ANNOUNCEMENTS
MTH 434/534 — Winter 2020

3/19/20
The final exam is available here.
3/16/20
For those of you taking the final exam:
If possible, use an actual scanner, or an app that generates PDF, not JPG.
(If it is open, you can scan your exam at the OSU library.)
You may want to test this technology before the exam starts.
The exam will be open book; detailed instructions will be included.
3/14/20
As you should already be aware, I have decided to offer two options for the final:
  1. No final.
    Your grade would be based on the weighted sum of your midterm and homework grades, curved as a single score. You should have already received an email message containing your course grade under this option.
  2. Take-home final (open book).
    The exam would be available as a PDF document via this webpage at noon on Thursday, 3/19/20, and would be due at noon on Friday, 3/20/20. Further details will be posted soon.
In the absence of a request to the contrary I will assume you are planning to take the final.
Once the exam has been posted, it is too late to change your mind.
3/13/20
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.
3/12/20
Friday's class will be delivered both remotely and in person.
You should have already received an email message with conection details for remote participation. If not, let me know.
There are three options currently under consideration for the final:
The first two options would likely require submission of either a scanned PDF, probably via email, or hard copy slipped under my office door. A photo (JPG, please) in lieu of scanned PDF would be acceptable if necessary, but the quality is rarely as good. The third option would likely result in the final simply being dropped from consideration in determining grades.
Comments on these options are welcome.
3/11/20
A clean copy of today's lecture notes can be found here, and solutions to last week's homework assignment can be found here.
In both cases, the computations are shown without explanation (which would cost me some presentation points...)
3/10/20
Information on using Zoom can be found here.
Here's the short version:
3/9/20
Wednesday's class will be delivered remotely, via Zoom. Stay tuned!
3/9/20
The "polarization" argument I was trying to make in class goes as follows: \begin{equation} S(\ev_1)\times\ev_2 + \ev_1\times S(\ev_2) = \bigl(S(\ev_1)+\ev_1\bigr) \times \bigl(S(\ev_2)+\ev_2\bigr) - S(\ev_1)\times S(\ev_2) - \ev_1\times\ev_2 \end{equation} after which taking the dot product with $\ev_1\times\ev_2$ yields \begin{equation} \bigl( S(\ev_1\times\ev_2)+(\ev_1\times S(\ev_2) \bigr) \cdot (\ev_1\times\ev_2) = \det(g+h) - \det(g) - \det h \end{equation} with $g$ and $h$ defined as in class in terms of the first and second derivatives of the patch $\xx$.
I did not state the first equality correctly in class... Sorry about that.
3/8/20
There may be a typo in the text:
The matrix of $S$ as given in Lemma 2.2 in §6.2 on page 270 should be: \begin{equation} \begin{pmatrix} \omega_{13}(E_1) & \omega_{23}(E_1) \\ \omega_{13}(E_2) & \omega_{23}(E_2) \end{pmatrix} \end{equation}
3/5/20
The final will be held in KEC 1003 on Friday, 3/20/20, at 9:30 AM.
3/4/20
I have posted a summary of both coordinate- and frame-based computations of the shape operator on a torus here.
I have also included an intrinsic computation of the Gaussian curvature that you won't learn until next week.
2/28/20
There still appears to be some confusion about the fact that, say, $\that[f]\ne\frac{\partial{f}}{\partial\theta}$ in polar coordinates.
2/21/20
As mentioned in class today, any surface in $M\subset\RR^3$ can be described as $g(x,y,z)=\hbox{constant}$, with $dg\ne0$.
There are two pieces to this statement, first that there is a function $g$ on $\RR^3$, and second that the surface is one of its level curves. The condition $dg\ne0$ applies on $\RR^3$, not merely on $M$.
As an example, consider the paraboloid $z=x^2+y^2+1$, which can be realized as a level curve of the function $g=z-x^2-y^2$. We then have $dg=dz-2x\,dx-2y\,dy$, which is a 1-form in 3 dimensions, and hence does not vanish anywhere (since the coefficient of $dz$ is never $0$).
Yes, of course, if we also use the relationship $g=1$, we get $dg=0$. But this computation treats $g$ as a 1-form in 2 dimensions, not 3, since $dz$ is no longer independent of $dx$ and $dy$.
More formally, consider the patch $p(u,v)=(u,v,u^2+v^2+1)$, and the function $g(x,y,z)$ as above. Then $dg\ne0$ as above, but $d(g\circ p)=d(1)=0$, with the latter computation being done in the $(u,v)$-plane.
2/18/20
Here are the answers to the midterm questions:
  1. (a) $xy$ (b) $y^2$ (c) $y\,dx\wedge dy\wedge dz$
  2. (a) $\kappa=\frac12$, $\tau=0$ (b) circle
  3. (see worked solution)
  4. (a) $\ee_1=\frac1h(u\,\xhat+v\,\yhat)$, $\ee_2=\frac1h(-v\,\xhat+u\,\yhat)$
    (b) $\sigma_1=h\,du=\frac1h(u\,dx+v\,dy)$, $\sigma_2=h\,dv=\frac1h(-v\,dx+u\,dy)$
    (c) $\omega_{12}=\frac{1}{h^2}(u\,dv-v\,du)$ $=\frac12\frac{x\,dy-y\,dx}{x^2+y^2}$
Worked solutions can be seen in my office, and will be discussed in class on Wednesday.
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.
2/14/20
A formula sheet will be available on the midterm. You can find a copy here.
Please let me know of any typos or desired additions as soon as possible.
2/12/20
I will hold extra office hours on Friday, 2/14/20, from roughly 9:30–10:30 AM, and 10:30–11:30 AM on Monday, 2/17/20.
I will of course also hold my regular office hour on Friday at 10:30 AM.
A printout of a Mathematica session that shows how to compute the frame, dual basis, and connection in toroidal coordinates can be found here.
I find Mathematica easier to use than Sage for the computations shown here, but harder to use for computations involving differential forms. That said, feel free to contact me if you wish to pursue this option further, as I do have a beta package that may help.
I have updated the strategies page.
2/10/20
The software demonstrated in class today uses the open-source computer algebra system Sage to work with frames and connections in spherical and toroidal coordinates.
Sage is freely available to install on most computers, although the installation is large (6 GB for my older version).
You can also run Sage in the cloud using the SageMath cloud server.
Both Mathematica and Maple (also embedded in Matlab) can also manipulate such geometric objects.
Each language has pros and cons; use the one you like the best. I will help if I can, although I am most comfortable with Mathematica.
2/7/20
As pointed out in class today – and as is in fact clearly stated in the assignment – your answers to parts d and e of this week's homework assignment should be given in terms of the dual basis.
The notation in the assignment has been updated to reflect our choice to use $\sigma_i$ rather than $\theta_i$ for the dual basis.
A summary of some strategies for solving problems involving frames can be found here.
The very last method for finding the connection models the use of the structure equations for this purpose, the possibility of which was briefly discussed in class today.
2/5/20
The midterm will be held in Kidd 364 on Monday, 2/17/20, at 3 PM.
2/4/20
Here is some further information about the midterm:
2/3/20
With apologies, here is the corrected argument from the last few minutes of class today:
Claim: The dual basis to $\ee_j=\sum a_{j\ell}\uu_\ell$ is given by $\sum\sigma_i=\sum a_{ik}dx_k$.
Check: $\sigma_i(\ee_j) = \sum\sum a_{ik}a_{j\ell} dx_k(\uu_\ell) = \sum\sum a_{ik}a_{j\ell} \delta_{k\ell} = \sum a_{ik}a_{jk} = \delta_{ij}$ since $AA^T=I$.
2/1/20
A list of essential formulas has been posted here.
An abbreviated form of this list will be included on the midterm, which will otherwise be closed book.
1/31/20
The midterm is confirmed for Monday, 17 February 2020 (Week 7).
1/28/20
The midterm is tentatively scheduled for Monday, 17 February 2020 (Week 7).
Please let me know immediately of any conflicts or strong preferences that might affect having the midterm on this date.
(Mark your calendars: Looks like our final is scheduled for Friday, 20 March 2020, at 9:30 AM.)
1/25/20
There appears to be a typo in the text in the statement of Theorem 2.1, as there is no $t$ on the RHS.
The correct statement is that $s(t)=\int_a^t ||\alpha'(u)|| \,du$.
1/22/20
An applet showing the example discussed in class of an object moving non-uniformly along a circle can be found here.
A static image with most of the same content can be found here.
1/17/20
A brief discussion of div, grad, curl in the language of differential forms can be found in the text, in problem §1.6:8.
Further discussion can be found §15.3 of my own text which can be read online as an ebook through the OSU library.
(There is also a freely accessible wiki version available, which is however not quite the same as the published version.)
1/16/20
You may enjoy – and benefit from – the discussion of mathematical writing I wrote for WIC students, which can be found here.
These suggestions are suitable for polished work, such as a journal article; homework assignments can be more informal. Nonetheless, they do provide an indication of what good mathematical writing should be.
1/13/20
As announced in class, please turn in hard copy for (future) homework assignemnts.
By all means, (also) send me a copy as an attachment (PDF preferred) if you prepared your assignment electronically.
(It's not worth sending me a scanned copy of handwritten work unless there is no alternative.)
1/9/20
My office hours have been posted on the course homepage.
I am often in my office on Wednesday mornings (after 10) and late Monday mornings (after 11).
1/1/20 (updated 1/14/20)
The official text for this course is the revised second edition.