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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 |
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| Instructor Prof David A Herron |
My Office Hours TBA and by appt |
E-mail me at David's e-address
My web page is at David's w-address |
Problem Session Wednesday during class time |
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 the text The Way of Analysis by Robert S. Strichartz to generate my lectures, and many of the HW problems will come from it. I believe that you can find a pdf of this somewhere on the internet.
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. Below I list some books that I know or have heard good things about. Also, do not overlook the web as a resource.
You can download a free pdf copy of Hoffman's book here.
The book by Trench is also free and available here. The book by Rudin is somewhat more sophisticated than the rest, but it's a great reference.
| S | Strichartz | The Way of Analysis |
| H | Hoffman | An Analysis in Euclidean Space |
| A | Apostol | Mathematical Analysis |
| DD | Dence & Dence | Advanced Calculus: A Transition to Analysis |
| F | Folland | Advanced Calculus |
| MH | Marsden & Hoffman | Elementary Classical Analysis |
| Ro | Rosenlicht | Introduction to Analysis |
| Ru | Rudin | Principles of Mathematical Analysis |
| T | Trench | Introduction to Real Analysis |
| W | Wade | An Introduction to Analysis |
General Syllabus Chapters 1-7 in The Way of Analysis.
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.
Especially, this course will present a rigorous in-depth treatment of calculus with an emphasis on the theory.
Unless I explicitly indicate otherwise, you should read everything in the text and try to work all of the exercises and problems 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.
Each week I will pass out a set of problems, many chosen from the text book. I will ask you write up and hand in solutions to certain of these exercises (aka, Homework); these will be graded and returned to you. Please be sure to check out my guidelines for writing up your HW solutions. Those problems not designated as Homework will be discussed at the weekly Problem Sessions. I will ask each of you to tell me which of the problems you have solved, and then I will select individuals to present their solutions at the board. I'll have more to say about this in class.
Here are some links to some interesting *things*; I will add to this list as the year progresses.
Writing Mathematics
Please be sure to follow the Homework Guidelines. Also, please read some of the following, at least the very first article!
Class Stuff