3D Dirac Delta

#Math

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

$\displaystyle \int_{\Omega} \delta^{3}(\vec{r}- \vec{r}_{0}) , \mathrm{d}v=1$

• In 3D, you need three delta functions for each dimension: $\displaystyle \delta(x-x_{0})\delta(y-y_{0})\delta(z-z_{0})=\delta^{3}(\vec{r}- \vec{r_{0}})$
• $\displaystyle \Omega$ is the volume to integrate over

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

• Fixes divergence theorem