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Messages - Victor Ivrii

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Home Assignment 2 / Re: Problem 2
« on: September 29, 2012, 02:36:32 PM »
for the last part, can we just assume the same solution as c) but state a few assumptions instead? it's because I dont think the general solution of the equation would vary since no other IC were stated.

You should check when and if the general solution is continuous at $r=0$ and adjust it respectively

Home Assignment 2 / Re: problem 1 typo?
« on: September 29, 2012, 09:04:56 AM »
Can we assume that for part (C) of Problem 1 that the Cauchy conditions are evenly reflected for x < 0?

Sure, you can but it will not be useful as your domain is $x>vt$ rather than $x>0$. Just use the general solution.

Misc Math / Re: Example 8b
« on: September 28, 2012, 05:36:19 PM »
If we are looking for $t>0$ then we are interested in $0<x<ct$ and $x>ct$ because the original problem is dealing with $x>0$. Continuation is the method to reduce it to IVP but we need to come back

Transitions in (9), (12) from the middle to the r.h. expression is due to the trigonometric formulae. Everything is ok

PS. Please use \sin,  \cos etc instead of sin, cos in LaTeX code as those are predefined macros.

PPS. In Lecture 9 we use \erf but as this is not a predefined macro, we define it by ourselves

Home Assignment 2 / Re: Problem 2
« on: September 28, 2012, 05:23:56 PM »
I think in the new variant (just posted) it will be more clear

APM346 Misc / Re: Midterm conflicts with schedule!
« on: September 28, 2012, 02:33:17 PM »
Just to open up the headline a bit:
Both midterm exams conflict with my schedule. So, what am I supposed to do?

There is now a poll attached to the announcement. Vote in it

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

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
\rho u_{tt} = \Bigl(\frac{u_x}{\sqrt{1+u_x^2}}\Bigr)_x
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.

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

Misc Math / Re: Lecture 6 example
« on: September 27, 2012, 04:49:15 PM »
I was going through the 6. lecture notes, where in the end an example is brought up that leads to an integral
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?

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.

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

It was fixed sometime ago but your computer could has cashed it--press "Reload" in your web browser.

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

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)

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

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?

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.

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
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)

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