Editing Poisson's equation (section)

=== Newtonian gravity ===
{{main|Gravitational field|Gauss's law for gravity}}

In the case of a gravitational field '''g''' due to an attracting massive object of density ''ρ'', Gauss's law for gravity in differential form can be used to obtain the corresponding Poisson equation for gravity. Gauss's law for gravity is
<math display="block">\nabla\cdot\mathbf{g} = -4\pi G\rho.</math>

Since the gravitational field is conservative (and [[irrotational]]), it can be expressed in terms of a [[scalar potential]] ''ϕ'':
<math display="block">\mathbf{g} = -\nabla \phi.</math>

Substituting this into Gauss's law,
<math display="block">\nabla\cdot(-\nabla \phi) = - 4\pi G \rho,</math>
yields '''Poisson's equation''' for gravity:
<math display="block">\nabla^2 \phi  =  4\pi G \rho.</math>

If the mass density is zero, Poisson's equation reduces to Laplace's equation. The [[Green's function for the three-variable Laplace equation|corresponding Green's function]] can be used to calculate the potential at distance {{mvar|r}} from a central point mass {{mvar|m}} (i.e., the [[fundamental solution]]). In three dimensions the potential is
<math display="block">\phi(r) = \frac{-G m}{r},</math>
which is equivalent to [[Newton's law of universal gravitation]].