Pre-Quantum Electrodynamics

\begin{equation} \delta^{(3)} ({\bf r} - {\bf r}') = \delta (x - x') \delta (y - y') \delta (z - z') \label{Gr(1.96)} \end{equation} \begin{equation} \int d\tau f({\bf r}) \delta^{(3)} ({\bf r} - {\bf a}) = f({\bf a}) \label{Gr(1.97)} \end{equation}

Resolution of divergence of \(\hat{\bf r}/r^2\) paradox:

\begin{equation} {\boldsymbol \nabla} \cdot \left(\frac{\hat{\bf r}}{r^2} \right)= 4\pi \delta^{(3)} ({\bf r}). \label{Gr(1.99)} \end{equation}

More generally,

\begin{equation} \boxed{ {\boldsymbol \nabla} \cdot \left(\frac{{\bf r} - {\bf r}'}{|{\bf r} - {\bf r}'|^3} \right)= 4\pi \delta^{(3)} ({\bf r}). } \tag{divdel}\label{divdel} \end{equation}

Since

\begin{equation} {\boldsymbol \nabla}_1 \left(\frac{1}{r_{12}}\right) = -\frac{\hat{\bf r}_{12}}{r_{12}^2} \tag{div1or}\label{div1or} \end{equation}

we have that

\begin{equation} {\boldsymbol \nabla}^2 \left( \frac{1}{|{\bf r} - {\bf r}'|} \right) = -4\pi \delta^{(3)} ({\bf r} - {\bf r}') \tag{Lap1or}\label{Lap1or} \end{equation}