Introductory Analysis is a two course sequence whose main purpose is to teach the basics of analysis in a rigorous and reasonably complete way. In our current setup, part I can be called Calculus of one variable done right. Part II covers the basics of Lebesgue integration. Together with the Introductory Algebra sequence it is supposed to prepare students for more advanced courses in mathematics. In our doctoral program, the qualifying exams are designed (or supposed to be designed) to test your readiness for taking these advanced courses; an immediate consequence of doing well in the Analysis sequence is that you will also do well in the analysis qualifier. It may be useful to mention that writing carefully, expressing yourself in coherent sentences in the English language (both in writing and orally), understanding concepts, and proving theorems are essential ingredients of the course.
A few historical remarks may be in order. For many centuries, millennia even, mathematics was a relatively static science. Mathematicians studied properties of geometric figures, of numbers, and made many marvelous and beautiful discoveries: The existence of irrational numbers, Eudoxus' Theory of Proportions (which effectively resolved the crisis caused by the discovery of irrational numbers), Euclid's proof of the infinitude of the set of primes, Archimedes' sequences to compute π, his Method in general, Apollonius' work on conics, Diophantus on Number Theory, the introduction of zero and negative numbers by Indian mathematicians. The list could go on. But then, beginning in the sixteenth century with a few baby steps, more in the seventeenth century and finally, coming to fruition with the work of Newton and Leibniz, a sea change occurred; mathematics became dynamic. Functions appeared. In his Principia Newton studied what he called fluents (functions), and explained how to find the fluxion (derivative) of a fluent and, more interestingly, how to find the fluent given the fluxion (integration). From then on functions played a central role in classical mathematics, in analysis as well as in algebra, even though it took over a century past Newton before the concept became truly formalized. Ironically enough, when rigor was taken to extremes already in the twentieth century, it was found that the function, this most dynamical of objects, had to be defined in a static way as a set of pairs. Not surprisingly, we will be spending a lot of time studying properties of functions, such as continuity, differentiability, integrability. We will, of course, also spend some time studying the sets on which these functions act, most particularly that extremely complicated and strange set known as the set of real numbers.
We have to be rigorous. Newton, Euler, had a tremendous intuition and knew exactly what they were doing. Most of us are not so fortunate and we need more guidance; rigor and precision provide that guidance. You can think of mathematics as a journey, an expedition, a trek. The journey at times can get quite difficult, but it is also very rewarding. Introductory Analysis and Introductory Algebra are the beginning of the journey, and a lot of time might be spent in doing the analog of fitness exercises, toughening you up for the difficulties that will come. You may find some of these exercises silly and boring, but there still is a point in doing them. You may find some of the going difficult but I can assure you that if you stick to it, put a serious effort into mastering the concepts, one day you'll look back and wonder how a course like this one could have ever given you any trouble. Anyway, mathematics is not supposed to be easy. It should, however, be enjoyable. If you do not find pleasure in solving exercises on your own, in grasping a new concept, then mathematics is perhaps not for you. It is, of course, useful knowledge, it has great applications, but working toward a Ph.D. or an M.Sc. should not be a constantly painful experience. There will (and there should) be some pain, but there also has to be some pleasure. I sincerely hope you will enjoy this course.
Course outline The plan is to cover most of Chapters 2, 3, 4, 5, and 6 of Rosenlicht's book; in other words, the following topics:
Click here (or on the link below) for a detailed schedule, including the exam dates.
- The Real Number System.
- Metric Spaces
- Riemann Integration
Grading Your final grade will depend on homework, two in class exams, class participation, and a comprehensive final exam. Homework will be assigned frequently and will usually be due about a week after it is assigned. Late homework will not be accepted. Homework accounts for 25% of your grade. While working with classmates and discussing homework with them is encouraged, the homework you hand in has to be your own work. Each in class exam accounts for 20% of the grade with the final exam accounting for 30% of the grade. Class participation takes care of the last 5% of the grade. Class participation takes place mostly during discussion sessions; which we will try to have on a biweekly basis.
There will be no make-ups. However, there will be ways in which you can make up for a missed exam or homework, as long as your grades are reasonably good.