Math 402/502 - Advanced Calculus II

### MATH 402/502 - Avanced Calculus IISpring 2013

This class is cross-listed as:

• Math 402 - Call # 25468 - Advanced Calculus II
• Math 502 - Call # 25473 - Advanced Calculus II

Textbook: Analysis II. 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 chapter of Tao's book in pdf format pdf file. 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 Chapters 11 through 15 and chapter 17 in Tao's books (Chapter 11 is the last chapter on Riemann integration in Book I). Chapter 16 is on a favorite topic of mine, Fourier series, but we will skip it, instead we will spend the last days discussing briefly the Lebesgue measure and the Lebesgue integral in Rn (Chapters 18-19) and how it compares to Riemann integration.

Course content: This is the second part of a first one year course in analysis, concerned mostly about analysis on metric spaces, particularly analysis on several variables. In the first part, Math 401/501, we 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. We 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 Riemann integration on bounded intervals (this is a topic we covered briefly in 401 both in Spring and Fall 2012, and it should help me to bring all students up to the same page, and to refresh our 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 will be a brief introduction to Lebesgue measure and integration in 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. Difficult as it may seem at the beginning, if you persist you will learn how to write a proper mathematical proof, you will learn how to read and understand other's proofs, and you will learn to appreciate and enjoy the beauty of an elegant argument.

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.

Americans with Disabilities Act: Qualified students with disabilities needing appropriate academic adjustments should contact me as soon as possible to ensure your needs are met in a timely manner. Handouts are available in alternative accessible formats upon request.

Last updated: Jan 07, 2013