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### Messages - Victor Ivrii

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2491
##### APM346 Misc / Re: TA office hours
« on: September 28, 2012, 02:31:54 PM »
What are the TA office hours? On the course outline they are still listed as TBD....

Working on it. Meanwhile you can drop to my office hours

2492
##### Misc Math / Re: 1-D Wave equation derivation
« on: September 28, 2012, 02:20:57 PM »
Starting with

\begin{equation} \frac{\partial}{\partial x} \left[ T(x,t) \sin{\theta (x,t)} \right] = \rho (x) u_{tt} \end{equation}

where $\rho$ is the density and $T(x,t)$ is the tension force, we made the assumption that the vibrations are small, which gave us a linearized wave equation. I can see why some of the other assumptions (i.e. full flexibility, and no horizontal tension component) make sense, but I don't think I understand the insight behind this one.

You mean that vibrations are small? Because usually they are. More general versions you find in some textbooks like
\begin{equation}
\rho u_{tt} = \Bigl(\frac{u_x}{\sqrt{1+u_x^2}}\Bigr)_x
\end{equation}
implicitly assume that displacement is only in the direction perpendicular to the string and that the density does not change--which is the case only for the small oscillations.

2493
##### APM346 Misc / Re: Extra help before term test
« on: September 27, 2012, 04:51:09 PM »
Yes special TA office hours or perhaps even a pre-test seminar similar to what Professor Moradifam used to offer last semester.

Let us think this out

2494
##### Misc Math / Re: Lecture 6 example
« on: September 27, 2012, 04:49:15 PM »
Hello,
I was going through the 6. lecture notes, where in the end an example is brought up that leads to an integral
$$I=\frac{1}{4}\int_0^t\cos(t')\bigl(\cos(x-ct+ct')-\cos(x+ct-ct')\bigr)dt'$$
I was trying to do that integral, but the only way that I could do it was to write out the cosines as complex exponentials, which lead me to eight terms in the end... Is there a cleverer way to do this integral?
Thanks!

In this example $c=2$ helps a bit but you could sea; without it.

The 1-st line -> 2nd (just integration) -> 3rd (formula $\cos (\alpha)-\cos (\beta)= 2\sin \bigl((\beta+\alpha)/2\bigr)\cdot \sin \bigl((\beta-\alpha)/2\bigr)$ and we ge the 4-th line.
Then formula $2\sin( \alpha) \cdot \cos(\beta)=\sin (\alpha+\beta)+\sin (\alpha-\beta)$ and integration.

2495
##### Home Assignment 2 / Re: Problem 3
« on: September 27, 2012, 01:41:03 PM »
Professor, could you fix this in the PDF file? I prefer the PDF file so I could work on the assignment without an internet connection.

Thank you

There was an error in both variants: it claimed to be "Home Assignment 1" in the title--now it fixed.

2496
##### Home Assignment 2 / Re: Problem 2
« on: September 27, 2012, 11:11:15 AM »
It looks like I should use the result from part 3 for part 4. However if so, how should I use the initial values?

Not really. You use parts (a),(b)

2497
##### Home Assignment 2 / Re: problem 1 typo?
« on: September 27, 2012, 11:10:32 AM »
Hi - In the pdf of home assignment 2 in problem 1 the inequalities throughout are different than the other version of the assignment (i.e. pdf version has 'greater than or equal to' and the other version in only '>').  Which one is correct?

Really does not matter, but I changed pdf to coincide

2498
##### APM346 Misc / Re: Extra help before term test
« on: September 27, 2012, 06:15:46 AM »
Will there be any extra help sessions prior to the term tests like there were last semester for ODEs?

Do you mean special TA office hours or something else?

2499
##### Home Assignment 1 / Re: Problem 4
« on: September 26, 2012, 09:40:44 PM »

Do you mean:
$$-u_\theta=r^2 \\ \Rightarrow u=-r^2\theta$$ and since it's not periodic we are done?

Basically yes, except certain things should be clarified: since our trajectories are closed each is parametrized by some parameter ($\theta$) running from $0$ to $T$ (in our case $T=2\pi$ but it may depend on trajectory) and equation looks like $\partial_\theta u= g(\theta,r)$. So we are looking for periodic solution
$u(\theta,r)=\int g(\theta,r)\,d\theta$.

Note that primitive of periodic function $g(\theta)$  is periodic if and only if average of $g$ over period is $0$:
$\int_0^T g(\theta)\,d\theta=0$. Otherwise this primitive is the sum of a periodic function and a linear function.

Finally, in (a) $g(\theta)=r^2 \sin(\theta)\cos(\theta)$ and integral over period is $0$; in (b) $g(\theta)=r^2$ and  integral over period is not $0$.

So, the source of trouble is: periodic trajectories.

2500
##### Home Assignment 1 / Re: Problem 4
« on: September 26, 2012, 04:34:04 PM »
Here we are talking about first order equation. There is no wave equation. Stick to the problem! How our original equation
\begin{equation}
yu_x-xu_y=f(x,y)
\label{eq-1}
\end{equation}
looks in the polar coordinates?

Hint: the l.h.e. is $-\partial_\theta u$. Prove it using chain rule  $u_\theta = u_x x_\theta + u_y y_\theta$ and calculate $x_\theta$, $y_\theta$.
Note that $\theta$ is defined modulo $2\pi \mathbb{Z}$ and all functions must be $2\pi$-periodic with respect to $\theta$ (assuming that we consider domains where one can travel around origin)

2501
##### APM346 Announcements / Term Tests
« on: September 26, 2012, 03:02:02 PM »
Both sections:

Term Test 1
16-OCT-12 EX 200 16:00--18:00 (TUE) Examination Facility http://osm.utoronto.ca/osm/exam_200.html

There is a poll attached. Answer it please and select the topmost answer which works for you. If nothing works contact class coordinator Prof. Colliander

Term Test 2  15-NOV-12 (THU) 18:00--20:00
GB 304 Galbraith Building  http://osm.utoronto.ca/class_spec/f?p=210:1:1579690136306710::NO::P1_ROOM:304
GB 404  Galbraith Building http://osm.utoronto.ca/class_spec/f?p=210:1:1579690136306710::NO::P1_ROOM:404
GB 412 Galbraith Building http://osm.utoronto.ca/class_spec/f?p=210:1:1579690136306710::NO::P1_ROOM:412

2502
##### Misc Math / Re: Heat Wave Equation Solution Integral
« on: September 26, 2012, 12:55:13 PM »
Sorry I meant the Heat Equation. What does the integral of the solution to that equation represent?

If we talking about heat equation then it is a total heat energy in $(-\infty,x)$. But for us it mainly a trick to get a a formula for a solution.

2503
##### Home Assignment 2 / Re: Problem 3
« on: September 26, 2012, 05:21:59 AM »
Part (a) and (b), part (c) and (d) the same questions? Thanks

There was misprint in (c) fixed almost immediately. Compare fourth lines in (a) vs (b) , (c) vs (d)

2504
##### Home Assignment 1 / Re: Problem 4
« on: September 26, 2012, 05:19:38 AM »
Peishan, the question now boils down to: why solution with r.h.e. $x^2+y^2$ does not exist and one really needs to use polar coordinates (or to use equivalent geometric motivation)

2505
##### Misc Math / Re: Heat Wave Equation Solution Integral
« on: September 26, 2012, 02:28:48 AM »
After we determined the solution to the heat wave equation, we took the integral from negative infinity to some x. The solution was in the form of a Normal Distribution function where in statistics the integral indicates the probability that we randomly choose a value within that interval. I was wondering what are the physical implications of this same integral in the wave equation? Does the area under the solution indicate the energy in the system up to that point or does it have some other meaning?

Heat wave equation? WTH are you talking about? There is a wave equation (and its ilk) and the heat equation  (and its ilk) describing either very different processes or the same process under very different assumption. The properties of these equations are really different.

PS Sure more complex phenomena can be described by systems containing both equations but for the purpose of this class ...

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