Instructor: Debra Lewis
Office: 359B Baskin Engineering
Phone: 459-2718
E-mail: lewis at ucsc dot edu (checked more often than voicemail or gmail) and/or DebraKLewis at gmail dot com
 

TIMES AND PLACES

Office hours: M, Th 4:00-5:00
Course web page: http://people.ucsc.edu/~lewis/Math208 (here)
 

TEXT

Introduction to Smooth Manifolds, John M. Lee, Springer.

Possible supplemental texts: An Introduction to Differentiable Manifolds and Riemannian Geometry, Boothby. Topology from a Diffential Viewpoint, Milnor.

Scanned supplemental material:
Grassmannians and projective space. Boothby (10/9/08)
Implicit Function Theorem. Marsden and Ratiu (10/9/08)
Local Surjectivity Theorem. Marsden and Ratiu (10/20/08)
Hairy Ball Theorem and Brouwer Fixed Point Theorem. Milnor, American Math Monthly, July 1978. (10/23/08)
Ruled surfaces Mathematica calculations: PDF, notebook. (10/28/08)

Some material on the phase flow and matrix exponential:

• My old lecture notes on the phase flow and matrix exponential
• Mathematica notebook with sample plots of flows of the unit square. Available as actual notebook or PDF firmcopy. (current version: 10/24/07)
• Scholarpedia Article on Dynamical Systems by James Meiss (dynamical systems big fish) of the University of Colorado. The section on Flows is the relevant part, but the whole article is nice.
• Wikipedia This actually looks like a reasonably good concise overview of the matrix exponential.
• Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later, by Cleve Moler and Charles Van Loan (SIAM Review, 48, 1). Cleve Moler is one of the designers of MatLab and Charles Van Loan is one of the authors of the classic Matrix Computations. This paper is more than we need, but fun/interesting reading, with a very clear, accessible review of the key material and lots of cute little examples that make the standard algorithms look bad. The dubiousness involves issues with floating point calculations and amplification of round-off errors.

• My old lecture notes on the Jordan normal form
• Mathematica notebook with sample 4x4 Jordan normal form and matrix exponention calculations. Available as actual notebook or PDF firmcopy. (current version: 10/24/07)
• MIT OpenCourseWare This looks like a good, fairly sophisticated treatment.
  
• Kahan's course notes William Kahan, of UC Berkeley, is one of the superstars of numerical analysis and King of Rigor. Lots of good information, beautifully presented, but some of the material is probably a bit advanced for our purposes.
• Wikipedia The first few sections are relevant, but possibly too cryptic to be very useful.

The midterm. (11/5/08)

Topics:
Definition of manifolds, tangent bundle, inverse and implicit function theorems, transversality, Sard's theorem and the Whitney embedding theorem, vector fields, flows, and Lie bracket, Frobenius' theorem.

COURSE WORK

• Weekly homework assignments. Most exercises will be from the text, but some may be taken from other sources (these will be provided as hard copies and online).
• Takehome midterm and final.
• Optional presentation.