This class is cross-listed as:
Here is a quick link to the homework.
Inverse Function Theorem vs Implicit Function Theorem handout.
Analysis II. Third (or Second) Edition
by Terence Tao. Text and Readings in Mathematics 38.
Hindustan Book Agency 2009. (required).
(If you were not in 401 in the last two semesters,
you might consider getting the first volume Analysis I .)
Here you will find the first four chapters of Tao's volume I in
pdf file, see also recent New York Times
article about Tao.
There are many other excellent introductory analysis books. Reading from other sources is always very valuable. I recommend two other books: Introduction to analysis by Maxwell Rosenlicht (a Dover book very cheap), and The way of analysis by Robert S. Strichartz.
Course Structure: There are 2 lectures per week. Tuesdays and Thursdays. The course will cover Chapter 11 (on Riemann integration in Volume I), Chapters 12 through 15 and chapter 17 in Volume II. Chapter 16 is on a favorite topic of mine, Fourier series, but we will skip it, instead we will use the time (if we have it) to discuss briefly the Lebesgue measure and the Lebesgue integral in Rn (Chapters 18-19) and how it compares to Riemann integration. Note that the third edition of the book came out on October 2014, and the chapter numbering in Volume II now starts at 1, for Chapter N in the second edition with N>11, it will be Chapter (N-11) in the third edition.
Course content: This is the second part of a first one year course in analysis, concerned mostly with analysis on metric spaces, particularly analysis on several variables. In the first part, Math 401/501, you covered the fundamentals of calculus in one variable, starting with the definition of the real numbers, sequences of numbers, series and working our way through the concepts of limits, functions, continuity and differentiability of functions on the real line, and perhaps you had time to lightly touch on Riemann integration and assuming all the properties of the integral prove the two fundamental theorems of calculus. You spent a good amount of time learning and practicing logical thinking. At this point I expect the students to have acquired the basic skills of mathematical reasoning, a deeper understanding of calculus, and to be ready to continue learning more analysis. We will start the semester discussing in detail Riemann integration on bounded intervals (if you covered this topic briefly in 401 both in Spring and Fall 2015, it won't hurt to see it again and it should help me to bring all students up to the same page, and to refresh your memory on some of the concepts and techniques learned there). Next topic of discussion will be metric spaces and point set topology, in particular the concepts of convergence of sequences, compactness, continuity and limits are revisited on metric spaces. Emphasis in the notion of uniform convergence will be made, and its crucial role in interchanging limit operations: differentiation, integration, series, power series. We will spend sometime discussing approximation of functions defined on the real line with polynomials: Taylor series, Stone-Weierstrass Theorem. Then we will plunge into several variable calculus: derivatives, partial derivatives, chain rule, and the celebrated contraction mapping, implicit and inverse function theorems. The last topic (time permitting) will be a brief introduction to Lebesgue measure, integration on Rn, and change of variables, paralleling the presentation of the Riemann integral at the begining of the semester.
Homework: The problems and exercises in the textbook are an integral part of the course. You should solve as many as possible. Homework will be assigned periodically, the problems in the homework will be carefully graded, and returned to you with feedback that will help you correct any errors. You are encouraged to discuss the homework with each other, but you should attempt the problems first on your own. You learn mathematics by doing, and there is no way around it, it is not enough to see your teacher or your friends solving problems, you have to try it yourself.
Exams: There will be one midterm, and a final exam or project.
Grades: The final grade will be determined by your performance on homeworks, the midterm, and the final exam or project. The grading policies will be discussed in class.
Prerequisites: Math 401/501 or permission from the instructor.
Accomodation Statement Accessibility Services (Mesa Vista Hall 20121, 277-3506) provides academic support to students who have dissabilities. If you think you need alternative accessible formats for undertaking and completing coursework, you should contact this service right away to assure your needs are met in a timely manner. If you need local assistance in contacting Accessibility Services, see the Bachelor and Graduate Programs office.
Academic Integrity The University of New Mexico believes that academic honesty is a foundation principle for personal and academic development. All university policies regarding academic honesty apply to this course. Academic dishonesty includes, but is not limited to, cheating or copying, plagiarism (claiming credit for the words or works of another from any type of source such as print, Internet or electronic database, or failing to cite the source), fabricating information or citations, facilitating acts of academic dishonesty by others, having unauthorized possession of examinations, submitting work of another person or work previously used without informing the instructor, or tampering with the academic work of other students. The University's full statement on academic honesty and the consequences for failure to comply is available in the college catalog and in the Pathfinder.
Return to: Department of Mathematics and Statistics, University of New Mexico
Last updated: Jan 18, 2016