MA387 (Real Analysis) Course Information

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Thanks to all for a great semester!

# Course Description

Broadly speaking, real analysis is a rigorous, extended version of calculus. This semester, you will study numbers, functions, continuity, and differentation in a new light. We will answer such questions as:

• Where does mathematics "begin"?
• Are some infinite sets "larger" than others?
• Can a sequence of numbers be rearranged to converge to a different number?
• How many discontinuities can a function have?
• You may understand 1, 2, and 3 dimensional objects. But what about 1.5-dimensional objects?
• What do voting, donuts, the streets in Manhattan, and Facebook have in common?

Answering these questions requires the rigorous, axiomatic approach to definition, theorem, and proof, which is the foundation of theoretical mathematics. Along the way, you will pick up more proof-writing skills, and also skills that enable you to better read, understand, and communicate mathematics. I will also describe how the field of analysis fits into the broader "picture" of mathematics.

## Textbook

The textbook for the course is Real Mathematical Analysis by Charles Chapman Pugh. All reading and exercises refer to this text, unless otherwise stated. The course will cover most of chapters 1-3 in this text. The specific material covered is as stated in the course catalog:

SCOPE A one semester course providing a rigorous introduction to the calculus of a single variable. The course is designed to introduce the student to the foundations of the calculus necessary for advanced undergraduate and graduate studies in applied mathematics and engineering. Course coverage includes a treatment of the structure of the real number system, sequences, continuous functions, and differentiation.

A recommended supplemental text is Principles of Mathematical Analysis by Walter Rudin. This is a classic text upon which most other real analysis texts are based.

# Course Requirements

The emphasis of this course is your ability to read, understand, and communicate mathematics, so most of the course grade will be based on homework assignments. There will also be three WPR's and two course projects.

The course will contain 1000 total points, divided as follows:

Graded Event(s) Date (if any) Points Percent
Graded Homework 250 25%
WPR I 12-Feb 100 10%
Mini-Project 13-Mar 50 5%
WPR II 14-Apr 150 15%
WPR III 2-May 100 10%
Project 15-May 100 10%
TEE 19-23 May 250 25%

Click here for my expectations on written work, including documentation, etc. Final grades will be assigned in accordance with USMA D/Math policies.

### Homework

The graded homework will be the primary means of evaluating and providing feedback on your progress. I expect you to complete all of the problems listed on the syllabus, and more if necessary for understanding. I will collect the homework and make every effort to provide prompt feedback on your work. Whenever you write up a problem, always (i) rewrite the problem, in your own words, (ii) state clearly the givens in the problem, and the goal, and (iii) use formal mathematical language. See also the page on how to write proofs.

Homework will be collected about once a week. See the expectations on homework and grading.

I recommend that you work on homework in groups, and come in for AI whenever necessary. You are strongly encouraged to ask questions on the discussion board. I will post answers there.

### Online Participation

Your grade will be based in part on your participation in the online wiki (as part of your homework grade). I will be able to see what contributions you have made to the site, and grade you accordingly.

### WPRs & TEE

There will be three midterms and a final. Each exam will cover a single chapter (or portion thereof). Problems on exams will be similar to those on homework assignments. For each exam, the class will collaboratively develop a glossary which they can use for the exam. The final will be comprehensive.

### Projects

The projects will involve researching and writing about a topic of your choosing. The projects will be posted on the course wiki, and reviewed by fellow classmates.

# Online Participation

In the Web2.0 world, more and more of reading, writing, and communicating mathematics occurs online. A major component of the course will be the course wiki at http://usma387.wikidot.com, which will provide an opportunity for you (as students) to collaborate together, and for me (the instructor) to make answers to your questions visible to all.

During the first week of class, take the following steps:

1. Go to the course wiki and apply for membership. You will need a Wikidot membership (please use your real name!)
2. Post some information about yourself (this will not be visible to anyone outside the class).

During the semester, you will use the wiki to:

1. Ask questions of your Professor and fellow students.
2. Collaboratively write a glossary of definitions for use on exams.
3. Post course projects.

There are substantial resources available online to help you learn the Wikidot markup, which is quite simple. You will also need to learn the basics of TeX, the standard language for typesetting in the mathematics community.

# Course Policies

• My door is always open for questions, help, or other matters. Feel free to stop by anytime.
• Schedule an appointment if you want to be sure I'm in my office.
• Feedback is extremely valuable to me. Please let me know if there's anything that irks you (or that you really like). I will also solicit feedback several times during the semester. Your honesty is always appreciated.
• It is extremely important to keep up with the work in this course. Late homework will be subject to a 10% per day deduction.
• Makeups for exams and other activities must be arranged at least a week in advance.
• Document all help received on homework and projects in accordance with USMA policies.
• WPR's and the TEE will be closed book, closed notes with the exception of the online glossary. No calculators will be permitted.

# Instructor's Final Comments

Real analysis abounds with the surprises and beauty that makes mathematics so inviting. Most students find that it is not an easy course… you should expect to put a lot into this course, but I expect that you will get a lot out of it as well.