## Math 4111Introduction to Analysis Fall, 2012

#### NEWS:

• "It is an established maxim and moral that he who makes an assertion without knowing whether it is true or false is guilty of falsehood, and the accidental truth of the assertion does not justify or excuse him."
Abraham Lincoln, chiding the editor of a Springfield, Illinois, newspaper, quoted from Antony Flew, How to Think Straight, p. 17

• Final Examination (Dec 18th, 6-8pm) will be in Crow Hall, room 201, rather than in our classroom.

### Syllabus

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 theorems 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 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 Duncker Hall, room 101.

Prerequisites. Math 310, 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:
 HW #1, due Tuesday, September 11 (Solutions) HW #2, due Tuesday, September 25 (Solutions) HW #3, due Tuesday, October 9 (Solutions) HW #4, due Tuesday, October 23 (Solutions) HW #5, due Thursday, November 8 (Solutions) HW #6, due Tuesday, November 20 (Solutions) HW #7, due Thursday, December 6 (Solutions)
Solutions are due at the end of class on the due date. Late homework will not be accepted.

Tests. Midterm examination: Thursday, October 25, 2012, in class. Cumulative Final Examination: Tuesday December 18, 2012, 6:00 PM - 8:00 PM, in Crow Hall, room 201.

Grading. One grade will be assigned for homework, one for the midterm examination, and one for the final examination. These three will contribute equally to the course grade. Letter grades, computed from the course score class average and standard deviation, will be at least the following:

 Course score at least: Letter grade at least: 90% 80% 70% 60% A B C D

Students taking the Cr/NCr or P/F options will need a grade of D or better to pass. Students auditing the course will need to attend at least 25 lectures to receive a successful audit grade.

Office Hours. See the instructor in Cupples I, room 105a, on Tuesdays and Thursdays from 11:30-12:00, namely after class, and Fridays from 3:00-4:00, or make an appointment by telephone or email.