ANNOUNCEMENTS
MTH 434/534 — Winter 2019
- 3/20/19
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Below are some of the answers to the final:
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2. $\beta_1$ is neither; $\beta_2$ is both
4. (a) $-1/a^2$ (b) $2\pi a^2$
5. $2\pi^2 r^4$
- 3/14/19
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Happy Einstein's Birthday! And Happy Pi Day!
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Extra office hours:
I should be in my office most of Monday, 3/18.
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I will probably be in my office from 10:30–11:30 AM.
I will definitely be in my office from 1:30–2:30 PM and from
3:30–4:30 PM (and later if there are people still waiting).
I may be in my office at other times, so don't hesitate to check,
or to send me a request for an appointment.
- 3/12/19
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A live computation using Sage of the curvature of the torus, as discussed
in class yesterday, can be found here.
- 3/11/19
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The final exam will be from 2–3:50 PM on Tuesday, 3/19/19,
in Weniger 212.
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There will be a review session during class on Friday, 3/15/19.
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A formula sheet will be available on the final. You can find a
copy here.
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- The exam is closed book.
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The exam will cover everything discussed in class through
Wednesday 3/13/19.
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The exam will cover material from the entire course, with an
emphasis on new material.
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Important new topics are the connection ($\omega^i{}_j$) and
curvature ($\Omega^i{}_j$).
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Important old topics are the exterior product ($\wedge$), Hodge dual
($*$), and exterior differentiation ($d$).
- 3/10/19
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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?
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Give up? Some hints can be found here.
- 3/3/19
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In HW 6, the index positions in part (c) should
really be
$\Omega^i{}_j=d\omega^i{}_j+\omega^i{}_k\wedge\omega^k{}_j$,
which however requires you to compute $\omega^i{}_j$ from $\omega_{ij}$.
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Yes, this step is trivial, but the wording in the homework is designed to
avoid this extra step.
My apologies for the confusion.
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As we will learn this coming week, $\Omega^i{}_j$ represents curvature.
- 2/24/19
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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.
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I have used most of these packages myself. Feel free to contact me for advice
and assistance.
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I have set up an experimental interface to Sage
here,
which should be fairly easy to adapt to other examples.
Some tips:
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The operations $d$, ${*}$, and $\wedge$ should all work, entered
as d(), star(), and wedge(), respectively.
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There is also an operation Wedge() that takes any number of
arguments.
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You can not add more boxes, but you can enter multiple lines of code in each
box.
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Only the last result will be printed.
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You may need to use the Show() command to see the result you expect.
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Computations in one box can be used in later boxes.
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This is beta software! Please do let me know if it does not work as expected.
- 2/22/19
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As expected, the class time will not be changed for next term.
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My apologies to students who have an unavoidable conflict.
- 2/20/19
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Based on the responses I have received so far, there appear to be only 3
insurmountable conflicts with the regular time (3 PM), and 2 PM appears to be
at least as bad. At this point, I do not expect that the class time will be
changed. Stay tuned.
- 2/19/19
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There is an unfortunate conflict next term between MTH 312 and MTH 437.
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Please come to class prepared to discuss the possibility of rescheduling
MTH 437/537.
(The only likely alternatives are MWF @ 8 AM, 2 PM, or 5 PM.)
- 2/16/19
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Here are the answers to the midterm questions:
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(a) $0$
(b) $dx\wedge dy\wedge dz$
(c) $0$
(d) $2\,dx\wedge dy\wedge dz$
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(a) $0$
(b) $2\,\alpha\wedge d\alpha$
(c) $d\alpha\wedge d\alpha$
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(a) $dx\wedge dy+dz\wedge dw$ (among others)
(b) $dx\wedge dy-dz\wedge dw$ (among others)
(c) no such forms exist
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(a) $e^{2uv}\,du\wedge dv$
(b) $e^{-2uv}\left(\frac{\partial^2f}{\partial u^2} +
\frac{\partial^2f}{\partial v^2}\right)$
(c) no change
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(a) ${*}(dh\wedge{*}dh)+h\,{*}d{*}dh$
(b) $\grad\cdot(h\grad h) = \grad h\cdot\grad h+h\grad\cdot\grad h$
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Worked solutions can be seen in my office, and will be discussed in class
on Monday
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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.
- 2/14/19
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A reminder that tomorrow's midterm is in Weniger 212.
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I plan to get there 10–15 minutes early so that I can return HW #4
before the exam.
- 2/13/19
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The formula sheet has been updated.
- 2/11/19
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The books mentioned in class were those by Flanders, Bishop & Goldberg,
and Misner, Thorne & Wheeler.
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An annotated list of related books, including these, is available
here.
(As pointed out there, several of these books are on reserve in the library.)
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Units:
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The electric field $E$ has dimensions ${ML}/{T^2Q}$, where $M=\hbox{mass}$,
$L=\hbox{length}$, $T=\hbox{time}$, and $Q=\hbox{charge}$. The magnetic field
$B$ has dimensions ${M}/{TQ}$.
(Note that $E$ and $cB$ have the same dimensions, where $c$ is the speed of
light.)
In the expression $F=\bar{E}\wedge dt+\bar{*}\bar{B}$, the first term
therefore has the dimensions ${ML}/{TQ}$. What about the second?
Well, $B$ is a 1-form, with dimensions ${M}/{TQ}$ as above. But
$\bar{*}$ takes spatial basis 1-forms, with dimensions $L$, to spatial basis
2-forms, with dimensions $L^2$. Thus, $\bar{*}\bar{B}$ also has dimensions
${ML}/{TQ}$!
- 2/3/19
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The midterm is confirmed for Friday, 2/15/19, in Weniger 212.
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There will be a review session during class on Wednesday, 2/13/19.
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A formula sheet will be available on the midterm. You can find a
copy here.
- 1/27/19
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The midterm is currently scheduled for Friday, 2/15/19, in class.
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If you have any concerns about this timing, please let me know immediately.
- 1/18/19
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You can find out more about the reasons we will use the "physics" convention
for the names of the spherical coordinates in our paper:
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Spherical Coordinates,
Tevian Dray and Corinne A. Manogue,
College Math. J. 34, 168–169 (2003)
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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.
- 1/16/19
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Here is an explicit example of "Einstein summation":
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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 $\det(A)$.
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The general solution is:
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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.
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Make sure that you can follow these "index gymnastics".
- 1/14/19
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There are two possible interpretations of Question 1(c) on HW #2. The first
is to assume that $\gamma$ is an unknown differential form. The second is to
assume that $\gamma=\beta$. Both are good questions, that are in fact related
to each other. You may answer either or both, but be clear about your
choices.
- 1/14/19
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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.)
- 1/13/19
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I have posted one possible solution to the first
homework assignment.
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If you don'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.
- 1/11/19
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Here is a lightly edited list of the basic linear algebra topics that arose in
class.
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- vector space
- linear (in)dependence
- basis
- change of basis
- span
- linear transformation
- eigenvalues
- eigenvectors
- determinants
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You should review these topics if you are rusty!
- 1/10/19
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The schedule has been updated, now also
including the scheduled final and (very) tentative midterm dates.
- 1/9/19
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A list of derivative rules in differential notation is available
here.
- 1/8/19
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Homework should be turned in at the beginning of class on the due date.
- 1/6/19
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The text
can be read online as an
ebook
through the OSU library.
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From off campus, you may need to first login to the library; try
this link.
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There is also a freely accessible
wiki
version available, which is however not quite the same as the
published version.
- 11/29/18
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MTH 434 and MTH 534 are both officially full.
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I expect that 5–10 additional spaces will be made available, but
possibly not until Week 1.
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To maximize your chances of getting in, make sure to get on the waitlist
and come to class.
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Waitlisting is not available until
Phase II.
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Please contact me if you have any questions about registration.
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Do let me know if a delay in registration poses a problem for you.
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By all means let me know of your desire to register for the course.