# Laplace and Poisson's equation.

1. Dec 10, 2013

### yungman

1. The problem statement, all variables and given/known data

I want to verify for $v=\ln( r)$, that

a)$\nabla^2 v=0$ for $v$ on a disk center at $(x_0,y_0)$. Therefore $v$ is Harmonic.

b)$\nabla^2 v=\frac{1}{r^2}$ for $v$ on a sphere center at $(x_0,y_0,z_0)$. Therefore $v$ is not Harmonic.

3. The attempt at a solution

a) For circular disk, $v=\frac{1}{2}\ln[(x-x_0)^2+(y-y_0)^2]$
$$\nabla v=\frac{\hat x (x-x_0)+\hat y(y-y_0)}{[(x-x_0)^2+(y-y_0)^2]^2}$$
$$\nabla^2 v=\nabla\cdot\nabla v=\frac{[(y-y_0)^2-(x-x_0)^2]+[(x-x_0)^2-(y-y_0)^2]}{[(x-x_0)^2+(y-y_0)^2]^2}$$=0

b)For a sphere, $v=\frac{1}{2}\ln[(x-x_0)^2+(y-y_0)^2+(z-z_0)^2]$
$$\nabla v=\frac{\hat x (x-x_0)+\hat y(y-y_0)+\hat z(z-z_0)^2}{[(x-x_0)^2+(y-y_0)^2+(z-z_0)^2]^2}$$
$$\nabla^2 v=\nabla\cdot\nabla v=\frac{[(y-y_0)^2+(z-z_0)^2-(x-x_0)^2]+[(x-x_0)^2+(z-z_0)^2-(y-y_0)^2]+[(x-x_0)^2+(y-y_0)^2-(z-z_0)^2]}{[(x-x_0)^2+(y-y_0)^2+(z-z_0)^2]^2]}$$
$$\Rightarrow\;\nabla^2 v=\frac{(x-x_0)^2+(y-y_0)^2+(z-z_0)^2}{[(x-x_0)^2+(y-y_0)^2+(z-z_0)^2]^2]}=\frac{1}{r^2}$$

So $v$ is Harmonic only for a two dimension disk, not in three dimension sphere.