Class: MWF, 12:00-1:00 PM, 310 Hearst Mining
Instructor: Ved V. Datar
Email: vv lastname at math.berkeley.edu, no spaces
Office: 1067 Evans Hall
Office hours: MW, 11:00AM-12:00PM
GSI: Harrison Chen, Office hours - Evans 959, Tuesday 4-7, Wednesday 3-7, Thursday 4-7. Check his webpage for last minute cancellations.
Text: Walter Rudin, Principles of mathematical analysis, (any edition)
Suplementary reading: Charles Pugh, Real mathematical analysis.
Homeworks - 20%, Midterms - 40%, Final - 40%
There will be 12 homeworks. The best ten will be counted towards the grade. There is no late submission of homeworks.
There will be two midterms of 40 points each and a final exam of 80 points. There will be no make-up exams. The midterm score will be calculated by the formula -
If you are a disabled student (with or without a document from the Disabled Students' Program) and require special accommodations of any kind, please e-mail the instructor as soon as possible, and no later than January 31st.
If you are representing the university on some official duties (say if you are an athlete or in a band), and if there is a conflict with any of the mid-terms, please let the instructor know before January 31st.
To do well in the course, it is important to be able to understand as well as write proofs. It might be useful to read Prof. Hutching's article on mathematical reasoning.
Most of our proofs will use standard set theoretc notation. Here are some notes from Prof. Bergman on set theory.
We will also often use mathematical induction. Here are some notes on that by Prof. Bergman.
Practice problems for first mid-term. Here are the solutions
Practice problems for the second mid-term.
Practice problems for the final
Final exam and solutions
The page numbers refer to the third edition of Rudin's book.
| Number | Date | Topic | Reading | Homework | Notes |
| 1 | W 1/18 | introduction, review of integers and rational number, ordered sets | 1-5 | A1 (due 01/30). | |
| 2 | F 1/20 | field axioms, construction of reals | 5-10 | ||
| 3 | M 1/23 | complex numbers, Euclidean spaces, cardinality | 12-17, 24-25 | Lectures 1-3 | |
| 4 | W 1/25 | cardinality (cont.), metric spaces | 26-31 | ||
| 5 | F 1/27 | metric spaces (cont.) | 16-18 | ||
| 6 | M 1/30 | metric spaces (cont.), compact sets | 32-36 | A2 (due 02/08). | |
| 7 | W 2/01 | compact sets | 37-40 | ||
| 8 | F 2/03 | compact sets (cont.), connected sets | 42 | Lectures 4-8 (Cardinality and metric spaces) | |
| 9 | M 2/06 | convergent sequences and subsequences | 47-52 | ||
| 10 | W 2/08 | Cauchy sequences and completeness | 53-55 | A3 (due 02/15). | |
| 11 | F 2/10 | monotonic sequences, limsup and liminf, examples | 56-58 | Sequences | |
| 12 | M 2/13 | infinite series, divergence test, comparison test | 59-60 | ||
| 13 | W 2/15 | p-series, geometric series, the number e | 61-65 | A4 (due 02/22) | |
| 14 | F 2/17 | root and ratio tests | 65-69 | Infinite Series - 1 | |
| M 2/20 | holiday | ||||
| 15 | W 2/22 | midterm-1 | |||
| 16 | F 2/24 | summation by parts, Dirichlet's test, absolute and conditional convergence | 70-72 | Infinite Series - 2 | |
| 17 | M 2/27 | rearrangements | 75-78 | A5 (due 03/08) | |
| 18 | W 3/01 | limits and continuity | 83-86 | Limits and Continuity (complete set) | |
| 19 | F 3/03 | equivalent formulations of continuity, examples | 87-89, 98 | ||
| 20 | M 3/06 | continuity and compact sets, uniform continuity | 90-92 | ||
| 21 | W 3/08 | continuity and connected sets, intermediate value theorem | 93-97 | A6 (due 03/15) | |
| 22 | F 3/10 | differentiability, basic properties | 103,105 | ||
| 23 | M 3/13 | chain rule, mean value theorems | 105-107 | Derivatives | |
| 24 | W 3/15 | mean value theorems (cont.), L'Hopital's theorem | 108-109 | A7 (due 03/22) | |
| 25 | F 3/17 | derivatives of higher order, Taylor's theorem | 110-111 | Exponentials and Log | |
| 26 | M 3/20 | Riemann integral | 120-121 | ||
| 27 | W 3/22 | sufficiency conditions for integrability, examples | 124-126 | ||
| 28 | F 3/24 | properties of integral, change of variables | 128-132 | A8 (due 04/05) | |
| M 3/27 | spring recess | ||||
| W 3/29 | spring recess | 91 | |||
| F 3/31 | spring recess | ||||
| 29 | M 4/03 | fundamental theorem of calculus, integration by parts | Integration | ||
| 30 | W 4/05 | midterm-2 | 133-134 | ||
| 31 | F 4/07 | improper integrals, integral test for series | A9 (due 04/12) | ||
| 32 | M 4/10 | sequences of functions, uniform convergence | 143-148 | Uniform Convergence | |
| 33 | W 4/12 | uniform convergence and continuity and integrals | 149-152 | A10 (due 04/19) | |
| 34 | F 4/14 | uniform convergence and differentiation | 152-154 | ||
| 35 | M 4/17 | equicontinuity and Ascoli-Arzela | 155-158 | Ascoli-Arzela, Stone-Weierstrass | |
| 36 | W 4/19 | Stone-Weierstrass | 159-161 | A11 (due 04/26) | |
| 37 | F 4/21 | power series | 69-70, 172-178 | Power Series | |
| 38 | M 4/24 | power series (cont.) | 69-70, 172-178 | ||
| 39 | W 4/26 | power series (cont.) | 69-70, 172-178 | A12 (due 05/03) | |
| 40 | F 4/28 | exp., log, trig functions | 179-184 | 41 | M 5/01 | RRR | 42 | W 5/03 | RRR | 43 | F 5/05 | RRR |