This class is cross-listed as:
Textbook (required):
Analysis I - Third Edition,
by Terence Tao. Text and Readings in Mathematics 37,
Springer 2016 (editions one and two
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 on his webpage.
Course Structure: Tuesdays and Thursdays will be devoted to lecturing on new material and occasional quizes. The recitation hour on Wednesdays will be used for problem solving and review of the material.
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 during 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).
UNM Administrative Mandate on Required Vaccinations:
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In accordance with University Policy 2310 and the Americans with Disabilities Act (ADA), academic accommodations may be made for any student who notifies the instructor of the need for an accommodation. It is imperative that you take the initiative to bring such needs to the instructor's attention, as I am not legally permitted to inquire. Students who may require assistance in emergency evacuations should contact the instructor as to the most appropriate procedures to follow. Contact Accessibility Resource Center at (505) 277-3506 or arcsrvs@unm.edu for additional information.
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Return to: Department of Mathematics and Statistics, University of New Mexico
Last updated: January 17, 2022