1

**APM346--Misc / Thank You!**

« **on:**December 22, 2015, 06:35:01 PM »

Thank you for a great course professor! Happy holidays!

This section allows you to view all posts made by this member. Note that you can only see posts made in areas you currently have access to.

1

Thank you for a great course professor! Happy holidays!

2

$g$ 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!

3

Also, in case anyone doesn't remember, note that $$\lim_{r \rightarrow 0} \frac{\sin (r)}{r} = 1$$

4

Thanks Vivian Tan! Great answer, as far as I recall I got the same!

6

Also, just to confirm---there won't be distributions on the exam, right?

7

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?

8

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.

9

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

10

Also for solution to P2, should you have $(y+1)^2$ in the denominator of Eq. 2.8, since $(y+1)$ is our $\alpha$?

11

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.

12

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.

13

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!

14

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.

15

Please correct if something is wrong, thank you.

Catch, I am very confused---why is $Y$ a function of $x$ in your last step?!