and epsilon-n arguments.
The ability to formulate and prove theorems that arise from the definitions and concepts of the course content, and the ability to apply those theorems to specific examples.
The ability to write up, and occasionally present orally, one’s mathematical proofs and arguments in a clear and compelling manner.

Grading

Three to five quizzes	75-125 points
Two or three "midterm" exams:	200-300 points
Two or three in class presentations	80-120 points
Final exam	200 points
Around 7 homework sets at ~20 pts each	~140 points
Total:	695-985 points

For each graded piece of work, I will post cutoff scores for grades of A-, B-, C, C-, and D. At the end of the term, if your point total is more than the total of the A- cutoffs, your grade will be an A- or better, and so on. Cutoffs will never be higher than: A-: 90% B-: 80% C: 70% C-: 67% D: 60% ... but they are often lower.

Exams: We will have a mix of in-class and take-home exams. Exams will be announced several days in advance. The "rules" for take-home exams will be provided with the exam.
Final exam date: Freday Dec. 16, 2-5 p.m. For borderline grades, I tend to pay more attention to the final exam score. The final may be take-home, in which case the above is the due date.
.
Exam makeup policy: Midterm make-ups or early midterms are given only for verifiable illness or for university-sanctioned intercollegiate activities. For collegiate activities, you must see me before you leave to arrange a makeup time. In any case, you must contact me in advance except in emergencies.If the final is in-class, it will not be given early; we will try to finalize this by mid-semester, but please plan accordingly.

Quizzes are 10 to 25 minutes long. Some will be primarily content quizzes, e.g. stating definitions and theorems from class and assigned reading; these may be given without notice. Other quizzes will be short reasoning and proof exercises, closely related to previously assigned homework.

Presentations: I will periodically assign individual presentations, typically proofs of moderate difficulty, to be given during class meetings. For the first presentation, I will ask students to give a preview in my office so we can be confident that things are going well.

Homework is assigned daily but collected sporadically. Approximately every other week, I will ask you to turn in a specific subset of the assigned work. If you have additional problems on the same page, it's fine if you turn them in as well, but the problems specified for collection need to be thoroughly legible and well written. If it is necessary to achieve legibility and coherence, students may choose to (re)write up a clean copy of their work to turn in (by the original deadline given).

Participation:
In advanced mathematics, the step from things that can be easily understood to things that may as well be spoken in the Martian language is a short step. To stay ahead of this "step," one needs to keep up with the concepts and definitions already covered. It is vital that students do so. The occasional "pop" quiz on definitions and theorems (see Quizzes above) and class discussion will help motivate students to keep up and help me assess how well students are doing. Please come prepared to discuss the homework and assigned reading from the previous class.

Oops! (added 9/1) As with all 400-level mathematics courses at Willamette, students of this course are expected to attend the department colloquium at least four times this semester. The purpose of this requirement is to instill an appreciation of the breadth and variety of mathematical topics, techniques, people and (academic) programs. If scheduling conflicts make colloquium attendance impossible, alternative short written assignments will be made. If a student anticipates needing this alternative for more than one colloquium session, they should contact the instructor well in advance so that suitable assignments can be prepared. Failure to complete this requirement will result in a modest grade penalty for the course.

Academic Honesty Expectations
All exams and quizzes are to be taken with books and notes closed (except as noted on the exam paper), completely on your own. When we have take-home exams, specific expectations will be distributed with those exams.

Plagiarism: any work copied or paraphrased from another source without proper written acknowledgement. Plagiarized work will recieve a zero grade.

In keeping with college policy, plagiarism will be reported to the dean (see student handbook). Systematic or organized plagiarism on exams or quizzes will result in course failure. If you are uncertain about some aspect of the academic honesty policy, it is your responsibility to get clarification from the instructor.

Last Modified Sept. 1, 2011.
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