Topics. This course is the first part of a rigorous introduction to mathematical analysis. We will cover basic inequalities for real and complex numbers, basic set theory, and basic point-set topology such as limits of sequences, the least upper bound property, compactness, connectedness, and continuity. We will study absolute continuity, uniform continuity, differentiability, bounded variation, and rectifiability, all in the context of metric spaces. We will prove the mean value theorem, Taylor's theorem, L'Hopital's rule, and other thereoems concerning differentiation and extrema of functions of n variables. We will finish with a rigorous development of the Riemann-Stieltjes integral, proving the second fundamental theorem of the calculus and other theorems concerning differentiation of integrals, changes of variable, and interchanges of the order of multiple integrals.
Time. Classes meet Tuesdays and Thursdays, 10:00 am to 11:30 am, in Lopata, room 301.
Prerequisites. Math 318, or the permission of the instructor.
Text. The lectures will follow the first half of the book Mathematical Analysis by Tom Apostol, second edition, published by Addison-Wesley, ISBN 0-201-00288-4 (1974).
Homework. You are encouraged to collaborate on homework, and to work additional exercises from the relevant problem sections, although the homework grade will be based only on the exercises listed below. Please return your solutions to the instructor by the end of class. Problem sets will be assigned as follows:
Tests. Midterm examinations: Test 1: Tuesday, September 25, 2001, in class.Test 2: Thursday, November 8, 2001, in class. Cumulative Final Examination: 3:30 pm - 5:30 pm on Thursday, December 13, 2001, in the classroom.
Grading. One grade will be assigned for homework, one for the combined midterm examinations, and one for the final examination. These three will contribute equally to the course grade. Students taking the Cr/NCr or P/F options will need a grade of D or better to pass.
Office Hours. See the instructor in Cupples I, room 105a, on Tuesdays from 9:30-10:00 or 11:30-12, namely before or after class, or make an appointment by telephone or email.