ANNOUNCEMENTS
MTH 435/535 — Spring 2020

6/6/20
I have posted a discussion of the extra-credit problem here.
Course grades will be posted as soon as that functionality is available, presumably early next week.
6/5/20
Slides from today's class (spacetime diagrams and paradoxes) can be found here.
A screenshot from the class discussion of the twin paradox can be found here.
6/3/20
Slides from today's class (the geometry of special relativity) can be found here.
We will analyze some standard paradoxes on Friday.
6/1/20
Today's class was presented somewhat differently from previous classes, using prepared slides and annotation, without video. Feedback on the pros and cons of this format would be welcome, via email or otherwise.
Slides from today's class (hyperbola trig) can be found here.
We will apply this geometry to special relativity on Wednesday.
I will accept the extra-credit problem through the end of this week.
If you're planning on turning it in, it wouldn't to let me know now. Ditto for requests for additional time.
5/29/20
Notes from today's class (regarding HW 8) can be found here.
The brief overview of special relativity planned for today has been postponed until Monday.
5/27/20
Notes from today's class can be found here.
On Friday, I will provide a brief overview of special relativity, again based on my own text, which should also be available to read online through the OSU library.
Regarding the extra-credit problem, you may also attempt §7.7: 6a, which isn't as difficult as I had originally thought.
For all three parts, what constraints does the Euler index of the sphere impose on the possible index of the vector field at the singular points?
5/23/20
The remainder of the course will be based on my own text, which can be read online as an ebook through the OSU library.
From off campus, you first may need to login to the library; try this link if the one above doesn't work.
There will be no pre-recorded mini-lectures for this part of the course.
There is also a freely accessible wiki version available, which is however not quite the same as the published version.
Chapters 17 and 18 in DFGGR correspond to Chapters 6 and 7 in the wiki, and §6.1 to §1.5 in the relativity wiki.
5/22/20
Notes from today's class can be found here.
As pointed out in class, there was an error in my presentation of the Poincaré–Hopf Theorem, which has been corrected.
5/21/20
As discussed in class, the notion of "$\alpha''$" for a curve in a surface $M\subset\RR^3$ depends on whether you use the connection in $\RR^3$ $(\alpha''=\frac{d\vv}{dt})$ or in the surface $(\alpha''=\nabla_\vv\vv)$. These derivatives differ by a component in the $\nn$ direction.
This difference does affect the argument in §7.4: 8, but not the conclusion. You may use the result of §7.4: 8 when solving §7.6: 6, thus avoiding the issue.
5/19/20
There is a minor typo in the statement of the first homework problem:
The lower limit of the second integral in §7.6: 6b should be $v_1$, not $u_1$.
5/18/20
Notes from today's class can be found here.
With apologies, the Gradescope listing for HW 7 (due today) was incorrect. (The homework page listing was correct.)
If you did the wrong problem, you may submit the correct one late without penalty — to me, via email.
HW 8 (due next week) will be the last homework assignment.
If you do the extra-credit problem, please email your solution to me separately.
5/15/20
Notes from today's class can be found here.
5/14/20
The slide that accompanies the mini-lecture for tomorrow's class can be found here.
Missing from that presentation is a reminder that a curve $\alpha:I\to M$ in a patch $\mathbf{x}:D\to M$ has coordinate functions $\alpha_i$ defined by $\alpha(t)=\mathbf{x}(\alpha_i(t))$. Equivalently, the curve is parametrized within the given patch by $u_i=\alpha_i(t)$. The tangent vector to the curve is then $\vv=\frac{d\alpha_i}{dt}\ev_i$, which is often written $\vv=\dot{u}_i\ev_i$.
5/13/20
Notes from today's class can be found here.
5/12/20
The computation of the change in the connection from yesterday's class (and mini-lecture) can be found here.
This handout includes notes for today's mini-lecture on covariant derivatives.
5/7/20
Notes from yesterday's class can be found here.
There is no new mini-lecture for tomorrow's class, as we didn't yet use the material from the one for Wednesday.
5/5/20
A reminder regarding homework submissions:
Please include your name in the filename!
Please select pages to associate with each problem!
The Gaussian curvature of both the helicoid $\{z=b\theta\}$ and the saddle $\{z=xy/b\}$ is $K=-\frac{b^2}{r^2+b^2}$.
Some other argument is needed to argue that the helicoid and saddle are not isometric...
5/4/20
Notes from today's class can be found here.
Last night's mini-lecture has been reposted on Canvas – the original upload was corrupted.
5/1/20
Is Earth smoother than a billiard ball?
Argument for: A billiard ball is $2.25\pm0.005$ in in diameter. That's an error of $\frac{0.005}{2.25}=0.0022$. Meanwhile, Earth's diameter is about 12700 km, and the difference between the top of Mount Everest and the bottom of the Maiana Trench is about 20 km. That ratio is $\frac{20}{12700}=0.0016$, which is less than the tolerance for a billiard ball.
Argument against: The imperfections in a billiard ball are only about 0.5 μm, which, compared to its diameter of 5.7 cm, is a ratio of only $\frac{0.00005}{5.7}=0.0000088$.
Roundness: Earth's polar diameter is shorter than its equatorial diameter. The difference is about 40 km, and $\frac{40}{12700}=0.0032$, which is in fact just barely within the tolerance of $\pm0.22$%, which yields a maximum difference of twice that amount. So Earth is as round as a billiard ball!
4/30/20
To resolve a minor confusion that arose at the end of class yesterday, yes, if $k_1=0\ne k_2$, then the Codazzi equations (Chapter 6, Theorem 2.6) do indeed imply that $\omega_{12}=0$.
However, $\omega_{12}$ is not an isometric invariant. A counterexample is provided by $\RR^2$, for which $\omega_{12}=d\theta$ in polar coordinates but is zero in rectangular coordinates, although these two patches are clearly (locally!) isometric.
4/29/20
Notes from today's class can be found here.
My apologies again for the technical glitches at the start of class.
4/28/20
The slide that accompanies the mini-lecture for tomorrow's class can be found here.
4/24/20
The GeoGebra applets I showed in class can be found here.
There wasn't enough on the shared whiteboard to post...
4/22/20
Slides from today's class can be found here, and the whiteboard can be found here.
Further details of the theorems can be found in §6.3.
The point of the whiteboard activity was that the coordinate basis $\{\Partial{\xx}{u},\Partial{\xx}{v}\}$ has dual basis $\{du,dv\}$. If the coordinates are orthogonal, then a frame $\{\ee_i\}$, and its dual basis $\{\sigma_i\}$, is obtained from the above by renormalization.
In polar coordinates, the coordinate basis is $\{\rhat,\frac1r\that\}$, with dual basis $\{dr,d\theta\}$, whereas the frame is $\{\rhat,\that\}$, with dual basis $\{dr,r\,d\theta\}$. Note the presence of normalization factors in both types of bases, but in different places.
To clarify the discussion at the end of class today, a compact surface in Euclidean $\RR^3$ is one that is closed and bounded.
4/20/20
Slides from today's class can be found here.
Please take a look at the computation on the second page, leading to the boxed equations at the bottom. I will go through the details of this computation on Wednesday, either in class or possibly in the mini-lecture, and the same style of argumentation will be used for other results.
4/19/20
A differential form $\phi$ (of any rank) is closed if $d\phi=0$, and exact if $\exists\xi:\phi=d\xi$.
$\xi$ is a potential for $\phi$.
An integral condition is given for 1-forms $\phi$ in §4.6:3:
$\phi$ closed ($d\phi=0$) implies $\oint_{\partial R}\phi=\int_R d\phi=0$ for every region $R\subset M$;
$\phi$ exact ($\phi=df$) implies, more generally, that $\oint_C\phi=\oint df=0$ for every closed curve $C\subset M$.
The converse of each statement is also true, although the condition can be difficult to verify in practice.
4/17/20
Slides from today's class can be found here, and the shared whiteboard can be found here.
Some interesting and fairly simple applications of Stokes' Theorem to topology are in §4.7, which I encourage you to read on your own.
You should also look up the definitions of closed and exact differential forms at the end of §4.4.
4/17/20
My MF morning office hours are not working well, so I'm going to change them.
I will hold an extra office hour today from 1:30–2:30 PM.
Going forward, I plan to start up the class Zoom session a half hour before class (2:30 PM MWF), and be available both then and after class for questions.
I am usually available most other times MWF, with that 1:30–2:30 PM slot being particularly good.
If the Office Hour Zoom session isn't already running, send me email to request an appointment, even on very short notice.
4/16/20
The slide that accompanies the mini-lecture for tomorrow's class can be found here.
4/15/20
The slides that accompany today's lecture can be found here.
4/13/20
The slides that accompany today's lecture can be found here.
The first page shown in class is essentially the same as the one used in the mini-lecture.
4/12/20
The slide that accompanies the mini-lecture for tomorrow's class can be found here.
Some earlier handouts have been reprocessed to generate letter-sized PDF output. Depending on your PDF viewer, you may not have noticed the odd size of the previous versions, but, if you did, try again.
4/11/20
If you are intrigued by the "cylindrical coordinates" approach to surfaces of revolution, you may want to look at my recent paper on embedding diagrams, which includes an application to black holes.
This paper will appear later this year in the American Mathematical Monthly.
4/10/20
Notes from today's class can be found here.
4/9/20
Please note the misleading notation (probably a typo) in the text in §5.7:8, which you will need for this week's homework.
A clarification has been posted on the homework page.
4/8/20
This might be a good time to review the handout from last term that computes the Gaussian curvature of the torus using both the shape operator and the structure equations.
You may want to try adapting at least one of these methods to surfaces of revolution, either the general case, or perhaps to one or both of the catenoid and the pseudosphere ("bugle").
4/7/20
The mini-lecture that precedes tomorrow's class will be posted as soon as it is ready on Canvas; my apologies for the delay.
The slides that accompany this lecture can be found here.
4/7/20
This week's posted homework assignment has been updated to reflect the correct problem numbers.
(The original post referred to the 2nd edition, not the 2nd revised edition...)
4/6/20
In addition to my regularly scheduled office hours, please feel free to request an appointment.
4/5/20
I have posted derivations of some properties of special surfaces here.
The slide used in today's mini-lecture on surfaces of revolution is included, as is Friday's derivation of the relative directions of asymptotic and principal directions on minimal surfaces.
4/3/20
I have posted a slide showing the catenoid here.
4/2/20
After some reflection, I have decided not to change the Zoom Meeting IDs for either class or office hours.
You should still be able to access the meetings within Canvas by clicking on the links in either the recent announcement about Zoom, or the previous one about Office Hours.
You can also join directly from the Zoom app, so long as you know the Meeting ID.
In either case, from now on you will need to sign in using ONID credentials.
Please let me know immediately if you are having difficulty connecting.
4/1/20
I have posted my class notes on the helicoid here and on Darboux frames here.
The Darboux formulas given in class were for unit speed curves, and require slight modification otherwise.
3/31/20
The Zoom instructions have been updated.
I believe I now understand why some of you couldn't find the Meeting ID for class in a timely fashion.
If anyone still hasn't found the Meeting ID, please let me know.
Classroom video from yesterday's session should now appear (as an ungraded assignment).
Automated processing through Canvas appears to be backed up, so I processed these videos manually.
Please let me know if you have any difficulties accessing these videos.
A review mini-lecture has been posted on Canvas.
Lecture notes for this review can be found here.
3/30/20
An interactive map of the spread of the coronavirus has been posted here.
3/30/20
Thanks to all who participated in today's class.
The video should be available later this evening.
Here are the important concepts from last term, as generated in today's breakout groups:
One student from last year added line elements and signature to this list.
We may have time to discuss these topics at the end of the term.
3/30/20
My tentative office hours (via Zoom) are MF 11–11:45 AM (starting today), and by appointment.
I am also available most days both before and after class. (You can check via Zoom.)
The Zoom Meeting ID for all office hours has been posted as a Canvas announcement.
3/29/20
Several resources from last term are listed below.
Please review them before class on Wednesday, 4/1/20, and come to class with any questions.
3/28/20
Assignments for the first week have been posted.
Further information about submitting assignments via Gradescope is available here.
Further information about using Zoom is available here.
These instruction pages will be updated as necessary.
A test mini-lecture has been published on Canvas.
3/23/20
You are encouraged, but not required, to use $\LaTeX$ when preparing your assignments.
You may wish to use Maple, Mathematica, Matlab, or Sage to support your computations.
Each language has pros and cons; use the one you like the best.
I am happy to help with $\LaTeX$ or Mathematica coding questions, but not with installation or editor-specific problems.
(I have forgotten most of what I knew about Maple, never learned Matlab, and am a novice at Sage.)
3/22/20
Welcome to remote teaching! Below is some information about how this course will be run.
Overview:
Technical Details:
Let me know if you have difficulties with any of these steps.
These instructions are likely to evolve as we adapt to this new format.
3/19/20
Yes, the class will run!
Further information will be posted both here and on Canvas.