The Nature of the Course: Your purpose in Math 4031 is to develop a thorough understanding of the differential and integral calculus, so thorough that within this subject area you

• know definitions precisely,
• can read proofs and other mathematical material critically, and
• can take hold of a mathematical statement and either devise a proof or give a counterexample.

When I was an undergraduate mathematics major, I took a one-year course similar to LSU's 4031 and 4032. It was for me a formative course, a gateway into mathematics. It contained a number of topics and results which still strike me as impressive and important. Even aside from the particular results, the course introduced me to the mathematicians' game--to techniques, points of view, ways of thinking, language. This semester I'll undertake to offer a modern version of that course which I took in 1959-1960.

This course will be defined by the lectures. I'll try to make the lectures mostly complete and self-contained. I advise you to take notes, and at least once a week to study your notes, perhaps rewrite them with more detail, perhaps organize them in a binder together with exercises and questions and applications. You can make all of this an easier process by staying up-to-date, right on top of things, so that you can get more questions answered and achieve more insight during class.

When there's a gap in your understanding, try to get it closed as soon as possible. You can help yourself by asking me questions either in class or outside of class.

We meet twice a week for 80 minutes. I believe that 80 straight minutes of lecture is inhumane. So we're going to take a 5-minute break each day.

In this course, almost every question and every exercise is a matter of writing a proof--that is, an argument that is convincing, clear, and sound (by the standards which it is my job to set forth). Most of you will be writing mathematical proofs for the first time. Some will be proofs that you've seen before, some will be proofs that you're devising on your own. The way to learn how is to practice, to write proofs, get them criticized, understand what's missing, write them again. Hence the importance in this course of the written exercises.

You don't understand a theorem (or lemma, or proposition) unless you know and understand its proof. One studies a theorem by asking questions about it: Why is each part of the hypothesis needed, and how is it used in the proof? Why couldn't the theorem still be true with a weaker hypothesis, or with a stronger conclusion? What variants on the theorem might be proved in a similar way? The easiest question I will ask you about a theorem on a test will be to prove it. You may ask, Should I memorize proofs? A facetious but just answer is, You should study the proof until you can't forget it. Actually, memorization may be a sensible first step in the study of a theorem; but then when you test your memorization by trying to write down the proof, you will find that it's a grasp of the concepts that is really needed.

Requirements: There will be three one-hour tests, a two-hour comprehensive final exam, and exercises to write up.

Please look at the schedule of hour tests and tell me as soon as possible if you have any problem with any of the dates. They are subject to change, but for the sake of fairness I will want to make changes only with ample notice, after we discuss it in class.

In grading, I use a numerical scale from 0 to 10, thus: 8.3 to 10, A; 6.3 to 8.29, B; 4.3 to 6.29, C; 2.3 to 4.29, D.

I will assign selected exercises to be written up and turned in, with definite deadlines. You will have a certain amount of choice as to which ones to do. I expect you to complete 20 of the assigned exercises during the semester with a grade of 7.00 or better on each. You may do up to 40 in all. Let N be the number of exercises that you complete with a rating of 7.00 or better. Your overall grade P on exercises will be the sum of your scores divided by the greater of N and 20. You should feel free to make attempts on difficult exercises, because if you receive a rating of less than 7.00 on an exercise, I will not count it at all--but then, of course, you will need to do another one.

Your written solutions to exercises should be easy to read, neat, organized, clear, and well-written - as well as correct. Your work must be your own, with the following exceptions: You may consult the text and other books. You may consult me for help. To a limited and reasonable extent, you may consult other persons, and even talk with others in the class. You may not in any way make use of the written work that another student will turn in, and you may not allow another student in any way to make use of the written work that you will turn in.

I may sometimes make copies of your solutions to exercises and test questions and distribute them to the class, with attribution.

In addition, I will frequently suggest other problems for you to do on your own (not to be turned in), as a further guide for study.

Grading Formula: Let T denote your average grade on the hour tests; P , your overall grade on exercises; and E , your grade on the final exam. Let X be the higher of T and E , except that X may be raised if you have done more than 20 exercises. Your overall grade in the course will be no lower than
 

Let me explain more precisely how X will be computed. Assuming that N is at least 20 but no more than 40, then
 

Being present: I expect you to attend class faithfully and to keep up with assigned work. When you are absent, or if you are late to class, or if you leave class early, I will assume that you have good reason; but please let me know why. If on occasion I ask you why you missed a class, I'm not fussing; it means only that I'm concerned about you. If you have difficulties of any kind that affect your work, I will be glad for you to tell me about them. Whenever there is some way you think I can help, please ask.

Tests: It will be best for you not to miss scheduled tests; but if you find that you are unable to take a test (or even the final exam) at the appointed time due to illness or other difficulty, please discuss it with me as soon as possible; there will be deadlines for taking a make-up. A make-up will be given a week or two later, and may cover more material than the test did. Once you have taken the final, or any other test, a re-take is not allowed.

When a test or problem set is graded and handed back to you, you should look it over carefully, and if you have any question or complaint about the grading, you should discuss it with me. It never hurts to ask.

Oral tests: If an individual student and I agree to do it, then an oral test may be substituted for any one of the three scheduled tests. If a student schedules a make-up, then I reserve the right to insist that it be an oral test. The final exam will not be given as an oral exam.

Read the book: Do not expect to read a math book fast. Read the book actively. Try to be convinced that you understand each line before going on to the next. An assignment to "read Section x" includes not just a thorough reading but also doing a few of the exercises at the end.

Personal advice: Have due respect for what you're up against in your academic program. Take care of your health and of your general well-being. You need to be in good physical condition to succeed (and to stay awake in class). Get the physical exercise that you need, and eat properly. Do what you need to do to assure that you have helpful conditions in the places where you live, work, and sleep. If you have emotional problems, get help, or at least find a friend to talk to; don't go it alone, and don't despair.