FINAL EXAM AND PRESENTATIONS:

The final exam will be take home, handed out (sealed) at the end of the last lecture and due not later than 1:00 PM Wednesday, December 12, in my office (359B JBE) or mailbox.
You are to complete the exam in a single four hour block. You can use a two page (two sheets of 8.5 X 11 paper) crib sheet; other than that, the exam is closed book. No calculators, computers, etc. can be used.

The final project presentations will be given 12-3 PM, Monday, December 10. Please let me know no later than this Tuesday if you intend to do a final project, and if the project will include a presentation.
Presentations should last at least 10-15 minutes; once I know how many presentations there will be, I can give you an upper bound on the available time for each presentation.
I expect everyone to attend the presentations; refreshments will be provided.
Presenters: please let me know if you'll be using the data projector, the overhead projector, the board, or some combination of these; if you need to set up a demo, or will be part of a multi-presentation team, let me know so I can schedule your presentation accordingly.

Practice problems for the final; the contour plots for problem 3 (with beta = 0).

There will be a review session Wednesday 6-8 PM, December 5. Room: 302 Baskin.

INSTRUCTOR

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
Lecture: TTh 12:00-1:45, 165 Baskin Engineering
Study hall/group work sessions: Tuesday, 6-8 PM, 302 Baskin Engineering and Wednesday 2:00-3:30 PM, 359B Baskin Engineering
Office hours: by appointment
Course web page: http://people.ucsc.edu/~lewis/Math106A/syllabus106a.html (here)
 

TEXT

Differential Equations, third edition, by Paul Blanchard, Robert Devaney, and Glen Hall.
Additional materials will be made available on line.
Those of you who have taken 105A and 117 or their equivalents and would like a more rigorous treatment of some of the key results and constructions may want to use the classic Differential Equations, Dynamical Systems, and Linear Algebra, by Morris Hirsch and Stephen Smale, as a supplemental text.
 

ONLINE MATERIALS

Some free online Java linear algebra and matrix manipulation packages:

• Simple linear algebra package with "friendly" user interface.
• Super-basic matrix arithmetic package (matrix multiplication, squaring of matrices, addition).
• Linear algebra routines from NIST (National Institute of Standards and Technology). Java versions of a classic collection of algorithms for a variety of linear algebra tasks, but no user interface. Any Java programmers in the class want to tackle this?

Some material on the phase flow and matrix exponential:

• My lecture notes on the phase flow and matrix exponential (current version: 10/20/07)
• Notes relating Chapter 3 techniques to the matrix exponential (current version: 10/28/07)
• 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. Maybe thinking about the relationship between discrete time dynamical systems and flows will be useful/interesting.
• 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 lecture notes on the Jordan normal form (current version: 10/17/07)
• 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.
  
• Some course notes These are from a course taught by G. Donald Allen, of Texas A&M. They're clear, but not too high level.
• Some other course notes These are from a course taught by Wolf Holzmann, of the University of Lethbridge.
• 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.

Some FYI/E material on beating and resonance:

• My Mathematica notebook from the November 1 lecture on beating and resonance. (Available as actual notebook or as a PDF firmcopy. (current version: 11/1/07)
• The Physics Classroom Some nice descriptions of the role of beating and resonance in music. If you've taken a physics series, you may have seen some of the experiments described here. Lessons 3-5 are most relevant, but you may want to look at the earlier ones to fill in some terminology and concepts used later.
• Nondestructive testing A brief overview of the use of resonance in detecting defects in structures. (Regularly scheduled nondestructive testing had been proposed/recommended for the Minneapolis bridge that recently collapsed, to monitor possible flaws, but was not implemented due to budget concerns.) Note the distinction between linear and nonlinear behavior. See also the New York Times article on bridge testing (not much detail, but topical).
• Is it live or is it Memorex? ad campaign: storyboard for the original commercial of opera singer Enrico Di Giuseppe breaking a wine glass (in the most famous version, the singer is Ella Fitzgerald, the jazz/pop vocalist), print ad, and technical report by the engineering consulting firm that developed the actual glassbreaking demo shown in the commercials. The Memorex ads involved amplification of the vocalists; an episode of the TV show Mythbusters featured an eventually successful attempt to shatter a glass without amplification.

Some material relevant to the midterm:

• Midterm solution suggestions (there are several different approaches to many of the problems; there is no implication that the ones I use here are 'best').
• Sample matrix exponential and JNF problems. These are representative of problems you might see on the midterm on these topics.

COURSE WORK

• Weekly homework assignments. Most exercises will be from the text, but some will be taken from other sources (these will be provided as hard copies and online). There will be computer-based exercises: you can use the provided software (CD in back of text) or other software (e.g. MatLab or Mathematica) with comparable capabilities. Printouts must be clearly labeled and coherently integrated with your written work. Writing qualitative descriptions of numerical/graphical results is difficult, but essential; this course will hopefully help you refine your scientific expository skills.

• Weekly in-class group problem solving sessions. Students will work in (hopefully self-selected) teams on extensions and applications of results derived in lecture. The goal is not to completely solve the problem in class (great if you can, of course), but to identify the crucial features of the problem, find a good match to some of those features among results or applications we've already studied, and propose a strategy for handling the novel aspects of the problem at hand.

• In-class midterm. Given the time limitations, the midterm will emphasize concepts, classifications, and strategies, rather than detailed number-crunching. Being an active participant in in-class discussions and informal problem-solving sessions will be great preparation for the midterm (and for research!).

• Final exam. Format TBD: might be given during the official exam time Monday, December 10, 12:00-3:00 PM, or as a take-home exam.

• Optional final project.

 
HOMEWORK POLICIES

Check your work! Whenever possible, verify that your analytically derived solution really is a solution.

Late homework will be discounted and, at the discretion of the grader and/or the instructor, may not be accepted.

Your homework should be neatly written and well-organized, with the pages securely fastened together and your name on every page. Many of the exercises involve several nontrivial steps; make it clear to your readers (and yourself!) what it is you're doing at each step.

Clearly number the exercises and try to submit them in numerical order; if any problems are out of sequence, indicate that at the beginning of the assignment. (You don't need to solve them in order, just submit them in order.) The grader should not have to hunt through several pages to find a particular problem.

Computer difficulties do not justify late or incomplete assignments.