VC5.43

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By divergence theorm,we have \iint _{S}F\cdot ndS=\iiint _{V}{\mathrm  {div}}FdV\,

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 ndS=\iiint _{V}(-4\pi \rho )dV=-4\pi \iiint _{V}\rho dV\,


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