University of Wisconsin-Madison
University of Wisconsin-Madison
Mathematics Department

Math 521
Advanced Calculus
A first course in mathematical analysis
Lecturer: Arun Ram

Fall 2004

About the course:

Math 521 is the first course in mathematics. The core topics are basic set theory, numbers, functions, basic topology, sequences and series, limits and continuity, derivatives, integrals, sequences and series of functions and special functions. Math 521 is particularly useful for future K-12 math teachers since one of the main points of this course is to explain the properties of the real numbers and functions which take real numbers as values-- why they do what they do, and how they can be sensibly modified. If you are going to be teaching math, then you will need to explain these things to your students. Along with Math 541 and Math 551 this course is a necessity for students considering going on to graduate school in mathematics. In order to do well in this course it will help to have (1) a good study ethic and (2) some experience with real valued functions such as that obtained in Math 221.

Special goal: One of the goals of this course is to teach everybody how to construct and write proofs.

Lectures: There will be three 50 minute lectures per week:

MWF 12:05-12:55 in B105 Van Vleck

Text: The textbook for the course is

Principles of Mathematical Analysis, by Walter Rudin, Third edition, McGraw-Hill, 1976.

I also recommend the following books:

Linear Operators, Part I: General Theory, by Dunford and Schwarz, Wiley, ISBN: 0-470-22605-6 or ISBN: 0-471-60848-3.

General Topology, by N. Bourbaki, Springer-Verlag.

Functions of a Real Variable, by N. Bourbaki, Springer-Verlag.

Theory of Sets, by N. Bourbaki, Springer-Verlag.

Undergraduate analysis, by S. Lang, Second Edition, Springer-Verlag, ISBN: 0-387-94841-4.

I strongly recommend you get one or all of these books (read them, study them, do problems from them, look things up in them, absorb them, osmose them, sleep with them under your pillow, and carry them with you everywhere you go).

Office hours: Wednesday 1:00-3:00 Van Vleck 711 and by appointment. For appointments, come up before or after class and make sure that I write your name and a time to meet in my date book. If you are having any problems, questions or concerns please come in to talk about it. If you do not keep me informed I cannot help.

Grading: The term grade will be based on homework and the exams as follows: Homework: 25% Vocabulary quizzes: 25% Midterms: 25% Final Exam: 25%. Final grades are computed by totalling the points from the homework, the midterms and the final. Grade letters will be assigned with the following curve as a guideline: 20% A's, 30% B's, 30% C's, 20% D's and F's.

Homework: Homework will be due weekly on Monday, in lecture. All claims that you make in your homework MUST BE PROVED in order to receive credit.

Homework assignments: Additional assignments will appear as the term progresses

Homework assignment 1: DUE September 13, 2004 pdf file

Homework assignment 2: DUE September 20, 2004 pdf file

Homework assignment 3: DUE September 27, 2004 pdf file

Homework assignment 4: DUE October 11, 2004 pdf file

Homework assignment 5: DUE October 18, 2004 pdf file

Homework assignment 6: DUE October 25, 2004 pdf file

Homework assignment 7: DUE November 1, 2004 pdf file

Homework assignment 8: DUE December 1, 2004 pdf file

Homework assignment 9: DUE December 15, 2004 pdf file

Vocabulary quizzes: One of the challenges when one is first starting to learn proof oriented mathematics, is to remember the rules of the game (the definitions). This EXTREMELY IMPORTANT part of doing proofs is easy since it is just regurgitation. To keep everybody in shape on this we will have a (10 minute) vocabulary quiz each Friday during class.

Quiz 1: September 10, 2004 pdf file

Quiz 2: September 17, 2004 pdf file

Quiz 3: September 24, 2004 pdf file

Quiz 4: October 8, 2004 pdf file

Quiz 5: October 15, 2004 pdf file

Quiz 6: October 22, 2004 pdf file

Quiz 7: October 29, 2004 pdf file

Lecture Notes:

Lecture 1: September 3, 2004: pdf file

Lecture 2: September 8, 2004: pdf file . Also look at the "Sets and functions" notes below pdf file

Lecture 3: September 10, 2004: See the "How to do proofs" notes below pdf file

Lecture 4: September 13, 2004: pdf file

Lecture 5: September 15, 2004: pdf file

Lecture 6: September 17, 2004: pdf file

Lecture 7: September 20, 2004: pdf file

Lecture 8: September 22, 2004: pdf file

Lecture 9: September 24, 2004: pdf file

Lecture 10: September 27, 2004: pdf file

Lecture 11: September 29, 2004: pdf file

Lecture 12: October 1, 2004: pdf file

Lecture 14: October 11, 2004: pdf file

Lecture 15: October 13, 2004: pdf file

Lecture 16: October 18, 2004: pdf file

Lecture 17: October 20, 2004: pdf file

Lecture 18: October 22, 2004: pdf file

Lecture 30: December 3, 2004: pdf file

Additional notes:

How to do proofs: pdf file

Sets and functions: pdf file

Exams: There will be three 50 minute in-class midterms: October 6, November 3, and December 1. There will be a 2 hour final exam at 7:25pm on Thursday December 16. The exams will be a random selection of homework problems from the homework assignments. The final exam will be cumulative.

Syllabus: The following is my plan. We can be flexible and change the syllabus if we feel like it.

(1) Numbers, Sept. 3: positive integers, integers, rationals, real numbers, complex numbers, algebraic numbers

(2) Sets and functions, Sept. 8: injective, surjective, bijective, cardinality, inverse functions.

(3) Proofs, Sept. 10: the proof machine

(4) Relations, Sept. 13: reflexive, symmetric, transitive, equivalence relations, equivalence classes, partitions.

(5) Numbers, Sept. 15: operations, commutative, associative, identities, inverses, monoids, groups, rings, fields, homomorphisms

(6) Numbers, Sept. 17: polynomials, formal power series, fields of fractions, evaluation homomorphisms

(7) Special functions, Sept. 20: exponential functions, logarithms, trig functions

(8) Derivations, Sept. 22: definitions, uniqueness, Taylor's theorem

(9) Derivations, Sept. 24: product rule, chain rule

(10) Cardinality, Sept. 27: integers, rationals, reals, the Cantor set

(11) Orders, Sept. 29: ordered sets, ordered groups, absolute values

(12) Orders, Oct. 1: ordered sets, posets, total orders, well orders, Zorn's lemma, axiom of choice

(13) Midterm I, Oct. 4, covers lectures 1-12 and homeworks 1-4.

(14) Topology, Oct. 6: open sets, closed sets, limit points, neighborhoods, interior, closure, connectedness

(15) Topology, Oct. 8: continuous functions and limits

(16) Topology, Oct. 11: compact sets

(17) Metric spaces, Oct. 13: boundedness, diameter

(18) Metric spaces, Oct. 15: The Heine-Borel theorem

(19) Sequences, Oct. 18: limits, subsequences, Cauchy sequences, completeness, completions

(20) Examples of completions, Oct. 20: real numbers, formal power series, p-adic numbers.

(21) Series, Oct. 22: convergence, radius of convergence, absolute convergence, rearrangements

(22) Series, Oct. 25: tests for convergence

(23) Continuity, Oct. 27: uniform continuity, discontinuity, examples, left and right continuity

(24) Review, Oct. 29

(25) Midterm 2, Nov. 1, covers lectures 15-25 and homeworks 5-8.

(26) Derivatives, Nov. 3: definitions, derivatives and continuity

(27) Derivatives, Nov. 5: The mean value theorem, Taylor's theorem

(28) Derivatives, Nov. 8: L'Hopital's rule

(29) Integrals, Nov. 10: Riemann integrals, Lebesgue integrals, Denjoy integrals

(30) Normed spaces, Nov. 12: The spaces C^k(X) and sup norms

(31) Integrals, Nov. 15: The Riesz representation theorem

(32) Integrals, Nov. 17: The fundamental theorem of calculus

(33) Integrals, Nov. 19: change of variable--the Jacobian

(34) Curves, Nov. 22

(35) Review, Nov. 24

(36) Midterm 3, Nov. 29: covers lectures 27-36 and homeworks 9-12

(37) Uniform convergence, Dec. 1: examples

(38) Uniform convergence, Dec. 3: theory

(39) Equicontinuity, Dec. 6

(40) The Stone-Weierstrass theorem, Dec. 8: part 1

(41) The Stone-Weierstrass theorem, Dec. 10: part 2

(42) Hilbert spaces, Dec. 13

(43) Fourier series, Dec. 15