ANNOUNCEMENTS
MTH 434/534 — Winter 2007

3/22/07

I expect to be in my office tomorrow from roughly 9–3. There is a fair chance I will have the exams graded by late morning, and I will certainly have solutions available for inspection.

3/18/07

Here are some comments/hints:

• Problem 1: It is useful to work in coordinates adapted to your surface.
  If you are having trouble choosing a surface, talk to me!
• Problem 2: "Euclidean R³" has coordinates (x,y,z); E and B have no w-component.
• Problem 3: "Flux" is essentially the same in all dimensions; reason by analogy.
• Problem 4: The commutator of two vector fields is also a vector field.
  Make sure to note the correction below!
• Problem 5: Proceed step by step. Start by setting up the integral. Do not try to evaluate it.

I will be in my office this afternoon (Sunday) at 3 PM, and will stay as long as there are questions.

If you are spending hours on this test, stop and ask me for help!

3/17/07

There is a typo in the statement of Problem 4!

The example in italics should say that the commutator of yk−zj and zi−xk is yi−xj.

Sorry about that. If the example still doesn't make sense, contact me.

3/16/07

I will be in my office until noon on both Monday 3/19 and Tuesday 3/20.

I expect to arrive by 9:30 AM, and most likely earlier.

Other times are possible by arrangement.

I will also check my email regularly this weekend.

3/3/07

The final in this class will be a take-home exam.

• Expect the exam to be comparable to, but somewhat longer than, a typical homework assignment.
• I plan to hand out the exam on Friday, 3/16.
• The exam will be due at noon on Tuesday, 3/20, although I would prefer to get it Monday, if possible.

If these arrangements will cause difficulties with your schedule, let me know as soon as possible!

2/28/07

Please note that my conventions for the connection 1-forms ωⁱ_j differ from those in Flanders by a factor of –1!

2/21/07

Here is some information about next term:

Information about textbooks has been posted here and here.

The class will again meet at 8 AM unless we are able to find an acceptable alternative...

2/20/07

My Mathematica package wedge.m can be used on any campus computer running Mathematica, such as those in the MLC.

Please note that this software has (still) not been extensively tested!

Some (old) instructions are available here, but they only tell you how to load the package on COSINe machines running Linux (e.g. app.science.oregonstate.edu).

For other machines on campus, the package must be loaded from \\poole\Class Folders\Math-Dray. For instructions on how to do this under Windows, follow the Getting Started section of this document from another class (but don't start GSP). Start Mathematica, then load the package with a command of the form "<<\\poole\Class Folders\Math-Dray\wedge.m".

2/13/07

I have graded roughly half of the HW. Of those, well under half are correct.

• The most common error is the use of a coordinate basis rather than an orthonormal basis.
• Several students calculated the gradient and Laplacian, but failed to calculate the divergence and curl.
• The remaining papers contained other computational errors, some serious.

I strongly encourage all students to stop by my office later today (Tuesday) to check your work.

2/12/07

I will be in my office tomorrow, Tuesday 2/13, at 11 and again at 1, to answer any questions you may have as you prepare for the midterm.

I will have to leave at 11:30 sharp, so you may wish to save long questions for the afternoon.

2/10/07

The midterm will be in class next Wednesday, 2/14.

• The exam is closed book.
• The exam will cover everything discussed in class through Friday, 2/9.
• Important topics are exterior product (∧), Hodge dual (∗), and exterior differentiation (d).
• There should be time on Monday for a review; come prepared to ask questions.

2/7/07

You can find a list of some orthogonal coordinate systems here, based on the Schaum Outline: Mathematical Handbook of Formulas and Tables, Murray R. Speigel, McGraw-Hill, New York, 1968.

2/5/07

I was right to express reluctance to use the formula ∗α = g(α, α) β, where α∧β = ω, although not for the right reasons. This formula will work in most cases, provided care is taken in the selection of β, which is not (yet) well-defined. However, difficulties arise if α is null.

For typical coordinate bases, which are orthogonal but not orthonormal, the formula does provide an easy, correct method for computing the Hodge dual, using the obvious choice of β.

2/4/07

Our trusty web reporter found an online article here which describes yet another way of visualizing differential forms (mostly in 2 dimensions).

2/3/07

There will be an in-class midterm during Week 6, tentatively scheduled for Wednesday, 2/14/07. :-)

2/2/07

Below are the two examples we didn't quite finish in class. You may find it helpful to verify these results before attempting the homework. Note especially the differences in signs.

• 2-d Minkowski space

  ds²=dx²-dt²; ω=dx∧dt

  ∗dx=dt; ∗dt=dx

  ∗1=dx∧dt;∗dx∧dt=–1

• 2-d Euclidean space

  ds²=dx²+dy²; ω=dx∧dy

  ∗dx=dy; ∗dy=–dx

  ∗1=dx∧dy;∗dx∧dy=1

You may also wish to use the following identity on the homework, which is derived (and explained) in my notes.

∗∗=(–1)^p(n-p)+s

1/26/07

When trying to reconstruct the inner product from the norm on the current homework assignment, you may want to first consider the analogous question for the ordinary dot product. That is, if you know u·u for all u, can you determine u·v?

HINT: Consider the relationship between the Law of Cosines and the dot product.

1/24/07

A sample writeup of the solution to (part of) the first homework assignment can be found here.

One student in the class found online notes here which proved helpful.

1/22/07

As pointed out on the book list, MTW (the last book on the list) discusses differential forms in Chapter 4, with pictures. Weinreich's book also discusses the geometry of differential forms.

1/16/07

Please note the logic in the current homework assignment: You are given u, and must find v and w.

1/15/07

I should be in my office Tuesday, 1/16, from 11–12 and 1–2 if you have questions about the homework due Wednesday, 1/17.

1/12/07

WARNING: At least one student has had font problems when printing the first homework assignment.

The first equation should read β=α₁∧…∧α₂.

The next line should contain α∧β+γ∧δ.

The second equation in problem 1 should be u=v×w.

The equation in problem 2 should be γ = A σ∧τ + B τ∧ρ + C ρ∧σ.

Please let me know if your copy looks different.

The symbol τ does not occur until problem 2.

1/10/07

Looks like I was too quick: Of the four properties mentioned in class (squares are zero, antisymmetry, distributive on the right, distributive on the left), the first two are indeed equivalent — so long as the last two hold.

1/2/07

A preliminary homework assignment has been posted on the homework page.

Make sure to read the ground rules as well.

12/8/06

The course time has indeed been changed to MWF 8 AM.

This has caused a glitch in online registration, which will be fixed ASAP.
There should be space for all those wanting to register; be patient.

11/18/06

The course time is almost certain to change.

The most likely time is MWF 8 AM.

If you are considering taking this class, please contact me.
(It would be useful to know the times when you could not take the course.)