ANNOUNCEMENTS
MTH 434/534 — Winter 2021

3/29/21

There will be an extra, optional meeting to go over the final exam from last term.

This session is tentatively scheduled for this Friday, 4/2/21, at 4 PM.

The session will be recorded, and posted on the Canvas page afterward.

If you want to attend live, please contact me for the Zoom ID.

3/22/21

Final exam scores have been published on Gradescope.

IF your grade were determined only by the raw score on the final exam, it would be:

• $\ge106$: A
• 87–105: B
• 79–86: B–
• 60–78: C
• 52–59: C–
• $\lt52$: F

The above ranges are a good starting point, but an effort was made to compensate for the difficulty of the exam by comparing several alternative scores, notably with and without the last question.

Course grades should be available online, but possibly not until tomorrow.

3/17/21

Several students have expressed concern about the length and difficulty of the final exam. That was certainly not my intention... Rest assured that the grading will, as always, take into account the difficulty of the questions.

There will be an opportunity to go over the exam as a group next term, and you should of course feel free to contact me privately about your own performance – after this week.

3/14/21

Happy Einstein's Birthday! And Happy Pi Day!

I will release the final exam at 8 PM on Monday, 3/15/21.

The exam is still due at midnight on Tuesday, 3/16/21.

I will hold office hours as usual at 4 PM on Monday, 3/15/21.

I should also be available by appointment most of Monday afternoon.

3/10/21

Notes from today's class can be found here.

The final exam will be a take-home exam that will be available all day (midnight to midnight) on Tuesday, 3/16/21.

There will be a review session during class on Friday, 3/12/21.

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

• The exam is closed book.
• The exam will cover everything discussed in class through Monday 3/18/21.
• The exam will cover material from the entire course, with an emphasis on new material.
• Important new topics are the connection ($\omega^i{}_j$) and curvature ($\Omega^i{}_j$).
• Important old topics are the exterior product ($\wedge$), Hodge dual ($*$), and exterior differentiation ($d$).

3/9/21

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.

3/8/21

Notes from today's class can be found here.

Live curvature computations using Sage, as discussed in class today, can be found here:

Euclidean 3-space in spherical coordinates
Sphere
Torus

3/5/21

Notes from today's class can be found here.

As was pointed out in class today, I posted an incorrect due date for this week's homework assignment on the course website. The correct due date is Monday, 3/8/21.

I believe the dates in Gradescope are correct... Please let me know if you were inconvenienced by this error.

3/3/21

Notes from today's class can be found here.

3/1/21

Notes from today's class can be found here.

The examples we drew at the end can be found here.

2/26/21

Notes from today's class can be found here.

2/24/21

Notes from today's class can be found here.

2/23/21

Starting next week, I will no longer be available on Monday mornings.

• My WF office hours at 4 PM are unchanged.
• I should be available before class most days.
  (I will try to start the classroom Zoom session by 2:45 PM.)
• I will also be available for appointments most Tuesday and Thursday afternoons.
  (Appointments at other times may be possible – ask!)
• As always, I can be reached at any time via email.

2/22/21

Partial notes from today's class can be found here.

The moral of HW 5 is not that Stokes' Theorem is subtle, but rather that orthonormal bases matter!

The easiest way to see that $\alpha=-r^2\cos\theta\,d\phi$ is not well-defined on the sphere is to express it as $\alpha=-(r\cot\theta)(r\sin\theta\,d\phi)$, whose component $\alpha_\phi=-r\cot\theta$ is badly behaved at the poles.
Alternatively, construct the vector field corresponding to $\alpha$, namely $\vf\alpha=-(r\cot\theta)\Hat\phi$, since $\alpha=\vf\alpha\cdot d\vf{r}$.

2/20/21

There are a variety of software packages capable of manipulating differential forms, including packages for both Maple and Mathematica. Another option is the open-source software SageMath, which is also available through a cloud server.

I have used most of these packages myself. Feel free to contact me for advice and assistance.

I have set up an experimental interface to Sage here, which should be fairly easy to adapt to other examples. Some tips:

• The operations $d$, ${*}$, and $\wedge$ should all work, entered as d(), star(), and wedge(), respectively.
• There is also an operation Wedge() that takes any number of arguments.
• You can not add more boxes, but you can enter multiple lines of code in each box.
• Only the last result will be printed.
• You may need to use the Show() command to see the result you expect.
• Computations in one box can be used in later boxes.

This is beta software! Please do let me know if it does not work as expected.

2/19/21

Notes from today's class can be found here.

2/17/21

Notes from today's class on integration can be found here and here.

2/15/21

My apologies for not being available during my office hours this morning.

I will have some free time tomorrow (Tuesday 2/16); contact me to make an appointment.

2/14/21

Here are the answers to the midterm questions:

1. (a) $0$ (b) $-dx\wedge dy\wedge dz$ (c) $0$ (d) $5\,dx\wedge dy\wedge dz$
2. (a) $0$ (b) $2\,\alpha\wedge d\alpha\ne0$ (c) $d\alpha\wedge d\alpha\ne0$ (d) $\alpha=x\,dy+dz$ (or similar) (e) $\beta=x\,dy+w\,dz$ (or similar)
3. (a) $\pm u^2\,du\wedge dv$ (b) $\frac{1}{u^2}\left(\frac{\partial^2f}{\partial u^2} + \frac{\partial^2f}{\partial v^2}\right)$
4. (a) ${*}(dx\wedge dy)=dz\wedge dw$ ... (b) $dx\wedge dy\pm dz\wedge dw$ (or similar)

Solutions to all problems will be discussed in class on Monday.

Should you have any questions about the midterm problems, you are strongly encouraged to try again on your own, then come to office hours.

IF your grade were determined only by your midterm, it would be:

• $\ge70$: A
• 62–69: A–
• 53–61: B
• 43–52: C
• 35–42: C–
• $\lt35$: F

2/11/21

Apart from computational errors, mostly minor, the most common errors on HW 4 were:

Failing to determine the orthonormal basis correctly;

Failing to express $\alpha$ in terms of an orthonormal basis: $\alpha = \sum_i\alpha_i h_i du^i$.

A sample solution for spherical coordinates can be found here.

2/11/21

The midterm will be available on Gradescope starting at 2:50 PM tomorrow, Corvallis time.

• You will have 70 minutes from when you start to complete and upload the exam if you start on time.
• Exams received after 4:15 PM will be considered late regardless of when you started.
• All exams must be uploaded by 4:30 PM regardless of when you started.
• All of these times are extended propotionally if you have a DAS extension.
• If you submit late, and/or if you have technical difficulties, send me an explanation via email afterward.
• Use the Office Hours Zoom session to ask me questions during the exam.
• You may use a word processor to typeset your responses, but this is not necessary or expected.
• Please shut down all other software during the exam, such as email, phone, browsers, etc.

To be as explicit as possible, these rules imply:

• If you start from 2:50–3:05 PM, you will get the full 70 minutes and will not be marked late.
• If you start from 3:05–3:20 PM, you may use the full 70 minutes, but will be marked late if you submit after 4:15 PM.
• If you start after 3:20 PM, you must still submit by 4:30 PM; you will be marked late if you submit after 4:15 PM.

Again, these times are extended proportionally if you have a DAS extension.

2/10/21

Notes from today's class can be found here.

An annotated copy of the chat session, with answers to the questions asked, can be found here.

You may want to read §6.1 in the text, the first part of which provides a good review of $\wedge$, $*$, and $d$.

An older document covering similar content can be found here.

A draft of the formula sheet that will be provided on the midterm is available here.

If you would like to suggest any additions to this page, please contact me.

2/9/21

A sample exam (from another class) has been posted on Gradescope.

• The sample exam has no real content; it is merely a test of the technology.
• The sample exam will become available at 5 PM today
• The sample exam is not posted as an assignment on Canvas.
• You will have 10 minutes from the time you start the exam to answer the single question and submit your work.
• The exam is due just after the start of class on Wednesday, 2/10/21.

Most of your work and answers should be done on paper you provide, then scanned and uploaded.

You do not need to copy the questions.
Make sure to clearly label which answer goes with which question.
Some questions involve annotating figures on the exam.

• If you have the capability to annotate PDF files, the best option is probably to download the exam and write on it electronically, then upload this electronic file when finished.
• If you have a printer, a good alternative would be to print the exam, write on it as you would an in-class exam, then scan and upload the exam when finished.
• You may also redraw the figure by hand on your answer sheet, so long as the resulting figure is clear.
• If you are not able to sign the exam as requested, you may certify your agreement with the House Rules by writing "I agree to the House Rules" on your answer sheet, then adding your signature.

Some of these instructions may be different for Friday's midterm.

Come to class on Wednesday prepared to offer suggestions for improvement.

One likely change is that the midterm will not be released until 3 PM (or slightly before) on Friday.

2/8/21

Notes from today's class can be found here.

Equation (15.87) on page 185 of DFGGR is incorrect: The numerator of each partial derivative should be $f$.

The wiki version here is correct.

A complete list of known typos in DFGGR is available here.

2/7/21

Be careful when completing HW 4 to note the subtle change in notation.

The textbook uses the symbol $\grad$ when writing div, grad, and curl, but the homework uses $\nabla$ (without the arrow).

These two operators are not the same! ($\grad$ acts on vector fields; but $\nabla$ acts on forms.)

2/5/21

Notes from today's class can be found here.

As stated on the homework page, suggestions for orthogonal coordinates can be found here.

2/3/21

Notes from today's class can be found here.

My office hour on Friday afternoon, 3/5/21, is canceled.

I should be available most of Friday morning. from 9:30–11 AM on Friday.
(If the Office Hours Zoom session isn't already running, send me an email message requesting an appointment.)

2/1/21

As is now clearly stated on the homework page, all assignments are due at the beginning of class.

Canvas and Gradescope do not synchronize this data. With apologies for the recent confusion, it was nonetheless true that Gradescope had the correct data from the beginning.

For this assigment, submission before midnight yesterday will not be considered late, despite what Gradescope says.

1/31/21

The midterm is currently scheduled for Friday, 2/12/21 (Week 6).

If you have any concerns about this timing, please let me know immediately.

The tentative format for the exam will be a traditional, timed, closed-book exam during the regularly-scheduled class period.

I will expect you to sign a statement confirming that the work you submit is your own.

Please let me know as soon as possible of any questions or concerns you have with these ground rules.

There will be a review session during class on Wednesday, 2/13/19.

A formula sheet will be available on the midterm. A copy will be made available earlier that week.

1/29/21

Notes from today's class can be found here.

1/27/21

Notes from today's class can be found here.

1/25/21

Notes from today's class can be found here.

1/22/21

Notes from today's class can be found here.

I believe the Gradescope settings for this class now show the correct term.

1/20/21

Notes from today's class can be found here.

Classroom video has now been posted in Canvas both in the Media Gallery and as ungraded assignments.

1/19/21

You can find out more about the reasons we will use the "physics" convention for the names of the spherical coordinates in our paper: Spherical Coordinates, Tevian Dray and Corinne A. Manogue, College Math. J. 34, 168–169 (2003)

The short answer is that most nonmathematicians will likely need to switch conventions at some point during their education, so this might as well be done sooner rather than later.

1/18/21

The shortened URL originally advertised for this website no longer works.

Please update your bookmarks to use the official URL, namely http://people.oregonstate.edu/~drayt/MTH434.

Update: As of 1/19, the shortened URLs (without "people") are working again...

1/17/21

Since tomorrow is a holiday, I will not hold my usual Monday office hours.

I am nonetheless available for discussions via email, both Monday and Tuesday, which can be moved to Zoom as needed.

HW 2 will be accepted without penalty through the end of the day on Tuesday, 1/19/21.

1/16/21

Here is an explicit example of "Einstein summation":

Let $\alpha\in\bigwedge^1(\RR^2)$ be a 1-form in two dimensions, and let $A$ be the linear map that swaps $dx^1$ ($=dx$) and $dx^2$ ($=dy$). Determine the matrix $(a^i{}_j)$ of $A$ in this basis. Then determine the action of $A$ on 2-forms, and compare with $\det(A)$.

The general solution is:

The components $(a^i{}_j)$ of $A$ are defined by $A(dx^i)=a^i{}_j\,dx^j$, where $i$ is fixed and there is a sum over $j$. So \begin{align} A(dx^1\wedge dx^2) &= A(dx^1)\wedge A(dx^2) = (a^1{}_i\,dx^i)\wedge (a^2{}_j\,dx^j) \\ &= ... = (a^1{}_1\,a^2{}_2 - a^1{}_2\,a^2{}_1) \,dx^1\wedge dx^2 = (\det A) \,dx^1\wedge dx^2 \end{align} where there is now a double sum over $i$ and $j$ in the third expression.

Make sure that you can follow these "index gymnastics".

1/15/21

Question 1(c) on the current homework assignment is still ambiguous. My intent was for $\gamma$ here to refer to the $p$-form you found in part (b).

You may also answer this question for the decomposable $p$-form $\gamma$ given at the top of the page. But you should then also comment on the indecomposable case.

Notes from today's class can be found here.

1/13/21

The typo in the homework assignment has been fixed.

In the first question, $\beta$ should be replaced by $\gamma$ (or vice versa).

Notes from today's class can be found here.

1/11/21

Notes from today's class can be found here.

The annotated image showing $dx+dy$ can be found here.
The remaining pictures are in the text, and are also available here.

You can find more pictures of differential forms (including higher-rank forms) in Chapter 4 of the book Gravitation by Misner, Thorne, and Wheeler. A very interesting discussion of stacks and similar (but less standard) geometric interpretations of forms can be found in the book Geometrical Vectors by Weinreich.

Both books are available in the OSU library.

Reminder: When submitting assignments to Gradescope, please follow the instructions here.

In particular, please make sure that the filename includes your name, and that you avoid uploading photos of handwritten work if possible – use a scanning app instead (preferably to PDF, not JPG).

1/10/21

I have posted one possible solution to the first homework assignment.
Can you spot the (minor) flaw in my logic?

If you don't get the score you were hoping for on this assignment, I encourage you to come to my office hours and/or to touch base with me via email about how things are going.

1/8/21

Here is a lightly edited list of the basic linear algebra topics that arose in class today.

You should review these topics if you are rusty!

• vector space
• linear (in)dependence
• basis
• change of basis
• span
• linear transformation
• eigenvalues
• eigenvectors
• determinants

Partial class notes from today's class can be found here.

1/7/21

Here are some of the criteria that will be used to assess your written work in this course:

Content:

• Correct logical reasoning;
• Correct computation;
• Correct answer to the question as posed;
• Assumptions clearly stated.

Presentation:

• Self-contained; should ideally be readable in 5 years without other sources;
• Clarity; the logical flow should be readily apparent;
• Typeset or carefully written;
• Good use of both inline and displayed equations;
• Figures are always a bonus.

1/6/21

A list of derivative rules in differential notation is available here.

Notes from today's class can be found here

We haven't yetdiscussed the derivatives shown on the second page.

1/1/21

Welcome to remote teaching! Below is some information about how this course will be run.

Overview:

• Class meetings will be held via Zoom at the scheduled time.
• Expect a combination of lecture, discussion, and both individual and group problem solving.
• Some "reading" assignments may involve watching short videos of me explaining a particular concept.
  Watch these videos before class, via Canvas.
• All class meetings will be recorded and available afterward to watch online via Canvas.
• All assignments will be submitted via Gradescope.

Details:

• General information about getting started with Zoom is available here.
• General information about submitting assignments via Gradescope can be found here.
• Further information can be found on my own information pages for Gradescope and Zoom.
• Each assignment exists in 3 places: on this website, in Gradescope, and on Canvas:
  • The assignment itself can be found (only) on the homework page.
  • Each assignment has a name, such as "Use Gradescope" or "HW 1".
  • When you have completed the assignment, export or scan it to PDF.
    Please do not take photographs of your work except as a last resort.
  • Upload your PDF to Gradescope, following the instructions here.
  • After grading, your corrected assignment will be available on Gradescope.
  • After grading, your score will be available on Canvas.

Let me know if you have difficulties with any of these steps.

These instructions are likely to evolve...

10/9/20

The text can be read online as an ebook through the OSU library.

From off campus, you may need to first login to the library; try this link.

There is also a freely accessible wiki version available, which is however not quite the same as the published version.