Toronto Math Forum
APM3462016F => APM346Lectures => Chapter 6 => Topic started by: Shentao YANG on November 05, 2016, 07:14:10 PM

Can any explain why we need the $\Theta $ function in section 6.4 (and onward) defined on $[0,2\pi ]$ instead of on $[0,2\pi )$ so that we can remove the periodic assumption of the $\Theta $ function and the boundary conditions related to $\theta $.
http://www.math.toronto.edu/courses/apm346h1/20169/PDEtextbook/Chapter6/S6.4.html
From Wikipedia, a standard convention for defining polar coordinate system to achieve Uniqueness of polar coordinates is restrict the domain to $[0, 2\pi)$ or $(âˆ’\pi, \pi]$.
https://en.wikipedia.org/wiki/Polar_coordinate_system

It is a really good question. In fact, in the "standard" settings $\theta$ runs $(\infty,\infty)$ but $\Theta$ must be $2\pi$periodic. So problem is
\begin{align}
&\Delta u=0,\label{eq1}\\
&u_{r=a}=g(\theta),\label{eq2}\\
&u(2\pi)=u(0), \ u_\theta(2\pi)=u_\theta(0).\label{eq3}
\end{align}
On the other hand, we can consider membrane also in the same shape, but it is "clumped" along $\{\theta=0\}$, in which case $\theta$ runs $(0,2\pi)$ and (\ref{eq3}) is replaced by
\begin{align}
&u(2\pi)=u(0)=0.\label{eq4}
\end{align}
We can also consider membrane also in the same shape, but it is "cut" along $\{\theta=0\}$ and both sides of cut are left free in which case $\theta$ runs $(0,2\pi)$ and (\ref{eq3}) is replaced by
\begin{align}
&u_\theta(2\pi)=u_\theta (0)=0.\label{eq5}
\end{align}
And so on. These are different problems. Separation of variables leads to different decompositions.

Just to make sure, in the "membrane" case you describe, we basically do not need the periodic assumption of $\Theta$ (Except the boundary condition), right?

Just to make sure, in the "membrane" case you describe, we basically do not need the periodic assumption of $\Theta$ (Except the boundary condition), right?
There is only one $2\pi$periodic case and it is the "standard" one. In this problem condition (\ref{eq3}) is equivalent to periodicity