College of the Holy Cross Mathematics and Computer Science

Mathematics 242 -- Principles of Analysis

Syllabus, Fall 2005

Professor: John Little
Office: Swords 335
Office Phone: 793-2274
email: little@mathcs.holycross.edu (preferred), or jlittle@holycross.edu
Office Hours: MWF 1 - 3pm, TR 11 am - noon, and by appointment
Course Homepage: http://math.holycross.edu/~little/Prin05/PrinHome.html

The Theory of Calculus

In your study of calculus to date, you have approached the subject from much the same intuitive point of view that Newton, Leibniz, and their contemporaries and immediate successors employed. You have learned how to apply the techniques of calculus to solve many types of problems, but the courses you have taken have not spent much time addressing exactly why those methods work (in the cases they do work), or their limitations (the exact hypotheses needed to make them work). Extremely clever, but intuitive and rather uncritical uses of infinite series, derivatives and integrals eventually led to some apparently nonsensical results that had to be clarified to ensure the reliability of calculus in its applications to physics and other areas of science.

In the education of each mathematician, as in the history of the subject itself, there comes a time when the intuitive treatment of calculus must give way to a deeper understanding of the theoretical foundations. For you as a student of mathematics, that time is now! In this class we will essentially repeat some of the historical developments of the 19th century and study the principles of analysis, the theory of calculus that emerged at that time. These steps were first taken by Cauchy, Riemann, Weierstrass and others whose collective work put calculus on a sound footing and gave the tools needed for complete proofs of the key results.

Text

The text is Understanding Analysis by Stephen Abbott, published by Springer. We will cover most of the material in Chapters 1 - 5 and 7 of the book over the course of the semester.

Topics to be Covered

We will be studying the following topics this term:

Unit I: (Chapter 1 of Abbott) Foundations of abstract mathematics; The real number system and its properties (about 9 days)
Unit II: (Chapter 2 of Abbott) Sequences and series of real numbers (about 9 days)
Unit III: (Chapter 3 of Abbott) Basic topology of the real line (about 4 days)
Unit IV: (Chapter 4 of Abbott) Limits of functions, continous functions, and their properties (About 7 days)
Unit V: (Chapters 5,7 of Abbott) The theory of calculus: The definitions of the derivative and definite integral reconsidered. "Interestingly pathological" examples -- functions that are non-differentiable, non-integrable. The Mean Value Theorem and its consequences. The Fundamental Theorem of Calculus. (About 7 days)

The remaining 3 class days will be devoted to in-class exams. There is a more detailed provisional day-by-day schedule posted on the course web site. As always, it may become necessary to add, subtract, or rearrange topics as we proceed into the course. Any changes will be announced in class and on the web site.

Comments About This Course

Together with the Algebraic Structures/Linear Algebra sequence, this course forms part of the "basic training" (think "boot camp"!) for mathematics majors. These intermediate level courses serve as bridges from the basic mathematics you have seen in high school and in calculus to the more advanced courses you will normally take in your junior and senior years. One of the major goals of these courses is to build your skill in developing and writing mathematical proofs.

There will almost certainly be times when the "boot camp" analogy seems painfully apt -- many Mathematics majors find these courses to be among the most challenging they take at Holy Cross because the whole way we work may well seem unfamiliar and baffling at first:

Many problems will not have a single number or formula as an answer and more generally, learning ways to perform calculations will not usually be the main focus.
You will definitely not master this material by learning rote procedures for solving a set collection of different types of problems.
A plausible reason for expecting something is true will not be sufficient.

Instead, in many problems the goal will be first to develop, and then to explain clearly, the complete logical argument that establishes the truth of a statement, starting from a clearly stated set of assumptions and using results established previously. You will have to think about the logical structure of the subject matter and understand the definitions of concepts and the statements and proofs of theorems to get the "skeleton" of the subject firmly in place in your mind. You will need to commit a collection of key examples to memory and be able to reason about their properties. You will need to document much more of your thinking about problems than you probably have done before. While a good intuition may guide you to correct statements, just making correct statements will not be enough.

At times, you may ask why we are doing this. The ultimate answer is that this "abstract" proof-oriented work is the way all mathematics is communicated. On a deeper level, it is even what most of advanced mathematics is really about. So you have to be prepared for it if you decide to continue! The concept of mathematical proof is the unique and distinctive feature of this branch of knowledge; I think it is no exaggeration to say that it is one of the crowning achievements of the human intellect. Even applied mathematicians (ones who work on problems directly inspired by questions in the real world) must develop new ideas to solve those problems, and then provide convincing evidence (proofs) that what they claim is true so that others can follow what they do.

Lest this sound too grim, be aware that you will have lots of chances to develop and practice these new skills, and I will always be willing to give you the benefit of my experience working with this kind of mathematics! Moreover, I think you will find our textbook to be an excellent resource, especially for the motivation behind the topics we study. Even if you find this difficult at first, persistence and openness to a different way of thinking will usually pay off in time.

Course Format

To get you more directly involved in the subject matter of this course, several times during the semester, the class will break down into groups of 3 or 4 students for one or more days, and each group will work together for a portion of those class periods on a group discussion exercise. The exercises will be made up by me. I will be present and available for questions and other help during these periods. At the conclusion of some of these discussions, at times the class as a whole may reconvene to talk about what has been done, to sum up the results, to hear short oral reports from each group, etc. Each group will be responsible for a write-up of solutions for the questions from each discussion day, and those will be graded and and returned with comments.

Grading

The assignments for the course will consist of:

Three in-class midterm exams, together worth 40% of course grade. The (tentative) dates are
- Friday, September 30,
- Friday, October 28, and
- Friday, December 2.
Final exam worth 25% of the course grade. The final exam for this course will be given Wednesday, December 14, at 2:30 p.m..
Individual problem sets, worth 20% of the course grade.
Group reports from discussion days, worth 10% of the course grade.
Weekly ``definitions quizzes'' -- given the first five minutes of each Friday class when an hour exam is not scheduled. The average of the best 5 out of the 9 quizzes will be count for the remaining 5% of your course grade.

If you ever have a question about the grading policy, or about your standing in the course, please feel free to consult with me.

Departmental Statement on Academic Integrity

Why is academic integrity important?

All education is a cooperative enterprise between teachers and students. This cooperation works well only when there is trust and mutual respect between everyone involved. One of our main aims as a department is to help students become knowledgeable and sophisticated learners, able to think and work both independently and in concert with their peers. Representing another person's work as your own in any form (plagiarism or ``cheating''), and providing or receiving unauthorized assistance on assignments (collusion) are lapses of academic integrity because they subvert the learning process and show a fundamental lack of respect for the educational enterprise.

How does this apply to our courses?

You will encounter a variety of types of assignments and examination formats in mathematics and computer science courses. For instance, many problem sets in mathematics classes and laboratory assignments in computer science courses are individual assignments. While some faculty members may allow or even encourage discussion among students during work on problem sets, it is the expectation that the solutions submitted by each student will be that student's own work, written up in that student's own words. When consultation with other students or sources other than the textbook occurs, students should identify their co-workers, and/or cite their sources as they would for other writing assignments. Some courses also make use of collaborative assignments; part of the evaluation in that case may be a rating of each individual's contribution to the group effort. Some advanced classes may use take-home examinations, in which case the ground rules will usually allow no collaboration or consultation. In many computer science classes, programming projects are strictly individual assignments; the ground rules do not allow any collaboration or consultation here either.

What are the responsibilities of faculty?

It is the responsibility of faculty in the department to lay out the guidelines to be followed for specific assignments in their classes as clearly and fully as possible, and to offer clarification and advice concerning those guidelines as needed as students work on those assignments. The Department of Mathematics and Computer Science upholds the College's policy on academic honesty. We advise all students taking mathematics or computer science courses to read the statement in the current College catalog carefully and to familiarize themselves with the procedures which may be applied when infractions are determined to have occurred.

What are the responsibilities of students?

A student's main responsibility is to follow the guidelines laid down by the instructor of the course. If there is some point about the expectations for an assignment that is not clear, the student is responsible for seeking clarification. If such clarification is not immediately available, students should err on the side of caution and follow the strictest possible interpretation of the guidelines they have been given. It is also a student's responsibility to protect his/her own work to prevent unauthorized use of exam papers, problem solutions, computer accounts and files, scratch paper, and any other materials used in carrying out an assignment. We expect students to have the integrity to say ``no'' to requests for assistance from other students when offering that assistance would violate the guidelines for an assignment.

Specific Guidelines for this Course

In this course, all examinations and quizzes will be closed-book and given in-class. No sharing of information with other students in any form will be permitted during exams and quizzes. On group discussion write-ups, close collaboration is expected. On the problem sets, discussion of the questions with other students in the class, and with me during office hours is allowed, even encouraged. BUT, if you do take advantage of any of these options, you will be required to state that fact in a "footnote" accompanying the problem solution. Failure to follow this rule will be treated as a violation of the College's Academic Integrity policy.