ANNOUNCEMENTS
MTH 434/534 — Winter 2013

3/18/13

The final exams have been graded, and course grades will be uploaded this afternoon (but may not be visible until Monday).

• Each problem was worth 20 points.
• The last part of problem 4 was counted as extra credit.
• The average on the final was 62 out of 80, which defined the A/B cutoff.

Thus, getting the first three problems correct was likely enough to get an A on the final.

You can get your exam from me next term; if you are taking MTH 437, I will return the exams in class.

Worked solutions can be seen in my office.

If there is sufficient interest, I will arrange an extra meeting to go over the exam in detail.

3/18/13

There is a typo in the formula sheet: The second equation should say d(f dα) = df ∧ dα .

The typo has been fixed and vector relationships in 3-dimensional Euclidean space added.
Download the revised formula sheet here.

3/15/13

A formula sheet will be available on the final. You can find a copy here.

3/11/13

Several sections have been added to the online text, notably one on the torus.

Follow the links in Week 10 of the schedule to see the material covered in class.
(Missing links will be added later this week.)

3/11/13

The final will be Tuesday 3/19/13 from 12–1:50 PM in Weniger 304 (our regular classroom).

• The exam is closed book.
• The exam will cover everything discussed in class through Wednesday 3/13/13.
• The final will be slightly less than twice as long as the midterm, and will cover material from the entire course.
• Important new topics are the structure equations and curvature.
• Important old topics are exterior product (∧), Hodge dual (∗), and exterior differentiation (d).
• Other topics discussed in class this past week should be regarded as supplementary material which is good to know but not required.
• There will be a review immediately after Friday's class; come prepared to ask questions.

3/9/13

Two mathematicians are talking on the telephone. Both are in the continental 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.

3/8/13

A summary of the geometric properties of a torus can be found at this website.
More detailed information (by a former student in this course!) is available here.

Follow the Curvature link. However, this page uses MathML, which did not render properly for me; you may prefer to look at the (more complete) PDF file. Note the amount of computation needed to compute the Gaussian curvature using tensor methods rather than differential forms!

3/7/13

There will be an exam review session next Friday, 3/15, at 4 PM, in Weniger 285.

3/6/13

A brief summary of some of the ideas in today's extra lecture can be found here.

3/5/13

Tomorrow's extra lecture on Clifford Algebras will be in Weniger 285 at 4 PM.

3/3/13

There will be an extra, optional lecture, tentatively scheduled for Wednesday, 3/6/13, at 4 PM, in a room to be announced. The topic will be Clifford Algebras, the mathematical description of the gamma matrices used in quantum field theory. A special case of these matrices are the Pauli spin matrices, which in turn are closely related to quaternions.

3/1/13

You do not need to use the machinery discussed in class today ("structure equations") when working on the homework due Monday. Yes, solutions that do use the structure equations are possible, but other methods are probably easier. Use what you know!

2/20/13

You are strongly encouraged to redo (some of) the midterm problems, and submit them to me as an extra homework assignment.

• Do not make corrections on your midterm.
• Do give me your midterm for comparison.
• Please be selective; I will likely only comment on significant changes from the work you originally did.

Turn in your revision in class on 2/25/13.

2/17/13

A formula sheet will be available on the midterm. You can find a copy here.

2/16/13

During today's review, we discussed the formula ∗σ^J = g(σ^J,σ^J) σ^K where σ^J∧σ^K = ω. This formula assumes that σ^J and σ^K are part of the orthonormal bases generated by a given orthonormal basis {σ^j} of 1-forms.

You can use this formula to determine ∗σ^J in an orthonormal basis; that's what it's for.

You cannot use this formula in general to determine ∗α.

You can however use this formula to get the scaling right if you have an orthogonal basis.

For example, in polar coordinates, ∗dφ = g(dφ,dφ)(−r dr) = −dr/r (correct scaling), but you have to know that the answer is proportional to dr, and not, say, dr+r dφ, even though dφ∧(dr+r dφ) is also proportional to ω. (You must also be able to compute g(dφ,dφ)…) Now try computing ∗(dx+dy) in Euclidean 2-space; you should get the correct answer. But the formula fails for null forms; try computing ∗(dt+dx) in Minkowski 2-space. (What should the answer be?)

2/14/13

There will be no class next Friday, 2/22/13.

There is class this Friday, 2/15/13.

2/13/13

My (old) 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 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 can 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".
• You can also run Mathematica using the OSU Virtual Computing Lab (Umbrella).

2/11/13

There will be a review session for the midterm on Saturday, 2/16, from 2–4 PM, in Weniger 304.

2/1/13

The midterm is scheduled for Monday, 2/18, in class.

• The exam is closed book.
• The exam will cover everything discussed in class through Friday 2/15/13.
• Important topics are exterior product (∧), Hodge dual (∗), and exterior differentiation (d).
• There will be a review session the weekend before the exam at a time to be arranged.

1/28/13

I have posted my old notes on the Hodge dual operator here.

In my opinion, the treatment in Chapter 3 of the text is better, so I encourage you to use these notes only if you find them helpful in clarifying the material in the text.

1/27/13

I have posted a sample solution to the second assignment here.

1/20/13

I have made several changes to the sections of the online text we have already covered, with significant changes to the section on pictures of differential forms, corresponding to Friday's lecture.

1/17/13

I have posted a sample solution to the first assignment here.

1/14/13

There was a typo on the acknowledgment page of the online text in the URL for our online vector calculus book, which has been fixed. (The PDF version of the book has not been updated.) The correct URL is http://www.math.oregonstate.edu/BridgeBook.

1/4/13

A rough schedule has been posted, and will be kept reasonably up-to-date.

Main readings are listed on the schedule, but see also the recommended readings on the homework page.

1/3/13

Please note the following schedule announcements:

There will be no class on Wednesday 1/9/13 or Friday 1/11/13.

These classes will be made up later, most likely as optional lectures on supplemental material.

Beginning Monday 1/14/13 we will meet in Weniger 304.

12/3/12

The mathematics department sent out the following message today:

Here is some further information regarding the two sections of MTH 434/534 that will run during Winter Term 2013.

      First of all, both sections will definitely run, and are scheduled at the same time. However, we hope to even out the numbers between the two sections as much as we can.

      Section 001, taught by Tevian Dray, will emphasize the use of differential forms to understand surfaces. While many examples will be drawn from 3-dimensional Euclidean space, examples in higher dimensions, and with non-Euclidean signature, will also be considered. Highlights of the course include discussions of curvature and geodesics, including a treatment of the Gauss-Bonnet Theorem; neither topology nor tensor language will be emphasized.

      Section 002, taught by Catherine Searle, will focus on the modern connection between geometry and topology as exemplified by Perelman's recent solution to the over 100-year-old Poincare conjecture. Although the topics in the course are necessary ingredients for an understanding of mathematical physics, the flavor of this course will be more geometric and less algebraic in nature. Differential forms as well as covariant tensors will be covered.

      If you are not planning on taking MTH 437 in the spring, please transfer to section 002. If you have not decided on MTH 437 in the spring, please come and talk to one of the instructors.

      Further information about the two sections can be found at:

• Section 001 (Tevian Dray)
• Section 002 (Catherine Searle)

Please don't hesitate to contact either instructor for help in deciding which section would be best for you.

Tom Dick — Chair
Christine Escher — Assistant Chair

11/26/12

The new section now appears in the online schedule.

11/20/12

There will be a second section of MTH 434/534, at the same time, taught by Dr. Catherine Searle. We are therefore seeking volunteers to transfer from my section (001) to hers (presumably 002).

Be warned that these classes may not be identical. My course is somewhat unconventional, and will emphasize differential forms; it is intended as a lead-in to my spring term course on general relativity. A more traditional course would emphasize curves and surfaces in three dimensions, which can be discussed with or without differential forms. There is certainly overlap between these two viewpoints, but they are not interchangeable.

IF YOU ARE PLANNING TO TAKE MTH 437/537 IN THE SPRING, YOU SHOULD PROBABLY REMAIN ENROLLED IN MY SECTION. IF NOT, YOU MAY PREFER TO SWITCH, AND WE ENCOURAGE YOU TO CONSIDER DOING SO.

Fine print: There is still a slight chance that the new section will be canceled. Anyone who switches is guaranteed reenrollment in my section should that prove necessary.

11/16/12

It is almost certain that additional space will open up in this class. The most likely scenarios are that the class size would be slightly increased, or that a second section would be opened (at the same time, but with a different instructor.)

11/14/12

There is currently no room left in this class. My understanding is that a waiting list is being established, but will not be available until November 29. I will post further updates here as they are available.