3D Dirac Delta

${\delta }^3(\vec{r})=\delta (x)\delta (y)\delta (z)$

${\int }_{\Omega }{\delta }^3(\vec{r}-{\vec{r}}_0)\mathrm{d}v=1$

• In 3D, you need three delta functions for each dimension: $\delta (x-x_0)\delta (y-y_0)\delta (z-z_0)={\delta }^3(\vec{r}-\overrightarrow{r_0})$
• $\Omega$ is the volume to integrate over

$\nabla \cdot \left(\frac{\hat{r}}{r^2}\right)=4\pi {\delta }^3(\vec{r})$

• Fixes divergence theorem