Honors Multivariable Calculus and Linear Algebra

Math 25a --- Fall 2004

Announcements

Date	Time	Announcement
Sun 19 Dec	7.15pm	Website change: posted homework 12.
Fri 10 Dec	9.30am	Website change: added the last question to homework 11.
Wed 8 Dec	12.00pm	Website change: posted a correction to today's lecture.
Mon 6 Dec	3.45pm	Website change: posted hints for the last problem on homework 10.
Wed 1 Dec	12.00pm	Website change: posted a de-mangled version of a result from today's lecture.
Mon 29 Nov	3.40pm	Website change: posted the reading for the rest of the semester, plus most of homework 11.
Mon 29 Nov	12.30pm	Website change: posted homework 10.
Wed 24 Nov	9.35am	Website change: posted the Challenge Problems.
Thu 11 Nov	11.00am	Website change: moved Q9 on p115 of Rudin from homework 8 to homework 9.
Tue 9 Nov	1.00pm	Website change: posted homework 8.
Sun 7 Nov	12.30pm	Website change: fixed a typo in question 2.2 on homework 7.
Wed 3 Nov	6.15pm	Website changes: posted the rest of homework 7; deleted old announcements.
Wed 3 Nov	1.45am	Website changes: new homework assignment posted; solutions to homework 4 now available.

Practice finals

Authors	Final	Solution	Solution authors
Matt Fasman and Sara Kate Heukerott			Elizabeth Goodman and Kristen Hendricks
Elizabeth Goodman and Kristen Hendricks			Matt Fasman and Sara Kate Heukerott
Vilsa Curto and Irena Wang			Gerardo Con Diaz and John Davies
Gerardo Con Diaz and John Davies			Vilsa Curto and Irena Wang
Seth Flaxman and Miriam Hinman			Yiyi Deng and Dave Galkowski and Adam Kapor
Yiyi Deng and Dave Galkowski and Adam Kapor			Seth Flaxman and Miriam Hinman
Connie Chao and Sam Lewellan			Maanit Desai and Yuanchen Zhu
Maanit Desai and Yuanchen Zhu			Connie Chao and Sam Lewellan
Takuya Kitagawa, Sam Lichtenstein, and Yiming Wang			Luca Candelori and Alex Waldron
Luca Candelori and Alex Waldron			Takuya Kitagawa, Sam Lichtenstein, and Yiming Wang
Silas Richelson and Stephanie Zhang			Dan Greenwald and Robin Walters
Dan Greenwald and Robin Walters			Virginia Fisher and Anastasia Artemyev
Virginia Fisher and Anastasia Artemyev			Silas Richelson and Stephanie Zhang

Essential information

Classes:	MWF 10--11 in 221 Science Center
Instructor:	Tom Coates
Office hours:	Mon 11.15--12.15 and Fri 11.15--12.15
Office:	230 Science Center
Email:	tomc@math.harvard.edu
Phone:	617 495 5340

Textbooks

Finite-dimensional vector spaces by Paul R. Halmos, published by Springer-Verlag
Principles of mathematical analysis by Walter Rudin, third edition, published by McGraw-Hill

Both of these are available from Harvard COOP, or from on-line bookstores.

I have also placed the following books on reserve in Cabot library:

How to Solve It by G. Polya. This is a classic text on strategies and techniques for solving mathematical problems.
Basic Topology by M. A. Armstrong. This has useful sections on metric spaces and point-set topology.

Course Assistants

The course assistants for the class are Yan Zhang and Anatoly Preygel.

	Anatoly Preygel	Yan Zhang
Email:	preygel at fas	yanzhang at fas
Problem session:	Thursday 8-9pm in 111 SC	Friday 3-4pm in 113 SC

Handouts

	Date	Title
	Mon 20 Sep	Policies and practicalities
	Wed 22 Sep	Supplement to Lecture 2
	Sat 25 Sep	A note on homework problem 3.5
	Tue 27 Sep	Supplement to Lecture 4
	Thu 30 Sep	A wonderful problem
	Mon 11 Oct	Another excellent problem
	Wed 13 Oct	Policy on working together and related issues
	Mon 25 Oct	Feedback form
	Wed 24 Nov	Challenge problems
	Wed 1 Dec	A clarification
	Mon 6 Dec	Hints for the last problem on homework 10
	Wed 8 Dec	Correction to today's lecture

Course Outline

Taylor's theoremcontinuous and monotonic functions are integrableno class (Thanksgiving)power seriesthe Fundamental Theorem of Algebra; the derivative of a function of several variablessubspaces; dual spaces

Date	Topics	Reading
Mon 20th Sep	Introduction and overview; sets and mappings	none
Wed 22nd Sep	Finite sets; cardinality	Rudin 1.1-1.10, 2.1-2.3
Fri 24th Sep	Equivalence classes; construction of the rational numbers	Rudin 1.12-1.35
Mon 27th Sep	Fields; ordered fields; properties of the real numbers	as last week
Wed 29th Sep	Countability; the uncountability of the real numbers	Rudin 2.5-2.14
Fri 1st Oct	Construction of the real numbers	Rudin pp17-21
Mon 4th Oct	Metric spaces; open and closed sets	Rudin 2.15-2.30
Wed 6th Oct	Properties of metric spaces	as Mon 4th Oct
Fri 8th Oct	Compactness	Rudin 2.31-2.42
Mon 11th Oct	No class (Columbus Day)	None
Wed 13th Oct	Compactness (continued); perfection	Rudin 2.36-2.44
Fri 15th Oct	the Cantor set; connectedness; sequences	Rudin 2.45-3.4
Mon 18th Oct	sequences (continued); sequential compactness; completeness	Rudin 3.5-3.14
Wed 20th Oct	examples; series	Rudin 3.15-3.30
Fri 22nd Oct	examples; comparison test	Rudin 3.31-3.37
Mon 25th Oct	comparison test; root test; ratio test	as Friday
Wed 27th Oct	power series; absolute convergence	Rudin 3.38-3.46
Fri 29th Oct	multiplication of series; rearrangement of series	Rudin 3.47-3.55
Mon 1st Nov	Continuity	Rudin 4.1-4.10
Wed 3rd Nov	Continuity and compactness	Rudin 4.10-4.21
Fri 5th Nov	Continuity (continued)	Rudin 4.22-4.34
Mon 8th Nov	Differentiability	Rudin 5.1-5.5
Wed 10th Nov	Rolle's theorem; the Mean Value Theorem; L'Hôpital's rule	Rudin 5.6-5.13
Fri 12th Nov	Rudin 5.15-5.19
Mon 15th Nov	Taylor's theorem; applications	none
Wed 17th Nov	Integration	Rudin 6.1-6.7
Fri 19th Nov	Rudin 6.8-6.11
Mon 22nd Nov	properties of integration	Rudin 6.12-6.19
Wed 24th Nov	the Fundamental Theorem of Calculus	Rudin 6.20-6.27
Fri 26th Nov	none
Mon 29th Nov	pointwise and uniform convergence	Rudin 7.1-7.12
Wed 1st Dec	uniform convergence, differentiation, and integration	Rudin 7.13-7.18
Fri 3rd Dec	Rudin 8.1-8.5
Mon 6th Dec	equicontinuity	Rudin 7.19-7.25
Wed 8th Dec	Stone-Weierstrass	Rudin 7.26-7.33
Fri 10th Dec	none
Mon 13th Dec	vector spaces; examples; linear combinations	Halmos 1-6
Wed 15th Dec	linear independence; bases; dimension	Halmos 7-10
Fri 17th Dec	Halmos 13-17
Mon 20th Dec	linear transformations	Halmos 32-38
next semester	direct sums; quotients; eigenvalues and eigenvectors; diagonalization; Jordan canonical form; the Spectral Theorem; Inverse Function Theorem; Implicit Function Theorem; calculus on manifolds; Stokes' Theorem