The instructor for this section is Associate Professor Eric Schechter of the Mathematics Department. You may call me "Dr. Schechter"; that is fewer syllables and will be more comfortable for both of us. My last name is pronounced "shek-tur."
| Office | (32)2-6651 | Located in SC 1529, in the top floor of the Math Building |
| Home | 662-4552 | Please call this number only between 9 am and 8 pm |
I have not yet chosen my office hours, but I will within a few days, and I will then announce them and post them on my web page.
You may send me email at schectex@math.vanderbilt.edu; I check my email on most days.
Personal background (if you're interested) -- I got my Ph.D. at University of Chicago in 1978. After that, I taught at Duke University for two years; I've been teaching at Vanderbilt since 1980. Most of my published research papers are in differential equations, a sort of branch of advanced advanced calculus. In 1996 I published a big fat book (900 pages) for beginning graduate students in mathematics. I'm also the webmaster for my department. When I'm not doing mathematics, I enjoy surfing the web or reading science fiction. I have a wife and two children (ages 8 and 11).
Our textbook has 30 chapters. I have not yet decided which chapters we'll cover. We might start at Chapter 2, or 6 or 12, or 19. This will depend on the background of the students. I will try to make my decision after I meet with the students.
A principal goal of this course is to cover the basics of real analysis, at the advanced undergraduate level. Among other things, this means proving rigorously many of the theorems that are stated without proof in a college calculus course -- e.g., proving that any continuous function on a compact interval is Riemann integrable.
Math 265 will include some computations, but the main emphasis will be on proofs. Our proofs will not be as tedious as in a formal logic course, but they will be far more rigorous than the handwaving and pictures that suffice in an introductory calculus course. Proofs have a style and rhythm that is entirely different from mechanical computations. This may be your first experience with rigorous proofs, and I don't know how quickly you'll pick it up. If necessary, we may digress from analysis for a few weeks, to materials such as Daniel Solow's How to Read and Do Proofs -- available at our library, QA9.S577.
We will have no quizzes, tests, or final exam. We will have lots of written homework, and perhaps some oral presentations in class. Your grade will be based entirely on your homework, and perhaps the oral presentations (I haven't decided yet). You will get an "A" if you learn everything that I think you should learn. If everyone works hard, it is possible for everyone in the class to get an "A", but I think it's more likely that we'll have one or two B's.
I don't know yet whether we'll cover Chapter 13 (The Gauge Integral) in DePree and Swartz, but I certainly hope to. I may as well mention now that that chapter is unusual, and it is one of the main reasons I chose this book. In fact, I'm very excited about it. Here is some historical background for that chapter:
Riemann devised his integral around 150 years ago. It worked well for many purposes, and became popular in mathematics. It is the "standard" integral of undergraduate mathematics; it is used in virtually all calculus books today. Its definition may be too difficult for most beginning college students, but it is much, much simpler than most other integrals in the literature.
Still, it had certain shortcomings -- e.g., many important functions are not Riemann integrable. Around 1900, Lebesgue introduced his integral, which was more general and more powerful (in most respects), but also more complicated. It has become the "standard" integral used in mathematical research; it is taught to virtually all beginning graduate students in mathematics.
About 10 years later, Denjoy and Perron independently devised new integrals, which were slightly more general and far more complicated. These integrals were studied extensively by a few mathematicians, but only a few -- these integrals did not become popular, because they are far too complicated. It turns out that the Denjoy and Perron integrals are equivalent to each other.
In the 1950's, Henstock and Kurzweil independently discovered a new kind of integral, which is called the gauge integral in DePree and Swartz's book. (Elsewhere in the literature, it is also called the Henstock integral, the Kurzweil integral, the generalized Riemann integral, and the Riemann complete integral.) The gauge integral is really remarkable. It is very simple to define -- in fact, its definition is just a very slight modification of the Riemann integral. It turns out to be equivalent to the Denjoy and Perron integrals.
The gauge integral is very simple to work with, very concrete, very intuitive -- probably more so than the Lebesgue integral. It is more general than the Riemann integral, and (at least for functions defined on a compact interval) it is more general than the Lebesgue integral. For many purposes and in many contexts, it is better than the Riemann or Lebesgue integral. For instance, it yields a simpler, stronger, and more elegant version of the Fundamental Theorem of Calculus.
If the gauge integral is so wonderful, why have you never heard of it before? Well, these things take time. The gauge integral did not catch on quickly, because Henstock and Kurzweil did not write for a wide audience. Most writing on the subject was very technical. The ideas began to spread to a wider audience in 1980, when McLeod published his little book, The Generalized Riemann Integral, but even this was not a textbook. In 1988, DePree and Swartz published their textbook which includes some material on the gauge integral. In 1994, Russell Gordon published a book comparing the many different kinds of integrals, but it's a rather technical book, suitable for advanced graduate students in mathematics. In 1996, Bartle published an expository summary article in the American Mathematical Monthly; I believe it was very widely read. In fall 1997 (I'm not sure exactly when), the Monthly will also publish an article by Gordon on other uses (besides integration) of gauges in real analysis; we might use excerpts from that article as a supplement to our course. The book by DePree and Swartz is still the only well-written textbook that introduces the gauge integral to undergraduates, but it will not hold that distinction for long; some other textbooks are presently being prepared (e.g., by Bartle and by Vyborny).
I think the gauge integral is now gaining momentum, and it may soon spread to more of the mathematical community. Personally, I believe it should replace the Riemann integral in our freshman calculus books (for reasons that I may make clear to you later in this semester); perhaps that will happen within a few more years. But in all honesty, I have to confess that the victory of the gauge integral over other integrals is not entirely inevitable. The advantages that the gauge integral offers over other integrals are small advantages, and might be outweighed by the conservatism of mathematicians -- i.e., their reluctance to replace an established working system with a new, unfamiliar system. Still, even if the gauge integral does not become the "standard" integral, it offers some advantages to the student: It may provide you with additional insight and intuition into the Lebesgue integral, the present "standard."