MATH 7900f08

MATH 7900 (Azoff)
Foundations for Graduate Mathematics
Fall 2008

This website was last updated on December 7.

Assignment 11 is due on Tuesday December 9. Be sure to include corrections to your second hour test.

The final exam is scheduled for Wednesday, December 17, from Noon - 3 P.M; it will be comprehensive.

Study aids for the exam include:

List of core topics found below.
Previous prelim exams found at http://www.math.uga.edu/graduate/prelim_info.html
Bolded class discussion problems
Notes to some problems from previous assignments gathered at http://www.math.uga.edu/%7Eazoff/courses/7900notes.pdf

Graded final exams should be available for pick-up on Thursday December 18.

Course Objective

From the Bulletin: "An intensive review of techniques and material essential for graduate study in mathematics, including background in calculus and linear algebra". Coverage of these topics will be integrated with underlying concepts from logic and set theory.
All aspects of the proof-writing process will be analyzed including

discovery and the role of examples
logical structure
rewriting for clear exposition

Texts and References

Author	Title	Web Links for Downloadable Material	Comments
Spivak, Michael	Calculus, 4th Edition		First main text (required)
Zakon, Elias	Basic Concepts of Mathematics	http://www.trillia.com/zakon1.html	For logic and set theory; also contains parallel treatment of some topics in Spivak's text (required; license purchased)
Azoff, Edward	Notes on Logic and Set Theory	http://www.math.uga.edu/%7Eazoff/courses/logic.pdf	Brief summary of preceding item (required & free)
Bond & Keane	Intro to Abstract Mathematics		More detailed version of previous item (optional)
Azoff, Edward	Sequences and Series	http://www.math.uga.edu/%7Eazoff/courses/3100s05.pdf	Parallel treatment of some topics in Spivak's text (free & optional)
Halmos, Paul	How to Write Mathematics	http://www.math.uga.edu/%7Eazoff/courses/halmos.pdf	Expository suggestions (free)
Shifrin, Theodore	Multivariable Mathematics		Second main text (required)
Courant & Hilbert	What is Mathematics?	http://www.math.uga.edu/%7Eazoff/ffds.pdf	Parallel treatment of many course topics (opitional, but everyone should read the preface)

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Exams

The final exam is scheduled for Wednesday, December 17, from Noon - 3 P.M; it will be comprehensive.

There were two "hour" exams during the term.

The first hour test was held on Friday October 3. It covered material from Assignments 1 through 5. The average score was 74.
The second exam was held on Friday December 5. It concentrated on material from Assignments 6 through 10. The average score was 72.
; a list of core topics can be found below.

It is also a good idea to look at relevant problems from previous prelim exams; see http://www.math.uga.edu/graduate/prelim_info.html
Notes to some problems from previous assignments are gathered at http://www.math.uga.edu/%7Eazoff/courses/7900notes.pdf

Here is a list of core topics for the tests; practice problems in boldface type below and relevant problems from old prelim exams could be used to help study for them. It was also a good idea to look at relevant problems from previous prelim exams; see http://www.math.uga.edu/graduate/prelim_info.html

Text	Chapter	Title	Core Topics
Zakon/Azoff	0 (Handout)	Logic	quantifiers
Spivak	1	Numbers	order axioms, inequalities
	2	Types of Numbers	induction, well-ordering, absolute value
	3	Functions	piecewise defintions
	4	Graphs	examples
	5	Limits	using the epsilon-delta definition
	6	Continuity	algebraic combinations of continuous functions; uniform continuity
	7-8	"Hard theorems"	least upper bounds, intermediate value and maximum value theorems
	9	Derivatives	definition and examples
	10	Differentiation	chain rule
	11	Significance of the derivative	mean value theorem and applications
	12	Inverse functions	increasing and decreasing functions
	13	Integrals	Definition via sups and infs
	14	Fundamental Theorem of Calculus	Statement, proof, and applications
	20	Approximation by polynomial functions	Taylor's theorem and applications
	22	Infinite sequences	Application of least upper bound principle; subsequences and Cauchy sequences
	23	Infinite series	Rationales for convergence tests
	24	Uniform convergence and power series	Domains of convergence, manipulation of power series, relation to Taylor's Theorem
Shifrin	1	Vectors and matrices	Dot products, norms, subspaces, linear transformations
	2	Functions, limits, and continuity	Real-valued functions of several values
	3.1	Partial and directional derivatives	Definition and computation
	4	Solutions of linear systems	Matrix algebra and relation to linear transformations
	5.5	Inner product spaces	Projections, least squares, orthonormal bases
	7.5	Determinants	Equivalent characterizations
	9	Eigenvalues and eigenvectors	Change of basis, diagonalization, spectral theorem
	8.3	Line integrals and Green's Theorem	Computing contour integrals both ways
	3.2 - 3.4	(Total) differentiability	Definition, chain rule, gradient

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Assignment Summary

Assign	Due	Text	Reading	Pages for Problems	Problems to Hand In	Extra Problems for Practice/Class Discussion
1	F 29 Aug	Zakon	Pages 1 - 14	9 -11	3a, 4, 11i, 17ii	3b, 5, 11ii, 17i;
		Azoff (Logic)	Pages 1 - 4	4	1, 2, 3	3
		Spivak	Chap 1	13-20	5v,6ab,11iv,12v,14, 20	3ii, 5viii, 6c, 11v, 13, 21
		Spivak	Chap 2	32	5, 12, 19	3, 12,13
2	F 5 Sept	Spivak	Chap 3	51-53	13,21,26	5,9,12,23,25
			Chap 4	72-73	11,17v	3,14,20
			Chap 5	108	3,8,12,13,24,37b	10cd,16,18,21b, 31
3	M 15 Sept	Spivak	Chap 6	119-121	3ab,6,12	4,10c,16
			Chap 7	130-131	7,10,17	3,5
			Chap 8	140-141	6c,12	1,2a,6ab,8a
			8 Appendix	146	1b,2bc	1c,2d
			Chap 9	163-166	14,22a	3,7,13
			Chap 10	181-186	9,16,28	1,24
4	M 22 Sept	Spivak	Chap 11	207-215	11,20,28,38,43,49,55,A,B	1vi,2vi,5,6,31,33,34b,41ab,53
4	M 22 Sept	Spivak	Chap 12	242-243	23	5a
5	M 29 Sept	Spivak	Chap 13	274-280	13,19,20,23ab,37	5,16,29,33,36
			Chap 14	298-299	8,18	3,4,5,7,16
			Chap 20	431-441	1iv,2iii,3iii,8,13,23ab	1,2,3ii,v,14
6	M 13 Oct	Spivak	Chap 22	460-470	9i,iii,11,14a,19,20,27abcd	2,3,4,5,15,23
6	M 13 Oct	Spivak	Chap 23	489-498	1i-x,xv-xviii,9a,15ab,16	6,12,23,26
7	M 20 Oct	Spivak	Chap 24	516-525	3i-ii,6,9,23,29a	1,5,11,14,23,25,28a
		Shifrin	Section 1.1	7-8	10	9,13
			Section 1.2	13-16	2g	4,5,7,11,15,17
			Section 1.3	22-23	1hi,7,12	3,5,6,8,9,10
			Section 1.4	39-43	8,20,30,33	2,3,5,6,17,19, 26,32,34,37
			Section 1.5	50-52	2,7b,9	6b,12,13,14
8	W 5 Nov	Shifrin	Section 2.1	61-64		4
			Section 2.2	70-71	3,9,14c	1,2,8,15
			Section 2.3	78-80	2,4,8bgi,9,12	3,6,7b,10,11,13
			Section 3.1	85-86	5,9,11,13	3b,4,8
9	W 12 Nov	Shifrin	Section 4.1	142-146	6b,12a,15,16,18	11a,12b,23
			Section 4.2	155-156	4	1b,2a,3c,5
			Section 4.3	168-171	1,3,6,8,9,10,13,15, 16,19,20,21,23	5,14c,22,25,26
10	W 19 Nov	Shifrin	Section 4.4	184-186	2,6,15,16,17	3d,4,7,9,12
			Section 7.5	321-324	5,6,12	1ac,11,19,21
			Section 9.1	420-422	3,4,9def,10	9,11
			Section 9.2	433-436	2,4,5,6,9,12gh,13,18	3,7,8,19,20abc,22,23,24,25
			Section 9.3	452-455	1,5	2,4
11	Tu Dec 9	Second Hour Test			Corrections
		Shifrin	Section 5.5	240-243	3,5,8,10	4,7,11,13,17,20
			Section 9.4	463-466	1e,3,9	4,5,6,8,13,14
			Section 8.3	362-366	3ac,6bc,10b	2,8,12,21
			Section 3.2	95-97	7,11a,18	1b,2b,3b,5,6,17
12	-	Shifrin	Section 3.3	102-104	8,12	3,5,11,13,17
			Section 3.4	106-109	6,11	2,10,14
			Section 5.1	201-202	2,3,6	4a,5,7,8,9,10
			Section 5.2	207-208	2,8,14	3,6,7,10,13
			Section 5.3	215-216	2,4,6ad	6c,7
			Section 5.4	222-225	1,7,10,29	2,5,6,9,15,16,23

Supplementary Problems
A. Suppose f:: R \to R is everywhere differentiable and satisfies lim_{x\to\infty} f(x) = \infty. Prove that f is not uniformly continuous.
B. Use Cauchy's Mean Value Theorem (= generalized mean value theorem) to give an epsilon-delta proof of l'hopital's rule (Theorem 11.9)

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MATH 7900 (Azoff) Fall 2008
Foundations for Graduate Mathematics
Course Syllabus

Call Number	11-730

Web Page	http://www.math.uga.edu/~azoff/courses/7900.html

Time & Place	MWF 1:25 - 2:15 P.M.	Room 410 Boyd

Texts	Calculus, by Michael Spivak, 4th Edition, Publish or Perish, 2008; ISBN 978-0-954098-91-1 Multivariable Mathematics, by Theodore Shifrin, Wiley, 2005; ISBN 0-471-52638-X.

Grading	Homework Hour Tests (2 @ 100 pts) Final Exam	100 points 200 points 200 points

	Homework will be collected about once a week; no late work will be accepted. The final exam is scheduled for Wednesday, December 17, from Noon - 3 P.M; it will be comprehensive.

Instructor	E. Azoff
e-mail Phone Office	azoff@math.uga.edu 542-2608 443 Boyd
Office Hours	2:30 - 3:30 Daily	No Office Hours on: Tu - W September 30 - October 1 Th October 9 Tu - W October 14 - October 15 Tu - W October 21 - October 22

First Assignment, due Monday August 25

1. Read Pages 1 - 14 of Elias Zakon's online text, Basic Concepts of Mathematics, available online at http://www.trillia.com/zakon1.html.
Work Problems 3a, 4, 11i, 17ii on Pages 9 - 11 of that text.

2. Read the four-page logic handout available online at http://www.math.uga.edu/%7Eazoff/courses/logic.pdf.
Work Problems 1, 2, 3 on Page 4 of that handout.

3. Read Chapter 1 of the Spivak text. Work Problems 5v,6ab,11iv,12v,14, 20 at the end of that chapter.

4. Read Chapter 2 of the Spivak text. Work Problems 5, 12, 19 at the end of that chapter.

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MATH 7900 (Azoff) Foundations for Graduate Mathematics Fall 2008

Contents

Course Objective

Texts and References

Exams

Assignment Summary

MATH 7900 (Azoff) Fall 2008 Foundations for Graduate Mathematics Course Syllabus

First Assignment, due Monday August 25

MATH 7900 (Azoff)
Foundations for Graduate Mathematics
Fall 2008

MATH 7900 (Azoff) Fall 2008
Foundations for Graduate Mathematics
Course Syllabus