College of the Holy Cross Mathematics and Computer Science


Mathematics 242, section 1 -- Principles of Analysis

Syllabus, Spring 2013

Professor: John Little
Office: Swords 331
Office Phone: 793-2274
email: little@mathcs.holycross.edu, or jlittle@holycross.edu
Office Hours: MF 10am - 12noon, TR 2 - 4pm, W 3 - 5pm, and by appointment
Course Homepage: http://mathcs.holycross.edu/~little/Principles13/PrinHome.html (also accessible through Moodle)

The Theory of Calculus

In your study of calculus to date, you have approached the subject in much the same way that Newton, Leibniz, their contemporaries, and their immediate successors in the 17th and 18th centuries did. You have learned how to apply the techniques of calculus to solve many types of problems. But the courses you have taken probably have not spent much time addressing exactly why those methods work (in the cases they do work), or their limitations (the situations where they do not work, or the exact hypotheses needed to make them work).

In the historical development of the subject, part of what happened after this initial exploratory period was that extremely clever, but intuitive and rather uncritical uses of infinite series, derivatives and integrals eventually led to some apparently nonsensical results. The situation 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 the mathematicians Cauchy, Riemann, Weierstrass and others. Their collective work put calculus on a sound footing and gave mathematicians the tools needed for complete proofs of the key results.

Text

The text for this course is An Introduction to Analysis, 2nd edition by Gerald Bilodeau, Paul Thie, and G.E. Keough, published by Jones and Bartlett (BTK in the following). We will cover most of the material in Chapters 1 - 6 of the book over the course of the semester.

Topics to be Covered

We will be studying the following topics this term: 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

  1. from the basic mathematics you have seen in high school and in the calculus sequence
  2. 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:

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 (those 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

Most classes will follow a lecture/discussion format. I will pause frequently and ask questions to make sure the class is following the development of the topic for the day and to keep you involved. To get you more directly involved in the subject matter of this course, several times during the semester, I may break the class down into groups of 3 or 4 students for one or more days to work together on a group discussion exercise.

Grading

The assignments for the course will consist of:

  1. Three in-class midterm exams, together worth 45% of course grade. The (tentative) dates are Since everyone can have a bad day occasionally, your lowest in-class exam score will be weighted less than the other two (9%,18%,18%).
  2. Final exam worth 25% of the course grade. The final exam for this course will be given at the time for MWF 8am classes (to be announced later in the semester by the Registrar). Please do not make any travel plans for the summer break before this is determined!
  3. Weekly individual problem sets, worth 20% of the course grade.
  4. 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 count for 5% of your course grade.
  5. The remaining 5% of your course average -- class participation and possible group discussion write-ups.

I will be keeping your course average in numerical form throughout the semester, and only converting to a letter for the final course grade. The course grade will be assigned according to the following conversion table (also see Note below):

Note: Depending on how the class as a whole is doing, some downward adjustments of the above letter grade boundaries may be made. No upward adjustments will be made, however. (This means, for instance, that an 85 course average would never convert to a letter grade of B- or below, but it might convert to a B+ in some cases.)

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

Advice On How To Succeed In This Class

A good "work ethic" is key but it may not be enough. Contrary to what you might believe, mathematics professors definitely want you to learn and do well in their courses! But you will also need to do your part by working hard. That is certainly necessary to succeed in this course. Unfortunately, though, hard work and dedication alone might not be sufficient. Mathematics at this level is not an easy subject for anyone. It requires a certain kind of very careful and logical thinking based on a deep knowledge of definitions of terms and basic examples that is simply not everyone's "cup of tea."

Come to class. Unless you are deathly ill, have a genuine family emergency, have to be away at a College athletic event, etc. plan on showing up here at 8:00am every Monday, Wednesday, and Friday this semester. If attending class wasn't important, all college courses would be online, and your tuition would be much lower!

Read the textbook. Don't just use it to look for worked problems similar to ones on the problem sets. (In this course, there won't be many of those in any case!) You will find alternate explanations of concepts that may help you past a "block" in your understanding. Reading a math book is not like reading a novel, though. You will need to read very carefully, with pencil and paper in hand, working through examples in detail and taking notes. Make a list of questions to ask in office hours or at the next class. One thing to bear in mind while reading your text is that the result of an example is often secondary to the process used in obtaining the result.

Take notes and use them. This may seem obvious, but in my experience too many students diligently copy down everything on the board, and then never look at their notes again. Used intelligently, your notes can be a valuable resource as you work on problem sets and prepare for exams. One method that works for some people is to create a second, polished set of notes for the course by recopying the notes you take in class, filling in details or providing other explanations as you go. You will need to do this carefully, though. The idea of this method is for you to think about what you are recopying as you do it, not just to act as a stenographer.

Set up a regular study schedule and work at a steady pace. It's not easy to play catch-up in a mathematics course, since every day builds on the previous one. You should expect to budget at least 6 hours in a typical week for work outside of class. The best way to use your time is to do a few problems, a little reading from the book, and reviewing or recopying of class notes every day.

Most importantly, if you are having difficulty learning something, get help as soon as possible. You can do this by asking questions during class (any time something isn't clear), or by seeing me during office hours.

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.