Advanced Calculus (15-Math-6001)	real number field, completeness axiom, limit superior and inferior, subsequences, Bolzano-Weierstrass and Heine-Borel theorems, continuity and differentiability, the Chain Rule, the Intermediate and Mean Value theorems, Taylor series and theorem, Weierstrass M-test, the Riemann integral	Department of Mathematical Sciences

This page is a work in progress! (Last revised 19 Sept 2012)

Below you can find information regarding: some suggested problems from the textbook, links to a number of interesting things, and the homework assignments.

Textbooks There are scores of Advanced Calculus/Introduction to Analysis texts available in the Geo-Math-Physics Library. Below I list just a few of the many many such texts. I will use Wade's book to generate my lectures, and most if not all HW problems will come from Wade. Sometimes it can be a good idea to look at more than one book, because often some author will say things in just the 'right' way. Also, do not overlook the web as a resource.

The last book, by Trench, is free and available here. The book by Rudin is somewhat more sophisticated than the rest, but it's a great reference.

W	Wade	An Introduction to Analysis
A	Apostol	Mathematical Analysis
DD	Dence & Dence	Advanced Calculus: A Transition to Analysis
F	Folland	Advanced Calculus
Ro	Rosenlicht	Introduction to Analysis
Ru	Rudin	Principles of Mathematical Analysis
T	Trench	Introduction to Real Analysis

General Syllabus Chapters: 1-8 and 10 in Wade's book.

Course Goals First and foremost, this is a course geared towards teaching you to think, act, and problem solve like a mathematician. Of particular importance, to me, is your acquiring the ability to communicate mathematical ideas effectively. This means learning how to read, and especially to write, mathematical proofs. In addition, you should come to understand that different environments lend to different communication requirements.

We shall study elementary analysis. A subgoal, at least for some of you, is preparation for the graduate program Qualifying Exam; I will have (much) more to say about this in the spring (at which time I may run a practice problem session). Here is a list of the Qualifying Exam topics and some practice problems.

Grades Your final course grade will be determined from your performance on two in class exams, a comprehensive final exam, your homework scores, your written assignments, and your classroom and problem session participation. Here is a precise breakdown:

• 30% --- final exam
• 30% --- 2 in-class hour exams
• 30% --- homework and assignments
• 10% --- class and problem session participation

The Final Exam is scheduled for Monday 10 December at 1:30-3:30. The in-class hour exams are (tentatively) scheduled for Friday 5 October and Friday 16 November.

You should be sure to work all of the exercises as well as the problems in the text. I may use some of these on the exams and quizzes.

I will ask you write up and hand in solutions to certain exercises from the textbook (aka, Homework); these will be graded and returned to you. See below for due dates. I'll say more about this in class.

Here are some links to some interesting *things* such as videos, power point files, etc. I will add to this list as the year progresses.

Writing Mathematics

• Homework guidelines you should follow.
• Mathematical Writing
• Suggestions for Writing Mathematics.pdf
• Guide2WritingMaths.pdf
• Tips4WritingMaths.pdf
• MathsWritingCheckList.pdf
• CommonErrorsWritingMaths.pdf
• LMS-WritingMaths.pdf
• WritingMathProofs.pdf
• WritingMaths.pdf

Class Stuff

• Homework guidelines you should follow.

Unless I explicitly indicate otherwise, you should read everything in Wade's book and work all of the exercises that you find as you read. Any of these exercises, as well as all of the "fill in the details" that I mention during lectures, are fair game as "easy" exam questions.

Below is the assigned homework with due dates. The "Problems To Hand In" are to be written up and handed in before class on the indicated dates. The "For Problem Session" problems will be discussed in that week's problem session. After the due date, you can find outlines for solutions (at least for some of the problems) by clicking the appropriate problem number.

Please be sure to check out my guidelines for writing up your HW solutions.

1. Use problem 1.4.1(c) to show that the sequence (x_n) does indeed converge. Be sure to indicate when the limit is 0 or 1 or in (0,1).
2. 2.4.7b is just what I called the BWT. Instead, please prove the corollary that I called the "BWT for Sequences".
3. follow

Below I list suggested problems for each indicated chapter in Wade's book (An Introduction to Analysis). Don't forget to look at (i.e., work through) the many exercises placed throughout the book.