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### Messages - Emily Deibert

Pages:  2 3 ... 6
1
##### 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!

2
##### Final Exam / Re: FE-6
« on: December 19, 2015, 12:00:58 AM »
Also, in case anyone doesn't remember, note that $$\lim_{r \rightarrow 0} \frac{\sin (r)}{r} = 1$$

3
##### 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! 4
##### Final Exam / Re: FE-3
« on: December 18, 2015, 11:36:20 PM »
What do you mean by nodd?

5
##### 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.

6
##### 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

7
##### 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 $\alpha$?

8
##### 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.

9
##### 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.

10
##### Test 2 / Re: TT2-P3
« on: November 19, 2015, 01:59:30 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?!
I guess she means y, but clerical error

Yes, you must be right! For a second I was worried that I had done everything wrong!

11
##### 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.

12
##### 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?!

13
##### Test 2 / Re: TT2-P1
« on: November 19, 2015, 01:35:52 AM »
I got the same answer as Fei Fan Wu, but unfortunately forgot to put the factor of $\frac{1}{2}$ during the test! Oops.

14
##### Test 2 / Re: TT2-P2
« on: November 19, 2015, 01:25:06 AM »
The final answer cannot be right if it does not satisfy the boundary conditions.
I don't understand neither, but maybe you can check the lecture notes, http://www.math.toronto.edu/courses/apm346h1/20159/PDE-textbook/Chapter5/S5.3.html

Okay, thank you for the link! It did not work for me before. I am checking the textbook now, I hope I can understand.

15
##### Test 2 / Re: TT2-P2
« on: November 19, 2015, 01:17:00 AM »
as Xi Yue Wang did in homework 7 and lecture notes here: http://www.math.toronto.edu/courses/apm346h1/20159/PDE-

Rong Wei, which section did you mean? The link does not work for me!

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