I am trying to show that the fundamental solution to the Laplacian in 2D satisfies
$$\Delta \phi(x) = \delta(x)$$
where $x = (x_1, x_2) \in \mathbb{R}^2$.
So the fundamental solution in 2D is $\frac{1}{2\pi}\ln|x|$ and we have
$$\Delta \frac{1}{2\pi}\ln|x| = \delta(x)$$
$$\int_B \Delta \frac{1}{2\pi}\ln|x| dx = \int_B\delta(x)dx$$
where $B$ is the unit ball.
$$\frac{1}{2\pi}\int_B \Delta \ln|x| dx = 1$$
$$\int_B \Delta \ln|x| dx = 2\pi$$
Using the divergence theorem we transform this to a surface integral
$$\int_{\partial B} \frac{\partial \ln|x|}{\partial \nu(x)} dx = 2\pi$$
where $\nu$ is the normal derivative. I can't see where to go from here?