This class is cross-listed as:
Textbook (required):
Analysis I - Fourth Edition,
by Terence Tao, Texts and Readings in Mathematics 37, Hindustain Book Agency (2022) (editions one, two and three
work equally well).
The first four chapters and the last two appendices, one on the basics of mathematical logic and one on decimal expansions, of Tao's book can be found here
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, and in 2019 he was the first recipient of the Riemann Prize. He also cares deeply about teaching,
and our textbook is an example of that. Here are links to his
webpage and his
wikipedia page,
see also a 2015 article in the New York Times
article about Tao.
There are many other excellent introductory analysis books. Reading from other
sources is always very valuable. For example: Elementary Analysis by Kenneth A. Ross and Analysis with an Introduction to Proofs By Steven Lay used recently by other faculty. Other good books are Calculus
by Spivak, 4th edition Publish or Perish, 2008, or The Way of Analysis
by Robert Strichartz, Jones & Bartlett Publishers, Revised Edition, 2000.
For a non-standard book, but a very lively one, full of historical anecdotes, see
A radical approach to real analysis by D. M. Bressoud.
Here is a resource for learning how to write proofs Book of Proofs by Richard Hammack. This book has been used as a companion to the main advanced calculus text in previous semesters. The author kindly makes it available through the American Institute of Mathematics (AIM) Open Textbook Initiative.
Course Structure: Mondays and Wednesdays will be devoted to lecturing on new material. The recitation hour on Tuesdays will be used to help you with the homework, for problem solving and review of the material as needed.
Course content: This is a first course in analysis. We will cover the fundamentals of calculus in one variable, starting with the construction of the real numbers, sequences of numbers and working our way through the concepts of limits, functions, continuity and basic properties of functions, we will then study carefully the theory of differentiation and integration. Basic calculus is a prerequisite, it provides you with computational skills and some intuition. Prior experience with mathematical abstractions and proofs will be helpful (for example exposure to at least one of Math 306, 317, 318, 319, 321, 322 or 327 will provide such experience). However we do not expect the students to be able to read, understand, and actually construct mathematical proofs at the begining of the course, although most of you have probably been already exposed to proofs. A great amount of time will be devoted to learn and practice logical thinking. At the end of the course we expect the students to have adquired the basic skills of mathematical reasoning, and a deeper understanding of calculus.
Homework: The problems and exercises in the textbook are an integral part of the course. You should attempt most of them. 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. The highlights of the solutions will be discussed in the Wednesday session. You are encouraged to discuss the homework with each other, but you should do the writing separately. 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. 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 two midterms on weeks 6 and 12, and a final exam.
Grades: The final grade will be determined by your performance on homeworks, midterms, and a final exam. The grading policies will be discussed in class.
Prerequisites: Math 264 and two MATH courses 300-level or above (or permission from the instructor).
Accommodations: 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 ( https://arc.unm.edu/) at arcsrvs@unm.edu or by phone at 505-277-3506.
Title IX: UAP 2720 and 2740. Our classroom and university should always be spaces of mutual respect, kindness, and support, without fear of discrimination, harassment, or violence. If you ever need assistance or have concerns about incidents that violate this principle, please access campus support resources. These include confidential services at LoboRESPECT Advocacy Center, the Women's Resource Center, and the LGBTQ Resource Center. The University of New Mexico prohibits discrimination on the basis of sex (including gender, sex stereotyping, gender expression, and gender identity). UNM faculty and graduate teaching assistants are considered "responsible employees." "Responsible employees" must communicate reports of sexual harassment, sexual misconduct and sexual violence to Compliance, Ethics and Equal Opportunity. For more information on the campus policy regarding sexual misconduct, reporting, and reporting for "responsible employees," please see UAP 2720 and UAP 2740.
Credit-hour statement: This is a four credit-hour course. Class meets for three 75-minute sessions of direct instruction for fifteen weeks during the Spring 2025 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: Cheating and plagiarism (academic dishonesty) are often driven by lack of time, desperation, or lack of knowledge about how to identify a source. Communicate with me and ask for help, even at the last minute, rather than risking your academic career by committing academic dishonesty. Academic dishonesty involves claiming that work created by another source is your own original work. It is a Student Code of Conduct violation that can lead to a disciplinary procedure. When you use a resource in work submitted for this class, document how you used it and distinguish clearly between your original work and the material taken from the resource.
Wellness:
If you do need to stay home due to illness or are experiencing a wellness challenge, please take advantage of the resources below. You can communicate with me at [crisp @ math . unm . edu ] and I can work with you to provide alternatives for course participation and completion. Let me, an advisor, or another UNM staff member know that you need support so that we can connect you to the right resources. UNM is a mask friendly, but not a mask required, community. If you are experiencing COVID-19 symptoms, please do not come to class.
Student Support: Student Health and Counseling (SHAC) at (505) 277-3136. TimelyCare: Free 24/7 virtual care services (medical, emotional support, health coaching, self-care, basic needs support). LoboRESPECT Advocacy Center (505) 277-2911: help with contacting faculty and managing challenges that impact your UNM experience.
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. 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: http://undocumented.unm.edu/.
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.
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Last updated: January 18, 2025