Laplace's Equation in Polar Coordinates I am trying to express Laplace's equation in terms of polar coordinates. That is,
$$
\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}=0,\\
x=r\cos\theta,\\
y=r\sin\theta.
$$
My book immediately concludes that it is
$$
\frac1r\frac{\partial}{\partial r}\left(r\frac{\partial u}{\partial r}\right)+\frac1{r^2}\frac{\partial^2u}{\partial\theta^2}=0,
$$
but leaves us with no insight as to how that was obtained.
Any hint would be greatly appreciated!
 A: My favorite method for this is to use Gauss's theorem in reverse, so to speak. For brevity, let me write $\Delta u$ for the Laplacian and $u_r$, $u_\theta$ etc for the partial derivatives. Note that $\Delta u$ is the divergence of $\nabla u$, so Gauss's theorem says
$$ \iint_\Omega\Delta u\,dx\,dy=\int_{\partial\Omega}\mathbf{n}\cdot\nabla u\,ds .$$
Apply this to the domain $\Omega$ given by $r_1<r<r_2$ and $\theta_1<\theta<\theta_2$, and note that


*

*On the boundary $r=r_2$, $\mathbf{n}\cdot\nabla u=u_r$ and $ds=r_2\,d\theta$

*On the boundary $r=r_1$, $\mathbf{n}\cdot\nabla u=-u_r$ and $ds=r_1\,d\theta$

*On the boundary $\theta=\theta_2$, $\mathbf{n}\cdot\nabla u=u_\theta/r$ and $ds=dr$

*On the boundary $\theta=\theta_1$, $\mathbf{n}\cdot\nabla u=-u_\theta/r$ and $ds=dr$


so Gauss becomes
$$\begin{aligned}\int_{\theta_1}^{\theta_2}\int_{r_1}^{r_2}\Delta u\cdot r\,dr\,d\theta
  &=\int_{\theta_1}^{\theta_2}\bigl(r_2u_r(r_2,\theta)-r_1u_r(r_1,\theta)\bigr)\,d\theta\\
  &\quad+\int_{r_1}^{r_2}\frac{u_\theta(r,\theta_2)-u_\theta(r,\theta_1)}{r}\,dr\\
  &=\int_{\theta_1}^{\theta_2}\int_{r_1}^{r_2}(ru_r)_r\,dr\,d\theta
   +\int_{r_1}^{r_2}\int_{\theta_1}^{\theta_2}\frac{u_{\theta\theta}}{r}\,d\theta\,dr\\
  &=\int_{\theta_1}^{\theta_2}\int_{r_1}^{r_2}\Bigl((ru_r)_r+\frac{u_{\theta\theta}}{r}\Bigr)\,dr\,d\theta
\end{aligned}$$
Since this holds for all choices of the limits, the integrands must be the same, so $$\Delta u\cdot r=(ru_r)_r+\frac{u_{\theta\theta}}{r}.$$
Now divide by $r$.
