VC5.43

By divergence theorm,we have $\iint_S F\cdot n dS=\iiint_V\mathrm{div}F dV\,$

Given that $F=\nabla\phi\,$ and $\nabla^2\phi=-4\pi\rho\,$

Now, $\mathrm{div}F=\nabla\cdot F=\nabla\cdot(\nabla\phi)\,$ from the above

= $\nabla^2\phi=-4\pi\rho\,$ ,from the above.

Therefore,from the first equality, $\iint_S F\cdot n dS=\iiint_V(-4\pi\rho)dV=-4\pi\iiint_V\rho dV\,$