ANNOUNCEMENTS
MTH 434/534 — Winter 2017

3/8/17

Here are the answers to the first four exam questions:

• 1. (a) ${*}1=\omega$, ${*}dt=-(r\,d\theta)\wedge (r\,\sin\theta\,d\phi)$, ${*}(r\,d\theta)=r\,\sin\theta\,d\phi\wedge dt$, ${*}(r\,\sin\theta\,d\phi)=dt\wedge(r\,d\theta)$, $**=-1$
  (b) $\displaystyle\frac{\partial^2 f}{\partial t^2} -\frac{1}{r^2}\frac{\partial}{\partial\theta} \left( \sin\theta\,\frac{\partial f}{\partial\theta} \right) - \frac{1}{r^2\sin^2\theta} \frac{\partial^2 f}{\partial\theta^2}$
• 2. (a) $\beta_1$ is closed; $\beta_2$ is not closed. (b) $\beta_1=d(y^2dz+z^2dx+x^2dy+df)$ is exact; $\beta_2$ is not exact.
• 3. (a) $2\pi^2$ (b) $\pi^2 a^4/2$
• 4. (a) $-1/a^2$ (b) $2\pi a^2$ (c) "$\alpha$" = 0 or $\phi$ = const

Worked solutions can be seen in my office next term.

3/6/17

Here are the title and abstract for Friday's guest lecture by Professor Hatton (see below):

Curvature and Locomotion
For many robot and animal systems, the relationship between changes in shape and changes in position can be modeled with a connection. The curvature of this connection describes the efficacy with which the robot or animal can turn cyclic shape changes (gaits) into large-scale motion through the world. In this lecture, I will demonstrate how the mathematical and physical concepts relate to each other, and how they inform the control of real robots.

And of course don't forget that the first guest lecture is today.

3/2/17

I will be in my office tomorrow from roughly 10 AM–2:30 PM, apart from a short lunch break.

If those times don't work for you, or if you want to request a particular time, let me know. But there aren't a lot of alternatives, so you may want to try sending me your questions via email.

3/1/17

There will be a second guest lecture, this one on Friday, 3/10, at the regularly scheduled class time (3 PM) in our regular classroom. Professor Ross Hatton, a mechanical engineer, will speak about his use of differential geometry to build robots.

Some further information about Professor Hatton's work is available here and here.

(There will be no class meeting on 3/8.)

2/28/17

With apologies for the short notice, I will be in my office both today and tomorrow from 1–2:30 PM.

Appointments late tomorrow morning may also be possible; inquire via email if you can't make the other times.

2/27/17

I will hold additional office hours this week; watch this page for updates.

You can of course also request an appointment via email.

2/25/17

There will be a guest lecture on Monday, 3/6, at the regularly scheduled class time (3 PM) in our regular classroom. Professor Eugene Zhang, a computer scientist, will speak about his use of the geometry of surfaces to model 3-dimensional images. A flyer is available here.

Some further information about Professor Zhang's work is available here and here.

2/23/17

I should be in my office this morning from roughly 11–11:45 AM.

I will likely also be there this afternoon shortly before my official office hour from 1:30–2:30 PM.

2/22/17

The second exam will be in class Friday, 3/3/17, in Gleeson 200.

There will be a review session after class on Wednesday, 3/1/17, in our regular classroom (Gleeson 100).

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

• The exam is closed book.
• The exam will cover everything discussed in class through Wednesday 3/1/17.
• 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 ($d$).

2/16/17

Some of you may be interested in the Astronomy Open House being held by the physics department tomorrow, Friday 2/17/17, from 7–9 PM, outside Weniger Hall at the corner of Monroe and 23rd.

2/15/17

As announced in class today, I will accept HW #4 without penalty until the start of class on Friday, 2/17.

You are encouraged to turn it in before then, either by slipping it under my office door or by sending me an electronic copy (PDF preferred, but a scan or photo is OK).

2/13/17

I will hold extra office hours on Wednesday before class (from 1:30 PM).

I may should also be available Wednesday morning, as well as late Tuesday afternoon; if you want to meet at one of those times, I encourage you to email me beforehand.

2/12/17

There is a significant typo in the text:

In both (17.4) and (17.5) on page 204, there should be no factors of r on the RHS

There is also a minor typo in the caption of Figure 17.1 on page 203:

θ-hat should be φ-hat.

Both errors have been corrected in the wiki version of the text

2/8/17

First of all, my apologies again for misreading the (incorrect) clock today... I have collated the data you provided today regarding class time. There were numerous objections to both MWF @ 8 AM and WF @ 4. I'm willing to continue the discussion, but at this point see no viable alternative to the currently scheduled time, MWF @ 3.

There appear to be only two students who definitely want to take MTH 437/537 and are unable to attend at 3 PM. I encourage those students to contact me privately to plead their case.

2/6/17

Should you have any questions about the midterm problems, you are strongly encouraged to try again on your own, then bring your second attempt to my office, along with your midterm, for discussion.

2/4/17

Here are the answers to the midterm questions:

• 1. (a) $dx\wedge dy\wedge dz$ (b) $0$ (c) $2\,dx\wedge dy\wedge dz$ (d) $4\,dx+3\,dy+dz$
• 2. (a) $0$ (b) $2\,\alpha\wedge d\alpha$ (c) $d\alpha\wedge d\alpha$
• 3. (a) $2\,dx\wedge dy\wedge dz\wedge dt$ (b) $x\,dy+z\,dt$ (other answers are possible) (c) no such forms exist
• 4. (a) $q\,dq\wedge q\,d\phi$ (b) $\frac{1}{q^2}\left(\frac{\partial^2f}{\partial q^2} + \frac{\partial^2f}{\partial\phi^2}\right)$
• 5. (a) ${*}(dh\wedge{*}dh)+h\,{*}d{*}dh$ (b) $\grad\cdot(h\grad h) = \grad h\cdot\grad h+h\grad\cdot\grad h$

Worked solutions can be seen in my office, and will be discussed in class on Monday

2/1/17

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

1/30/17

Extra office hours this week: W 9:30–11:30 AM & 1:30–2:30 PM.

I also expect to be available Friday morning; details follow.

1/27/17

It was pointed out to me after class today that there was a minor typo on HW #2.

Question 1(b) asks for an "indecomposable 2-form $\gamma\in\bigwedge^p(\RR^n)$," but either $p$ should have been $2$ or vice versa. For 2-forms, it is indeed true that $\gamma\wedge\gamma=0$ if and only if $\gamma$ is decomposable; the counterexample below is a 3-form.

1/26/17

The first exam is confirmed for Friday, 2/3/17, in Gleeson 200.

The review session will be after class on Wednesday, 2/1/17, in our regular classroom (Gleeson 100).

1/23/17

Can indecomposable forms square to $0$? Not in $\RR^4$! However, if $\gamma$ is an indecomposable element of $\bigwedge^2(\RR^4)$, then surely $\gamma\wedge dx^5$ is still indecomposable as an element of $\bigwedge^3(\RR^5)$. But all 3-forms in 5 dimensions square to 0...

1/22/17

The first exam is tentatively scheduled for Friday, 2/3/17, in class.

• The exam is closed book; there will be a formula sheet.
• The exam will cover everything discussed in class through Wednesday 2/1/17.
• Important topics are exterior product ($\wedge$), Hodge dual ($*$), and exterior differentiation ($d$).

There will be a review session. Good times are Wednesday 2/1 after class or in the evening; I will also consider times on Thursday. Please be prepared to discuss these choices during class this week.

1/21/17

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.

1/20/17

Here's the general solution to the summation example:

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", and that you can relate the general formula above to the explicit example given below.

1/19/17

Here is the promised list of basic linear algebra topics. class.

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

You should review these topics if you are rusty!

I also encourage you to work through an explicit example of "Einstein summation", such as the following:

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 $\det(A)$.

We can go over this problem after class tomorrow (Friday).

1/18/17

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. (I also have copies in my office.)

1/17/17

I have posted one possible solution to the first homework assignment.

If you didn't get the score you were hoping for on this assignment, I encourage you to stop by my office to touch base with me about how things are going.

1/16/17

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.

1/14/17

My official office hours have been posted on the website.

I am usually in my office MWF mornings from 9:30–11:30 AM. These are good times to suggest for an appointment, but do also feel free to take your chances and just drop in. Ditto for W afternoon from 1:30–2:30 PM. Other times may also be possible; ask.

In addition, I will stay in Gleeson 100 after class on Fridays to answer questions, and occasionally to address supplemental topics; we have the room until 5 PM.

1/13/17

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.

1/11/17

I have requested registration overrides for all students present in class today who indicated they wish to register.

You should be contacted tomorrow morning via email; if not, let me know.

I will be in my office most of the day on Friday.

Feel free to stop by with questions or just to chat.

If you're feeling rusty with vector calculus, you might want to take a look at our online text.

1/9/17

Registration for MTH 434 will be sorted out Wednesday after class.

There should be enough seats in the new classroom to accommodate everyone who has contacted me. The catch is that graduating senior math majors will be given priority, which could reduce the number of available seats.

1/8/17

Despite Monday's class being canceled, this week's homework assignments are still due as listed.

12/17/16

The class has been moved to a room that seats just over 45. There are currently 30 students registered and 10 on the waiting list, so there should still be room for a few more.
If you want to register for this class, but have not yet contacted me, please do so as soon as possible.

It would be helpful to know your major(s), and what year you are.
Make sure to come to class the first day!

Be warned that registration will not be possible until Week 1; if this poses a problem for you, please let me know.

11/21/16

Correction:

It is likely that at least 5–10 additional spaces will be made available.

11/15/16

MTH 434 is officially full.

It is likely that 5–10 additional spaces will be made available.

To maximize your chances of getting in, make sure to get on the waitlist and come to class.

Waitlisting is not available until Phase II.

11/6/16

Please note the unusual scheduling:

We will meet MW from 3–3:50 PM and F from 3–4:20 PM.

Please let me know immediately if this scheduling causes a conflict or other difficulty for you.

• It is possible in principle to reschedule the extra class meeting, but in practice this may be difficult.
• It may be possible to meet longer on some Fridays than others...