Math 402/502 - Advanced Calculus II

MATH 402/502 - Avanced Calculus II
Spring 2024

This class is cross-listed as:

Here is a quick link to the homework.

Textbook (required): Analysis II (Texts and Readings in Mathematics Book 38) by Terence Tao. Springer, Fourth Edition, 2022 (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 and the last two appendices in pdf format pdf file. Terence Tao is an exceptional mathematician who has earned almost all possible awards a mathematician can get, including the 2006 Fields Medal (the equivalent of a Nobel Price in Mathematics), the 2014 Breakthrough Prize in Mathematics, he was the first recipient of the Riemann Prize in 2019 and the Grand Medaillefrom the French Academie of Sciences. He also cares deeply about teaching, and our textbook is an example of that. Here are links to his wikipedia page, see also a 2015 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 in the third and fourth editions 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 or fourth editions.

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 (if at all) 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. In case you need to brush up on your proof skills, here you can find a copy of Richard Hammak's Book of Proof, third edition.

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 gorup project.

Grades: The final grade will be determined by your performance on homeworks, the midterms, and the final group project. The grading policies will be discussed in class.

Prerequisites: Math 401/501 or permission from the instructor.

Accomodation Statement: UNM is committed to providing equitable access to learning opportunities for students with documented disabilities. As your instructor, it is my objective to facilitate an inclusive classroom setting, in which students have full access and opportunity to participate. To engage in a confidential conversation about the process for requesting reasonable accommodations for this class and/or program, please contact Accessibility Resource Center ( ( at or by phone (505) 277-3506.

Credit-hour statement: This is a three credit-hour course. Class meets for two 75-minute sessions of direct instruction for fifteen weeks during the Spring 2024 semester. Please plan for a minimum of six hours of out-of-class work (or homework, study, assignment completion, and class preparation) each week.

Responsible Learning and Academic Honesty: We all have shared responsibility for ensuring that learning occurs safely, honestly, and equitably. Submitting material as your own work that has been generated on a website, in a publication, by an artificial intelligence algorithm (AI), by another person, or by breaking the rules of an assignment constitutes academic dishonesty. It is a student code of conduct violation that can lead to a disciplinary procedure. Please ask me for help in finding the resources you need to be successful in this course. I can help you use study resources responsibly and effectively. Off-campus paper writing services, problem-checkers and services, websites, and AIs can produce incorrect or misleading results. Learning the course material depends on completing and submitting your own work. UNM preserves and protects the integrity of the academic community through multiple policies including policies on student grievances (Faculty Handbook D175 and D176), academic dishonesty (FH D100), and respectful campus (FH CO9). These are in the Student Pathfinder ( and the Faculty Handbook (

Title IX: To meet obligations under Title IX, UNM faculty, Teaching Assistants, and Graduate Assistants are considered "responsible employees." 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 Compliance, Ethics and Equal Opportunity ( For more information on the campus policy regarding sexual misconduct and reporting, see:
Support: LoboRESPECT Advocacy Center , the Women's Resource Center , and the LGBTQ Resource Center all offer confidential services.

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 university's website:

Land Acknowledgement: Founded in 1889, the University of New Mexico sits on the traditional homelands of the Pueblo of Sandia. The original peoples of New Mexico Pueblo, Navajo, and Apache since time immemorial, have deep connections to the land and have made significant contributions to the broader community statewide. We honor the land itself and those who remain stewards of this land throughout the generations and also acknowledge our committed relationship to Indigenous peoples. We gratefully recognize our history.

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

Last updated: Jan 11, 2024