**
MATH 403
A Real Analysis II Spring 2005**

**Professor**:
George L. Ashline

**Office**:
261 Jeanmarie Hall, Phone: 654-2434

**Office Hours**:
M, W from
1:30 to 3:00 PM and T, Th from
1:30
to 2:30 PM; also, feel free to set up appointments
with me for other times.

**Class Meets**:
M and W from
8:05 to 9:20 AM in
Jeanmarie 380

**
Textbook**:
Bressoud, David, *A Radical Approach to Real Analysis*, MAA, 1994; on
library reserve are Rudin’s *Principles of Mathematical Analysis*, 3^{rd}
edition and Marsden and Hoffman’s *Ele**mentary
Classical Analysis*,
2^{nd} edition. Comments on the textbook:

Bressoud’s book offers an interesting departure from a typical analysis textbook. In it, rather than presenting concepts beginning with axioms and then in the order of the final logical development, he takes a historical approach and highlights some problems which led mathematicians to develop the analysis concepts. The hope is to make real analysis seem more natural by considering how it actually developed.

One reason why analysis can be challenging is that by only seeing it in its final form, it can be difficult to see where the subject is leading or why it is posed in a certain way. This semester we will start with some important problems and then consider the concepts developed along the way before reaching the final forms of the solutions. The challenge of this historical treatment is that you will need to remind yourself of what is known and what are the final objectives. You will have seen some of these analysis concepts before in Calculus and Real Analysis I, and hopefully your efforts in this course will enhance and extend your understanding.

Through Bressoud's book, you will be engaging in your own exploration of analysis and its history, and will be aided by a variety of exercises and the Maple computer algebra systems. As always, your work on these problems will play a critical role in the success of your exploration, and please let me know if you have questions or concerns about them.

**
Technology**:
As mentioned above, some questions will require your use of Maple. Please
consult the handout *General Information: Maple Under Windows* to review some
of Maple’s basic commands, which you will be expanding upon in your work this
semester.

**
Home
Page**:
You can access online information about this course and the other courses I
teach at
http://academics.smcvt.edu/gashline. I have listed there a number of
Internet sites on Real Analysis, and you may find these and other sites helpful
in your work on your final presentation and paper.

**
Homework**:
Problems will be assigned regularly. Each problem set will have a specified due
date. You are strongly encouraged to keep up with the material on your problem
sets. If you are having difficulty with some of the new concepts, try to
resolve your questions early on before they have a chance to grow. Of course,
you are welcome to stop by my office to ask questions and discuss any
difficulties you may encounter.

**Exams**:
There will be two in-class exams during the semester
tentatively scheduled on Wednesday, February 16 and Wednesday, April 6. More
information on these will be forthcoming.

**
Presentation/Paper**:
Each of you is to prepare a thirty to forty minute presentation for the end of
the semester
and a corresponding final paper on a real analysis topic of your own choosing.
A list of some possible topics will be distributed later in the semester,
as well as a handout with suggestions for your work on this. By Wed., March 2,
you are to submit your topic choice, including a typed paragraph or two with a
tentative title, abstract, and resources that you intend to use. By Wed., March
23, you are to submit an outline of your paper. In this outline, you should
include your topic, your paper/presentation title, your paper focus and outline,
which part(s) of your paper you intend to present, a working bibliography of
sources you are using, and any questions you may have. Please see
me
before then if you have any questions about your project. Finally, you are
strongly encouraged to consider giving a version of your presentation at the
2005 Hudson River Undergraduate Mathematics Conference (HRUMC) at Williams
College. See
http://www.skidmore.edu/academics/mcs/hrumc12in.htm for more details.

**Grading**:
Your grade will be based on homework,
your presentation/paper, and your exams according to the following distribution:

Homework 150 points

Highest semester exam 140 points

Lowest semester exam 70 points

Final presentation 50 points

Final paper 90 points

Thus, your final course grade will be based on a total of 500 points.

**Summary of
Important Dates**:

Exam 1 W February 16

Presentation topic due W March 2

Exam 2 W April 6

Presentation outline due W March 23

Presentation dates W Apr. 20, M Apr. 25, W Apr. 27, M May 2

HRUMC S Apr. 30

Final paper due W May 4

If you are aware
of a conflict with these dates, let
me
know of it as soon as possible *beforehand*.

**Learning Disabilities**:
Any student having a documented learning
disability that may affect the learning of mathematics is invited to consult
privately with me
during the first week of the semester
so that appropriate arrangements
can be made.

**Academic Integrity**:
You are reminded of the academic integrity policy of St. Michael's College.
Simply stated, academic integrity requires that the work you complete for this
class is your own. Some examples of
offenses against academic integrity include plagiarism, unauthorized assistance,
interference, and interference using information technology. Details about
academic integrity offenses and the possible sanctions resulting from them have
been distributed at the beginning of the academic year and also can be found in
the Associate Dean's office.

**Class Attendance**:
The following is taken from pp. 48-49 of the St. Michael's College 2003-2005
Catalogue:

“Students should understand that the main reason for attending college is to be guided in their learning activities by their professors. This guidance takes place primarily in the classroom and the laboratory.

The following policies have been established:

1. Members of the teaching faculty and students are expected to meet all scheduled classes unless prevented from doing so by illness or other emergencies.

2. The instructor of a course may allow absences equal to the number of class meetings per week. Additional absences will be considered excessive.

3. The instructor may report excessive absences to the Associate Dean of the College, who may warn the student.

4. If absences continue, the Associate Dean of the College may remove the student from class with a failing grade.”