For the temperature distribution $u(x,y,t)$. I know the heat equation on a plane is $\frac{\partial u}{\partial t}=c(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2})$. But if we consider the heat equation on a circle, we can express $u(x,y,t)=u(cos2\pi s,sin2\pi s,t)$ where $s\in [0,1]$. I am trying to derive the equation $\frac{\partial u}{\partial t}=c'(\frac{\partial^2 u}{\partial s^2})$ given in Fourier analysis by Stein.
I computed $\frac{\partial^2 u}{\partial s^2}=4\pi ^2(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2})-4\pi ^2cos(2\pi s)\frac{\partial u}{\partial x}-4\pi^2sin(2\pi s)\frac{\partial u}{\partial y}$. And I am stuck here. Any suggestion will be appreciated.
The circle here is modelled exactly as a rod, i.e., with the one-dimensional heat equation $u_t = c u_{xx}$, the only difference being that the solution is required to be periodic in $x$. (Imagine the rod being bent around, and the ends joined, to make a ring.)
It has nothing to do with the heat equation for a disk.