Mathematics 131BH - Honors Analysis - Winter 2017 - UCLA

Time: 10-10:50 MWF, plus 10-10:50 Tu for discussion.
Place: Lectures: MS 5138. Discussion: MS 5138.

Instructor: Burt Totaro.
E-mail: totaro@math.ucla.edu.

Course web page: www.math.ucla.edu/~totaro/131bh.1.17w/

Office hours: 3-3:50 M and 2-2:50 F, in my office, MS 6136.
TA: Dimitrios Ntalampekos (dimitrisnt@ucla.edu).
TA office hour: 1:00-2:00 F in MS 3975.

Book: W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, third ed., ISBN: 978-0070542358.

Material to be covered: The Riemann integral; sequences and series of functions; power series, and functions defined by them; differential calculus of several variables, including the implicit and inverse function theorems. That amounts to Chapters 6-9 of Rudin, plus some of Chapter 10 as time permits.

Homework will be due each week in discussion, and returned the next week. There will be no makeup or late homework accepted, but the lowest homework grade will be dropped.

In this class you will be required to write precise mathematical statements in a clear logical order, and present pictures or examples as necessary to illustrate your work. Acquiring these skills is impossible without steady practice: it is essential that you do the homework problems carefully and promptly.

You may discuss homework problems with other students, the TA, or me, before they are turned in. I do expect, though, that: (i) you should make a serious effort to do the exercise yourself before discussing it with anyone, and (ii) you should write up the solution yourself after understanding it thoroughly, without following someone else's written version. Otherwise, homework will not help you to prepare for the exams. Identical solutions to a source will get zero credit.

Homework 1. Due Tuesday, January 17.
Homework 2. Due Tuesday, January 24.
Homework 3. Due Tuesday, February 7.
Homework 4. Due Tuesday, February 14.
Homework 5. Due Tuesday, February 21.
Homework 6. Due Tuesday, March 7.
Homework 7. Due Tuesday, March 14.


Midterm Exams: We will have two midterm exams. The dates are Monday, January 30 and Monday, February 27. There will be no makeup exams.
Sample Midterm 2.


Final Exam: The final exam is on Monday, March 20, 2017 from 11:30 AM to 2:30 PM. You must take the final to pass the class! If you have a documented reason that you are unable to take the final, you will receive an Incomplete.

Evaluation: Grading: Grades will be assigned based on the higher of the following two schemes:
10% homework + 25% first midterm + 25% second midterm + 40% final
10% homework + 35% (best of two midterms) + 55% final Miscellanea: Catalog description: 131BH. Analysis (Honors). (4) Lecture, three hours; discussion, one hour. Required: courses 33B, 115A, 131A. Honors sequence parallel to course 131B. P/NP or letter grading. Derivatives, Riemann integral, sequences and series of functions, power series, Fourier series.

Tentative schedule of lectures, in terms of the book:
1/9: Ch. 6. The Riemann and Riemann-Stieltjes integrals. 1/11: Integrability of continuous functions. 1/13: Properties of the integral.
1/16: Martin Luther King holiday. 1/18: Integration and differentiation. 1/20: Integration of vector-valued functions.
1/23: Ch. 7. Uniform convergence. 1/25: Uniform convergence and continuity. 1/27: Uniform convergence and integration.
1/30: Midterm 1. 2/1: Uniform convergence and differentiation. 2/3: Equicontinuous families of functions.
2/6: The Stone-Weierstrass theorem. 2/8: Ch. 8. Power series. 2/10: The exponential and logarithmic functions.
2/13: Trigonometric functions. 2/15: Fourier series. 2/17: Convergence of Fourier series.
2/20: Presidents' Day holiday. 2/22: The fundamental theorem of algebra. Introduction to the gamma function. 2/24: Ch. 9. Linear transformations.
2/27: Midterm 2. 3/1: Differentiation in several variables. 3/3: The contraction principle.
3/6: Inverse function theorem. 3/8: Implicit function theorem. 3/10: Rank theorem.
3/13: Determinants. 3/15: Differentiation of integrals. 3/17: Review.