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General Course Description
Math 401 is an upper undergraduate level introductory course in Analysis. It introduces the students to logic and basic techniques of proof, applied to a rigorous study of single-variable calculus. Topics include the definition and topology of real numbers, sequences, limits, functions, continuity, differentiation, and integration. Basic calculus is a prerequisite; it provides one with computational skills and some intuition. We do not expect the students to be able to read, understand, and actually construct mathematical proofs at the beginning of the course. Students will learn to read, understand, and construct mathematical proofs. A great amount of time will be devoted to learning and practicing logical thinking. At the end of the course, students are expected to have acquired the basic skills of mathematical reasoning, plus a deeper understanding of calculus.
Student Learning Outcomes (SLOs) and Rubric for Math 401
Textbook
Steven R. Lay, Analysis with an Introduction to Proof, 3rd or any later edition (Required)
3rd Edition, 4th Edition, 5th Edition
2nd Edition may work as well.
Students are fully expected to read the sections in the textbook since both the lectures and the text's sections will reinforce one another. The text will cover some material not covered in lectures, and the lectures will cover some material not covered in the text. Of course, there will be some overlapping content.
Students should strive to read the corresponding sections in the book before the lectures so that they can ask questions for clarification. See the syllabus for a tentative schedule.
Sections 9 (Axioms for Set Theory), 15 (Metric Spaces), and 24 (Continuity in Metric Spaces) are all optional sections. Students are welcome to read these sections; in fact, students are encouraged to read these sections along with section 34 (power series) and chapter 9 on Sequences and Series of Functions (pointwise and uniform convergence, applications of uniform convergence, and uniform convergence of power series) - all of which are not covered in the course.
Supplementary Literature
There are many other excellent introductory analysis books. Reading from other sources could be very valuable:
1) Terence Tao, Analysis I, any edition (Recommended);
Chapters 1 - 4 (PDF File)
Appendix: Logic (PDF File)
Terence Tao is an exceptional mathematician who won the 2006 Fields Medal. New York Times article about Tao.
2) Jerrold E. Marsden, Elementary Classical Analysis
3) Richard Hammack, Book of Proof
The PDF file comes from Hammack's webpage. The author has kindly made it available on there.
5) Walter Rudin, Principles of Mathematical Analysis, 3rd Edition
Course Material - Lecture Notes:
Please note: the posted lecture notes were originally intended for the instructor's purposes only and may not be complete. By no means are the posted lecture notes meant to serve as a substitute for attendance in class and the reading of the text. It is not guaranteed that they are without error either. Should there be any errors, notify and inform the instructor of the error immediately or as soon as possible.
Set Theory: Set Operations and Relations
Functions and Cardinality
Ordered Fields, The Completeness Axiom
Sequences: Convergence and Theorems
Cauchy Sequences, Subsequences
Continuous Functions and their Properties
The Derivative, The Mean Value Theorem
L'Hopital's Rule, Taylor's Theorem
Taylor Polynomial Approximations:
f(x) = sin x and f(x) = 1/(1 - x)
The Riemann Integral & Properties
The Fundamental Theorem of Calculus
Infinite Series: Convergence Tests
Homework
Homework problems come from the different editions of the textbook and the supplementary literature.
Project
Project: Construction of the Natural Numbers, the Integers, and the Rational Numbers
It will be beneficial to use the first four chapters of Terence Tao's Analysis I for reference.
Exams
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