Mathematics 131AH - Honors Analysis - Fall 2016 - UCLA
Time: 11-11:50 MWF, plus 11-11:50 Tu for discussion.
Place: Lectures: MS 5138. Discussion: MS 6627.
Instructor: Burt Totaro.
E-mail: totaro@math.ucla.edu.
Course web page: www.math.ucla.edu/~totaro/131ah.1.16f/
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: Rigorous treatment of the foundations
of real analysis, including construction of the rationals and reals;
metric space topology, including compactness and its consequences;
numerical sequences and series; continuity, including connections
with compactness; rigorous treatment of the main theorems
of differential calculus. That amounts to Chapters 1-4 of Rudin,
plus some of Chapter 5 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, September 27.
Homework 2. Due Tuesday, October 4.
Homework 3. Due Tuesday, October 11.
Homework 4. Due Tuesday, October 25.
Homework 5. Due Tuesday, November 1.
Homework 6. Due Tuesday, November 8.
Homework 7. Due Tuesday, November 22.
Homework 8. Due Tuesday, November 29.
Midterm Exams: We will have two midterm exams.
The dates are Monday, October 17 and Monday, November 14.
There will be no makeup exams.
Sample Midterm 2.
Final Exam: The final exam is on Friday, December 9, 2016
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:
- Every exam will include at least one problem taken from the homework,
possibly with minor variations.
- It is your responsibility to know how to do the problems.
Practicing that is an essential part of studying for the exams.
- A grade of 'F' will be assigned to any student who misses the final. Incompletes are reserved for those who have completed all of the work for the class, including both midterms,
but who, for a legitimate, documented reason, miss the final.
- Exams (or copies) will be returned, but I will keep
copies (or originals) of the exams, as required by the math department.
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
- 10/17 - First midterm exam.
- 11/11 - Veterans Day holiday. No class.
- 11/14 - Second midterm exam.
- 11/25 - Thanksgiving holiday. No class.
- 12/9 - Final exam. The final will be
from 11:30 AM to 2:30 PM on Friday, Dec. 9.
Miscellanea:
- If you wish to request an accommodation due to a disability, please contact the Office for Students with Disabilities as soon as possible at A255 Murphy Hall,
(310) 825-1501, (310) 206-6083 (telephone device for the deaf). Web site: www.osd.ucla.edu.
Catalog description:
131A. Analysis (Honors). (4) Lecture, three hours; discussion, one hour. Required: courses 32B and 33B, with grades of B or better. Recommended: course 115A. Honors sequence parallel to course 131A. P/NP or letter grading. Rigorous introduction to foundations of real analysis; real numbers, point set topology in Euclidean space, functions, continuity.
Tentative schedule of lectures, in terms of the book:
9/23: Ch. 1. Mathematical logic, induction.
9/26: Ordered fields. 9/28: Axioms for R; supremum. 9/30: Applications
of the axioms.
10/3: Complex numbers, Euclidean spaces. 10/5: Construction of R.
10/7: Ch. 2. Finite and infinite sets.
10/10: Countable sets. 10/12: Metric spaces, open and closed sets.
10/14: Compact sets.
10/17: Midterm 1. 10/19: More on compactness. 10/21: Perfect sets. Connected sets.
10/24: More on connectedness. 10/26: Ch. 3. Convergent sequences. 10/28: Subsequences, Cauchy sequences, completeness.
10/31: Lim sup, lim inf. 11/2: Some special sequences. 11/4: Series.
11/7: Absolute convergence. 11/9: Ch. 4. Limits of functions. 11/11: Veterans Day holiday.
11/14: Midterm 2. 11/16: Continuous functions. 11/18: Continuity and compactness.
11/21: Continuity and connectedness. 11/23: Ch. 5. The derivative. 11/25: Thanksgiving holiday.
11/28: Chain rule. Mean value theorem. 11/30: Taylor's theorem. 12/2: Review.