Honors Multivariable Calculus and Linear Algebra

Math 25b --- Spring 2005

Announcements

Date	Time	Announcement
Sun 8 May	9pm	Added orientations (which I had forgotten to specify) for many curves and surfaces on homework 12.
Sat 30 Apr	11.30pm	Corrected a typo in the last question on homework 11.
Fri 22 Apr	6.30pm	Corrected a serious error in question 3 on homework 10.
Thu 21 Apr	11.50am	Made a correction to the first question on integration on homework 10.
Mon 18 Apr	12.10pm	Posted an addendum to the lecture on Fubini's Theorem.
Sun 17 Apr	2.20pm	Made a correction to the Lagrange multipliers question on homework 9.
Sun 17 Apr	1.25pm	Added links to the homework solutions.
Sun 17 Apr	1.10pm	Made a correction to the footnote on page 2 of homework 9.
Fri 15 Apr	12.45am	Made a correction to the third question on homework 9.
Wed 6 Apr	1.45pm	Made a correction to the last question on homework 8.
Wed 6 Apr	12.10pm	Made a correction to the first question on homework 8.
Wed 9 Mar	12.30pm	Posted homework 6.
Wed 9 Mar	1.35pm	Posted homework 5.
Tue 1 Mar	9.10pm	Posted homework 4.
Wed 23 Feb	6pm	Corrected a typo in question 1 on homework 3.
Mon 21 Feb	7pm	Modified and completed the plan for the course. Posted homework 3.
Tue 1 Feb	6.30pm	Started new website. The website for Math 25a is now available here.

Essential information

Classes:	MWF 10--11 in 222 Science Center
Instructor:	Tom Coates
Office hours:	Monday 2--3pm and Friday 1--2pm
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
Calculus on Manifolds by Michael Spivak, published by Westview Press

These are available from Harvard COOP, or from on-line bookstores.

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:	TBA	TBA

Homework

Date due	Handed in?	Returned?	Solutions
Tuesday 15 February	Yes	Yes
Tuesday 22 February	Yes	Yes
Tuesday 1 March	Yes	Yes
Tuesday 8 March	Yes	Yes
Tuesday 15 March	Yes	Yes
Tuesday 22 March	Yes	No	Not yet
Friday 25 March	Yes	No	Not yet
Tuesday 12 April	Yes	No	Not yet
Tuesday 19 April	No	No	Not yet
Tuesday 26 April	No	No	Not yet
Tuesday 3 May	No	No	Not yet
Friday 13 May	No	No	Not yet

Handouts

	Date	Title
	Wed 2 Feb	Office hours and feedback form
	Wed 16 Feb	Tensor products
	Mon 18 Apr	Fubini's Theorem

Course Outline

Date	Topics	Reading
Wed 2 Feb	review; bases and dimension	none
Fri 4 Feb	subspaces; dual spaces	Halmos 10-14
Mon 7 Feb	dual spaces (continued)	Halmos 15-17
Wed 9 Feb	direct sums	Halmos 18-22
Fri 11 Feb	quotient spaces	Halmos 18-22
Mon 14 Feb	quotient spaces (continued)	none
Wed 16 Feb	bilinear forms; tensor products	Halmos 23-25
Fri 18 Feb	tensor products (continued)	handout
Mon 21 Feb	no class: Presidents' Day	none
Wed 23 Feb	permutations; multilinear forms	Halmos 26-31
Fri 25 Feb	linear transformations	Halmos 32-38
Mon 28 Feb	invariant subspaces; projections	Halmos 39-43
Wed 2 Mar	adjoints; change of basis	Halmos 48-52
Fri 4 Mar	quotient transformations; rank-nullity; tensor products (again)	Halmos 53-55
Mon 7 Mar	determinants; eigenvalues	Halmos 56-57
Wed 9 Mar	triangular form; nilpotence	Halmos 58
Fri 11 Mar	Jordan form	Halmos 58
Mon 14 Mar	inner product spaces	Halmos 59-61
Wed 16 Mar	orthogonality; completeness; projections (again)	Halmos 62-66
Fri 18 Mar	Riesz representation theorem; naturality	Halmos 67-70
Mon 21 Mar	self-adjointness; isometry; change of basis (again)	Halmos 71-74
Wed 23 Mar	projections; the Spectral Theorem; polar decomposition	Halmos 75-83
Fri 25 Mar	recap and overview of linear algebra	none
Mon 28 Mar	no class: Spring Break	none
Wed 30 Mar	no class: Spring Break	none
Fri 1 Apr	no class: Spring Break	none
Mon 4 Apr	derivatives; the Chain Rule	Spivak 15-25
Wed 6 Apr	partial derivatives	Spivak 25-33
Fri 8 Apr	Inverse Function Theorem	Spivak 34-40
Mon 11 Apr	Implicit Function Theorem	Spivak 41-43
Wed 13 Apr	integration	Spivak 44-56
Fri 15 Apr	Fubini's Theorem	Spivak 57-61
Mon 18 Apr	partitions of unity; change-of-variable	Spivak 62-74
Wed 20 Apr	change-of-variable	none
Fri 22 Apr	tensors	Spivak 75-85
Mon 25 Apr	differential forms	Spivak 86-90
Wed 27 Apr	exterior differentiation	Spivak 91-94
Fri 29 Apr	Poincaré Lemma; chains	Spivak 95-100
Mon 2 May	Stokes's Theorem	Spivak 101-104
Wed 4 May	manifolds	Spivak 109-121
Fri 6 May	Stokes's Theorem on manifolds	Spivak 122-137