ANNOUNCEMENTS
MTH 674 - Spring 2005
- 6/3/05
Read the Notes on page 287 for an indication of what you can do with
the material you have learned in this course!
- 5/27/05
-
I will giving a talk aimed at physics juniors next Friday, 6/3/05, at 1 PM in
Winger 304F, which you may find interesting. I will discuss the role of the
metric tensor, although the level will be more elementary than this class.
More info is available
here.
- 5/25/05
-
Oops... I misstated one of the results about de Rham cohomology, which says
(only) that Hn(M) is nonzero for compact orientable
manifolds (without boundary). The proof given in class (correctly) constructs
a homomorphism from this space onto R; I (incorrectly) claimed this map was an
isomorphism.
- 5/24/05
-
Oops... The homework due this week was posted incorrectly -- my fault. I had
intended to assign problems from §5.8 on page 219, not from §5.5. I
will of course grade the assignment as originally posted, but encourage you to
look at the other set of problems, especially 5.
- 5/23/05
-
The final will be handed out on Friday, 6/3, and may be completed during any
consecutive 48-hour period you wish, with a latest possible due date of Friday
morning, 6/10, at 11 AM.
- 5/22/05
-
There will be a take-home final. Since I know you have other constraints
during finals week, I will allow flexible scheduling of the exam; we'll
discuss this further in class.
- 5/16/05
-
The geometry of covectors (and much more) is described in the book
Geometrical Vectors
by Gabriel Weinreich.
-
Read
my review
of the book, which appeared in
Am. J. Phys. 67, 553-554 (1999).
(From off-campus, you will need to use the
OSU proxy server to access the review.)
-
The geometry of differential forms is also discussed in the book
Gravitation, by Charles W. Misner, Kip S. Thorne, and John
Archibald Wheeler, the classic introduction to relativity.
-
Chapter 4 has a nice introduction to differential forms, including some great
pictures.
- 5/13/05
-
A copy of the "computational" problems I handed out today can be found
here. While these problems have no formal due
date, you should attempt them as soon as reasonably possible. Although they
will not be graded, please do turn them in so I can verify your mastery of
these skills.
- 5/11/05
-
Strange but true: The 13th of the month is more likely to be a Friday than
any other day of the week!
- 4/27/05
-
The take-home exam will be handed out Friday 4/29, due Monday 5/2 in class.
- 4/23/05
-
There will be a take-home midterm handed out in class either Friday 4/29 or
Monday 5/2, due back at the next class period. I have so far had one vote for
each alternative, so please let me know if you have a preference.
-
I will be giving a talk entitled The Geometry of Special Relativity at
Western Oregon University on Monday afternoon, 5/2, at 4 PM, which students in
this class might be interested in — and which could affect the vote.
- 4/22/05
-
Mea culpa: The final part of the cube demonstration in class today
failed because I confused the left and right actions... The eigenvector
computed for a rotation of the form P(π/2) from the homework corresponds to
clockwise rotations, rather than counterclockwise, which
explains why I had the wrong axis. Counterclockwise rotations correspond to
P(-π/2).
- 4/21/05
-
As I mentioned at the end of yesterday's class, you can check the
correspondence between SU(2) and SO(3) by comparing their actions. To do
that, use the Lie algebra elements rk and
sk from the homework, and the corresponding rotations
Rk(θ) and Sk(θ) obtained by
exponentiating them. Then check that Sk(θ) X
Sk†(θ) induces the same rotation as
Rk(2θ) x, where x is in R3 and
X is the corresponding 2×2 Hermitian matrix as discussed in
class.
-
Start with sz, which is easy to exponentiate, then try the others.
If you want a computational challenge, try exponentiating an arbitrary
antihermitian matrix (to get an element of SU(2) — or simply start with
a unitary matrix), then computing the action on X, showing that this is
in SO(3).
- 4/20/05
-
An analog to the rotation I demonstrated in class today with a coffee cup is
the "Dirac belt" rotation, which you can find at
this
site.
- 4/18/05
-
The due date for this week's homework has been moved up. Please try to
complete it by Friday if possible, so that we can discuss it in class.
- 4/15/05
-
Yes, there is a counterintuitive sign which shows up on the current homework
assignment. This is not a computational error. Do spend some time thinking
about where it might come from, but also feel free to simply label it
surprising. We'll discuss this further in class on Monday.
- 4/6/05
-
Take a good look at the second paragraph in §3.3, at the top of page 65.
- 4/5/05
-
Rather than my announcing official office hours, feel free to drop in any time
I am in my office; if I am busy, I'll say so. Good times to catch me are MWF
before class (after 10 on M; probably earlier WF), and MF from roughly 2-3:30.
Other times are possible; email is the best way to make an appointment.
- 4/4/05
-
The optional text by Bishop & Goldberg should be available in the
bookstore this afternoon.
- 3/27/05
-
Please note that the class has been moved to Kidder 356.
-
Brief descriptions of several geometry texts can be found
here.
- Note that these are from a different course taught last year.