General information about Math 311, spring 2003

Here is the catalog description of this course.

01:640:311. Advanced Calculus I (4)
Prerequisites: CALC 4 and 01:640:300
Introduction to language and fundamental concepts of analysis. The real numbers, sequences, limits, continuity, differentiation in one variable.
The course develops the material of calculus as much as time permits, starting from the foundational assumptions of the real numbers as a complete ordered field (please note: an explanation of that phrase is the object of the initial segment of the course!). Students entering the course must have thorough background writing proofs in detail. This course has substantial content, and cannot also serve as an initial introduction to proof construction and exposition. The most obvious material that students must know includes mathematical induction ("The sum of the first n integers is ...") and "proof by contradiction" (analysis of the contrapositive: the most familiar example is the traditional proof that there's no rational number whose square is 2). Students should also be familiar with quantified statements and how to negate them (What is the negation of "All horses that dance are blue"?).

Instructor
S. Greenfield, e-mail: greenfie@math.rutgers.edu.

Meeting time(s) and place(s)
The course meets three times a week: Monday and Wednesday 1:10-2:30 (fourth period) in SEC 205, and Thursday 1:10-2:30 (fourth period) in SEC 212 (all on Busch Campus). Students are expected to attend all classes. The appointed date and time for the final exam is Tuesday, May 13, from 12 to 3 PM.

Text(s)
The official text is Introduction to Real Analysis, 3rd edition, by Robert G. Bartle and Donald R. Sherbert, John Wiley & Sons (ISBN 0-471-32148-6). Please note that this book is available today (1/8/2003) at the Rutger University Bookstore for the price of $106.75 new and $80.00 used. New copies can be bought from various vendors on the web for less than the price for a used copy at the bookstore.

Several other books covering similar material can be recommended.

The book Understanding Analysis by Stephen Abbott was recently (2001) published by Springer Verlag (ISBN 0-387-95060-5). It has received enthusiastic reviews such as this one.. The list price is $39.95.

A Radical Approach to Real Analysis by David Bressoud is published by the Mathematical Association of America (ISBN 0-88385-701-4). The list price is $38.75. The author is a master of exposition. The text carefully explains the reasons why close study of the foundations of real analysis/calculus became historically necessary. The examples are especially rich. The preface is available through this link, and reading it may reassure students who perceive the subject as monumental and inhuman by explaining that one important reason the subject started was because some intelligent and talented people got very confused!

Other material
Most of the pedagogy of this course will be lectures. Listening to mathematical lectures is difficult. T. W. Körner has written a short essay on how to listen to a mathematical lecture which may be useful.

Students likely believe (and the instructor surely does!) that abstraction can be difficult to understand. Edsger W. Dijkstra was a mathematician who made some of the most important contributions to theoretical and applied computer science in the last forty years. He died in 2002. Here he states an argument for the primary importance of abstraction. The title is Why Johnny can't understand.

Almost surely the most important concept in Math 311 is the limit of a sequence. Students may not realize how complicated sequences can be. One "real" sequence is the decimal digits of  pi. Consider the problem of computing perhaps the initial trillion (that's 1012) of these decimal digits. A discussion of the methods needed for such investigation may help you appreciate the difficulties involved and the need for understanding the theory underlying sequences. This is the content of Math 311.

Syllabus
I'll try in the first week or so to gauge how quickly we will cover material. Here is a list of recommended textbook problems. Most of chapter 1 should be known to students. The real work is in chapters 2 and 3. Then we'll go on and cover as much of chapters 4, 5, 6, and 7 as we can. Certainly some "triage" will need to be made: a severe selection of material to be covered from the later chapters. [MORE TO COME!]

Grading
 There will be two exams in class and a three-hour cumulative final. There will be graded homework, both workshop writeups from material to be handed out and problems from the text. There will be quizzes in class of various types, some of which may not be announced. Students may be asked to present material. All of this will be blended to create a number to be translated into a term grade. The likely weight of these components is now (before the semester begins -- things might change!) 15% for each in-class exam, 30% points for the final, 20% for class participation, and 20% for homework.

Office hour(s)
My office is in Hill Center: Hill 542, telephone number: (732) 445-3074. I usually check e-mail several times a day so it is probably the best way to communicate with me: greenfie@math.rutgers.edu [MORE TO COME!] It is my job and my pleasure to teach and interact with students. Please feel free to visit me! I also encourage you to ask questions via e-mail or after almost any class or to make an appointment at a mutually convenient time.

Maintained by greenfie@math.rutgers.edu and last modified 1/15/2003.