Analysis 1 - Fall 2006
Isaac Newton
1643-1727 |
Augustin-Louis Cauchy
1789-1857 |
Karl Weierstrass
1815-1897 |
Georg Friedrich Bernhard Riemann
1826-1866 |
Images from Keith Lynn's "Pictures of Mathematicians" webpage and the The MacTutor History of Mathematics archive.
COURSE: MATH 4217/5217
Call # 33004/33005
TIME AND PLACE: 11:15-12:35 TR in Room 231 of Lamb Hall
INSTRUCTOR:
Dr. Robert Gardner
OFFICE HOURS: 9:00-10:00 TR
OFFICE: Room 308F of Gilbreath Hall
PHONE: 439-6979 (308F Gilbreath), Math Department Office 439-4349
E-MAIL:
gardnerr@etsu.edu
WEBPAGE: See my webpage (
www.etsu.edu/math/gardner/gardner.htm) for an online copy of this syllabus with homework assignments and any changes which might arise.
TEXT: An Introduction to Analysis, 2nd edition, by J. R. Kirkwood, Published by PWS Publishing Company and Waveland Press, Inc. 1995.
PREREQUISITES: It is assumed that each student has some experience with
proof proving (at the level of MATH 2800 - Math Reasoning, for example).
Of course, you should feel comfortable with references to results from
freshman calculus.
ABOUT THE COURSE: In this course, we give a rigorous development of
calculus and a study of the topology of the real line. Several of the
results which we will see will be familiar from your freshman calculus
classes (in fact, a calculus book will make good supplementary reading).
I will occasionally assign problems and cover material not in the text. I
will rely on the following sources:
- Topology, a First Course, by J. R. Munkres. A readable
introduction to general topology. This text has been used in the past in
our graduate Topology class (MATH 5350).
- Real Analysis, by H. L. Royden. This is a standard text
for a first graduate course in real analysis.
It includes the more advanced topics of
measure theory, Lebesgue integration and Lp spaces.
Students registered for MATH 5217 will be given extra homework problems
and an extra problem on each test.
GRADING: Homework (H) will be assigned and collected regularly. We will
have a midterm (M) and a final (F). Your
average will be computed as follows:
AVERAGE = (2H + M + F)/4.
Grades will be assigned based on a 10 point scale with "plus" and
"minus" grades being assigned as appropriate.
A NOTE ABOUT HOMEWORK: While I suspect that you may work with each other on the homework problems (in fact, I encourage you to), I expect that the work you turn in is your own and that you understand it. Several of the homework problems are fairly standard for this class, and you may find proofs online. However, the online proofs may not be done with the notation, definitions, and specific methods which we are developing and, therefore, are not acceptable for this class.
COURSE OUTLINE: We will cover:
- Chapter 1 = Sets and Functions, Real Numbers as a Field,
Completeness Axiom.
- Chapter 2 = Sequences of Real Numbers, Subsequences,
Bolzano-Weierstrass Theorem.
- Chapter 3 = Topology of Real Numbers.
- Chapter 4 = Limits and Continuity, Monotone and Inverse Functions.
- Chapter 5 = Differentiation, Mean Value Theorems.
IMPORTANT DATES:
- Friday, September 1 = Last day to late register.
- Monday, September 4 = Labor Day holiday.
- Friday, September 8 = Last day to drop without a "W."
- Monday, September 11 = Last day for 75% refund.
- Monday, September 25 = Last day for 25% refund.
- Monday and Tuesday, October 16-17 = Fall Break holiday.
- Monday, October 23 = LAST DAY TO DROP.
- Thursday, October 26 = Midterm (1.1, 1.2, 1.3, 2.1)
- Thursday and Friday, November 23-24 = Thanksgiving holiday.
- Wednesday, December 6 = Last day to withdraw from all classes.
- Tuesday, December 12 = final, 8:00 a.m. - 10:00 a.m.
HOMEWORK
| ASSIGNMENT NUMBER |
PROBLEMS | DUE DATE |
POINTS | SOLUTIONS |
| HW 1 |
1.1.7a, 1.1.11g, 1.1.13c, 1.1.15, G-1, BONUS 1 (Airplane Problem) | THR 9/7 | 3+3+3+3+(6)+[3]=12+(6)+[3] | PDF PS |
| HW 2 |
1.2.1a, 1.2.3, 1.2.7a (give a proof by contradiction) | THR 9/14 | 3+3+3=9 | PDF PS |
| HW 3 |
1.2.8a, 1.2.18a, 1.2.19a, G-2 | THR 9/21 | 3+3+3+(4)=9+(4) | PDF PS |
| HW 4 |
1.3.2a, 1.3.3, 1.3.5 | THR 9/28 | 3+3+3=9 | PDF PS |
| HW 5 |
1.3.12, 1.3.13a, 1.3.14, G-3, G-4, Bonus 2 | THR 10/5 | 3+3+3+(3+3)+[3]=9+(6)+[3] | PDF PS |
| HW 6 |
2.1.1c, 2.1.5, 2.1.8a, G-5 | THR 10/12 | 4+3+3+(3)=10+(3) | PDF PS |
| HW 7 |
2.1.12a, 2.1.12d (in this one, you may assume 12b and 12c), Problem 1 | TUE 10/24 | 3+3+3=9 | PDF PS |
| HW 8 |
2.2.10, 2.2.12b | TUE 11/7 | 3+3=6 | PDF PS |
| HW 9 |
2.3.12, 2.3.16, G-6 (2.3.11b), Bonus: 2.3.11a | TUE 11/14 | 3+3+(2)+[4]=6+(2)+[4] | PDF PS |
| HW 10 |
3.1.4, 3.1.5, Bonus: 3.1.6a | TUE 11/28 | 3+3+[3]=6+[3] | PDF PS |
| HW 11 |
3.1.9, 3.1.13, G-7 (3.1.15a,b,c), Bonus: 3.1.20 (without using Heine-Borel) | TUE 12/5 | 5+3+(9)+[3]=8+(9)+[3] | PDF PS |
| - |
- | TOTAL POINTS | 93+(30)+[16] | - |
NOTICE: The number of POINTS in the third column are for the undergraduate homework assignments, with additional graduate requirements in parentheses and bonus problems in square brackets.
PROBLEMS
- Problem 1. Prove that a convergent sequence is Cauchy.
GRADUATE PROBLEMS
- G-1. 1.1.19
- G-2. 1.2.18b and 1.2.19b.
- G-3. A real number is algebraic if it is the zero of a polynomial with rational coefficients. Show that the algebraic numbers form a countable set.
- G-4. 1.3.18a.
- G-5. 2.1.8c.
- G-6. 2.3.11b.
- G-7. 3.1.15a,b.c.
BONUS PROBLEMS
- Bonus 1. Airplane Problem: PDF, PostScript.
- Bonus 2. Show that if, for set X, we have |X|=n, then the power set of X satisfies |P(X)|=2n. You may either show this using induction, or with a countaing argument which uses the Binomial Theorem (which itself is proved by induction).
SOME WEBSITES
- Bertrand Russell's and Alfred North Witehead's Principia Mathematica: (1) A
description of the work from the Stanford Encyclopedia of Philosophy, (2) A copy of the text in PDF from the University of Michigan Historical Math Collection.
- My outline of the Nova episode A MATHEMATICAL MYSTERY TOUR, part of which we watched on September 5, 2006.
- Proof of the existence of nth roots (Exercises 1.3.9a and 1.3.9b): PDF and PS.
Return to
Bob Gardner's home page