Math 402/502 - Advanced Calculus II

MATH 402/502 - Avanced Calculus II
Spring 2019

This class is cross-listed as:

Textbook (required): Analysis II (Texts and Readings in Mathematics Book 38) by Terence Tao. Springer, Third Edition, 2016 (previous editions work as well).

If you did not use Tao's first volume in Math 401 you might consider getting or borrowing the first volume Analysis I . Here you will find the first four chapters of Tao's Analysis I in pdf format pdf file. Terence Tao is an amazing mathematician and teacher, here is a New York Times article about Tao.

Recommended texts (not required): There are many other excellent introductory analysis books. Reading from other sources is always very valuable. I recommend two other books that include the material discussed in this class: 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 on Tuesdays and Thursdays. The course will cover Chapter 11 (in Tao's Analysis I), Chapters 12-15, Chapter 17, and time permitting, we will touch on Chapters 18-19 (in Tao's Analysis II).
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 R^n. In the first part, Math 401, the fundamentals of calculus in one variable were covered, starting with the definition of the real numbers, sequences of numbers, series and working your way through the concepts of limits, functions, continuity and differentiability of functions on the real line, and lightly touching on Riemann integration. A good amount of time was spent learning and practicing logical thinking. At this point I expect the students have acquired the basic skills of mathematical reasoning, a deeper understanding of calculus, and are ready to continue learning more analysis.

We will start the semester discussing in detail Riemann integration on bounded intervals. 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 on the notion of uniform convergence will be made, and its crucial role in interchanging limit operations: differentiation, integration, series, power series. 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 integration on R^n, 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 this. It is not enough to see your teacher or your friends solving problems, you have to try it yourself.

Exams: There will be two midterms, and a final exam or project.

Grades: The final grade will be determined by your performance on homeworks, the midterms, 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 accesible 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.

Gender Discrimination: In an effort to meet obligations under Title IX, UNM faculty, Teaching Assistants, and Graduate Assistants are considered "responsible employee" by the Department of Education (see pg 15 - http://www2.ed.gov/about/offices/list/ocr/docs/qa-201404-title-ix.pdf ). This designation requires that any report of gender discrimination which includes sexual harassment, sexual misconduct and sexual violence made to a faculty member, TA, or GA must be reported to the Title IX Coordinator at the Office of Equal Opportunity (oeo.unm.edu). For more information on the campus policy regarding sexual misconduct, see: https://policy.unm.edu/university-policies/2000/2740.html

Citizenship and/or Immigration Status: All students are welcome in this class regardless of citizenship, residency, or immigration status. Your professor will respect your privacy if you choose to disclose your status. As for all students in the class, family emergency-related absences are normally excused with reasonable notice to the professor, as noted in the attendance guidelines above. UNM as an institution has made a core commitment to the success of all our students, including members of our undocumented community. The Administration's welcome is found on the website https://undocumented.unm.edu

Return to: Department of Mathematics and Statistics, University of New Mexico

Last updated: Jan 8, 2019