MTH 434/534 Announcements

ANNOUNCEMENTS
MTH 434/534 — Winter 2009

3/16/09

Here are some comments and hints for the final:

Problem 1: The Gaussian curvature can be calculated both extrinsically and intrinsically.
Problem 2: Work through the example! What sort of object does each equation refer to?
Problem 3: The integration hint refers to something specific from class (which is also in the textbooks).
Problem 4: There are (at least) two ways to factor this line element; one works better than the other.

3/13/09
Strange but true: The 13th of the month is more likely to be a Friday than any other day of the week!: Sound familiar? Look below.

3/12/09
The final will be handed out during class tomorrow, and will be due in my office at 2 PM on Tuesday, 3/17/09.: I expect to be in my office on Saturday from 1–2:30 PM, on Monday from 9–11 AM and from 1:30–3 PM, and on Tuesday from 9–10 AM, from 11–11:45 AM, and from 1–2 PM. These times are approximate and subject to change; watch the website for updates and/or double-check with me via email or by phone.
You are strongly encouraged to contact me for help or advice, in person and/or via email.

3/10/09
The Murder Mystery Method for finding potential functions is described here.

3/10/09
The following announcement is on behalf of the Math Club:: OSU's Math Club and the student chapter of Pi Mu Epsilon will host an undergraduate conference at OSU late Spring term. Undergraduates are strongly encouraged to begin thinking now of a topic about which they could present a brief (15 min) talk to peers. Course projects, past WIC papers, summer research, COMAP solutions or Putnam problems are all excellent examples of topics for an undergraduate level talk.
For further information, contact Nathan Gibson.

3/9/09
Two mathematicians are talking on the telephone. Both are in the continential United States. One is in a West Coast state, the other is in an East Coast state. They suddenly realize that the correct local time in both locations is the same! How is this possible?: Give up? Some hints can be found here.

2/28/09
There will be an Integration Bee on Tuesday, 3/3/09, at 6 PM in Weniger 151.: Further details will hopefully be available soon, but both contestants and spectators are welcome.

2/22/09
For those of you considering taking my relativity course next term (MTH 437/537), please let me know whether the scheduled time (1 PM) is good or bad. There is some possibility it can be changed.

2/18/09
This week's HW assignment will be accepted until Friday 2/20.: If you turned it in today, you are welcome to submit an addendum on Friday.

2/13/09
Strange but true: The 13th of the month is more likely to be a Friday than any other day of the week!: Give up? Further information is available here.

2/9/09

My Mathematica package wedge.m can compute ∧, d, and ∗, and should work on any 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".

2/8/09
My office hour for Wednesday 2/11/09 is canceled.: I should be available Tuesday after 2 PM.
To be sure to catch me, arrange an appointment by email.

2/7/09

There were several common errors and misconceptions on the current homework assignment, which are discussed below.

Make sure you know the difference between using an orthonormal basis and using a coordinate basis.
The gradient is a basis-independent object. Which parts of it you identify as the basis, and which parts are the components, does of course depend on the basis, but df itself does not change.
Curl and divergence are defined on vector fields, not scalars, so you must start with a 1-form to compute them, not a 0-form.
The main advantage of using an orthonormal basis {σ¹,σ²,σ³} is that ∗σ¹=σ²∧σ³. It's certainly useful to know how to compute the Hodge dual from the formula, but you had better obtain this answer.
The main advantage of using a coordinate basis {dxⁱ} is that the general product rule simplifies to d(f dxⁱ)=df ∧ dxⁱ. This product rule does not hold in an arbitrary basis.

2/2/09

The midterm will be in class next Wednesday, 2/11.

The exam is closed book.
The exam will cover everything discussed in class through Friday, 2/6.
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.

1/27/09
The line element is a good way to determine an orthonormal basis — by regarding it as an expression of an infinitesimal Pythagorean Theorem.: You can change from one coordinate system to another using either elementary calculus or geometric reasoning about measuring distance.
Be careful when working in a non-orthonormal (but typically still orthogonal) basis! The coefficients in the line element are not the (squared) magnitudes of the basis — this would yield incorrect answers when applied to the corresponding orthonormal basis vectors, which must have squared magnitude ±1 in any basis. Rather, the coefficients in the line element are the reciprocals of the squared magnitude of the corresponding basis elements. (For a non-orthogonal basis, a matrix inverse is needed.)

1/16/09

The pictures shown in class today come from MTW (the last book on the book list).

Weinreich also discusses the geometry of differential forms, and in particular the terminology "stacks".

Here are some comments about the homework assignment due next week:

When trying to reconstruct the inner product from the norm, 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?
An inner product must have certain properties, which you can read about in this week's suggested reading. Among these are linearity (the product must distribute over addition), and symmetry (it shouldn't matter which vector is "first").
Note that u and v in the second problem are not the same as in the first problem; you can not assume the third component is zero.

1/15/09
Given the, shall we say, lack of clarity of yesterday's discussion of this week's homework assignment, I have posted a detailed solution here.

11/9/08
There is some possibility of changing the course time.: 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.)