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APM346--Misc / Thank You!
« on: December 22, 2015, 06:35:01 PM »
Thank you for a great course professor! Happy holidays!
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Final Exam / Re: FE-7
« on: December 19, 2015, 12:06:50 AM »$g$ is a degree 3 polynomial, so $P$ will be a degree 1 polynomial. $g$ also has rotational symmetry about the z axis ($x^2+y^2=s^2$ in cylindrical coordinates)
Thank you Vivian for explaining why the symmetry exists!
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Final Exam / Re: FE-6
« on: December 19, 2015, 12:00:58 AM »
Also, in case anyone doesn't remember, note that $$\lim_{r \rightarrow 0} \frac{\sin (r)}{r} = 1$$
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Final Exam / Re: FE-3
« on: December 18, 2015, 11:43:13 PM »
Thanks Vivian Tan! Great answer, as far as I recall I got the same! 
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APM346--Lectures / Re: Should we also study HA10 for the exam?
« on: December 11, 2015, 09:20:07 AM »
Also, just to confirm---there won't be distributions on the exam, right?
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APM346--Lectures / Should we also study HA10 for the exam?
« on: December 07, 2015, 09:40:24 PM »
Hi Professor,
On the course website, you say that studying HA1-10 is a good preparation for the final exam, but in class you said that HA10 will not appear on the exam. Could you clarify whether or not we should study this section?
On the course website, you say that studying HA1-10 is a good preparation for the final exam, but in class you said that HA10 will not appear on the exam. Could you clarify whether or not we should study this section?
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Web Bonus = Oct / Re: Web bonus problem : Week 4 (#5)
« on: November 30, 2015, 12:28:32 PM »
To prove the energy conservation law, we have the time derivative of $E(t)$. So:
\begin{equation}
\frac{\partial E(t)}{\partial t} = \frac{1}{2} \int_0^{\infty} \left[ u_{tt}^*u_t + u_t^*u_{tt} + c^2 \left( u_{xt}^*u_x + u_x^*u_{xt} \right) \right] dx
\end{equation}
We make use of the wave equation to rewrite the equation:
\begin{equation}
\frac{\partial E(t)}{\partial t} = \frac{1}{2} \int_0^{\infty} \left[ \left( c^2u_{xx} \right)^*u_t + u_t^*\left( c^2 u_{xx} \right) + c^2 \left( u_{xt}^*u_x + u_x^*u_{xt} \right) \right] dx
\end{equation}
We then notice that these terms can be combined as a derivative, since $u_tu_{xx} + u_xu_{xt} = \frac{d}{dx}u_tu_x$ So:
\begin{equation}
\frac{\partial E(t)}{\partial t} = \frac{c^2}{2} \int_0^{\infty} \left[ \frac{d}{dx} \left( u_x^*u_t \right) + \frac{d}{dx} \left( u_t^*u_x \right) \right] dx
\end{equation}
\begin{equation}
\frac{\partial E(t)}{\partial t} = \frac{c^2}{2} \left( u_x^*u_t |_0^{\infty} + u_t^*u_x |_0^{\infty} \right)
\end{equation}
We neglect the terms at $\infty$, since we assume the function is fast decaying. We then make use of the boundary condition to rewrite this:
\begin{equation}
\frac{\partial E(t)}{\partial t} = -\frac{c^2}{2} \left( -i \alpha u_t^*u_t + i \alpha u_t u_t^* \right) = 0
\end{equation}
Thus we have proven the energy conservation law.
\begin{equation}
\frac{\partial E(t)}{\partial t} = \frac{1}{2} \int_0^{\infty} \left[ u_{tt}^*u_t + u_t^*u_{tt} + c^2 \left( u_{xt}^*u_x + u_x^*u_{xt} \right) \right] dx
\end{equation}
We make use of the wave equation to rewrite the equation:
\begin{equation}
\frac{\partial E(t)}{\partial t} = \frac{1}{2} \int_0^{\infty} \left[ \left( c^2u_{xx} \right)^*u_t + u_t^*\left( c^2 u_{xx} \right) + c^2 \left( u_{xt}^*u_x + u_x^*u_{xt} \right) \right] dx
\end{equation}
We then notice that these terms can be combined as a derivative, since $u_tu_{xx} + u_xu_{xt} = \frac{d}{dx}u_tu_x$ So:
\begin{equation}
\frac{\partial E(t)}{\partial t} = \frac{c^2}{2} \int_0^{\infty} \left[ \frac{d}{dx} \left( u_x^*u_t \right) + \frac{d}{dx} \left( u_t^*u_x \right) \right] dx
\end{equation}
\begin{equation}
\frac{\partial E(t)}{\partial t} = \frac{c^2}{2} \left( u_x^*u_t |_0^{\infty} + u_t^*u_x |_0^{\infty} \right)
\end{equation}
We neglect the terms at $\infty$, since we assume the function is fast decaying. We then make use of the boundary condition to rewrite this:
\begin{equation}
\frac{\partial E(t)}{\partial t} = -\frac{c^2}{2} \left( -i \alpha u_t^*u_t + i \alpha u_t u_t^* \right) = 0
\end{equation}
Thus we have proven the energy conservation law.
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Textbook errors / Broken Link in S 10.2
« on: November 23, 2015, 11:36:59 AM »
Clicking the arrow to go to the next section at the bottom of this page:
http://www.math.toronto.edu/courses/apm346h1/20159/PDE-textbook/Chapter10/S10.2.html
leads to a 404 missing page error.
Fixed
http://www.math.toronto.edu/courses/apm346h1/20159/PDE-textbook/Chapter10/S10.2.html
leads to a 404 missing page error.
Fixed
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Test 2 / Re: Solutions to TT2
« on: November 20, 2015, 09:35:29 AM »
Also for solution to P2, should you have $(y+1)^2$ in the denominator of Eq. 2.8, since $(y+1)$ is our $\alpha$?
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Test 2 / Re: Solutions to TT2
« on: November 20, 2015, 09:31:59 AM »
Professor, I think you have a slight error in the solution to P1. In Eq. 1,7 you have a term $\cos(2nt)$ at the end, but should this not be a function of $x$? As you have Eq. 1.7 right now, there is no $x$-dependence.
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Test 2 / Re: TT2-P3
« on: November 19, 2015, 02:03:25 AM »I did it a different way. But by Catch's method shouldn't there be the additional constraint that $\lambda=\lambda_1 +\lambda_2$?
Actually I think Catch did mention that, near the top right of the page.
You're right, but then in the end shouldn't the final eigenvalues be $\lambda_n=\left(\frac{n\pi}{a}\right)^2+\left(\frac{n\pi}{b}\right)^2$?
Indeed, good point.
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Test 2 / Re: TT2-P3
« on: November 19, 2015, 01:59:30 AM »I guess she means y, but clerical errorPlease correct if something is wrong, thank you.
Catch, I am very confused---why is $Y$ a function of $x$ in your last step?!
Yes, you must be right! For a second I was worried that I had done everything wrong!
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Test 2 / Re: TT2-P3
« on: November 19, 2015, 01:59:03 AM »I did it a different way. But by Catch's method shouldn't there be the additional constraint that $\lambda=\lambda_1 +\lambda_2$?
Actually I think Catch did mention that, near the top right of the page.
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Test 2 / Re: TT2-P3
« on: November 19, 2015, 01:52:18 AM »Please correct if something is wrong, thank you.
Catch, I am very confused---why is $Y$ a function of $x$ in your last step?!