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Newtonian 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 ∇ ⋅ g = − 4 π G ρ. {\displaystyle \nabla \cdot \mathbf {g} =-4\pi G\rho.}

Since the gravitational field is conservative (and irrotational), it can be expressed in terms of a scalar potential ϕ: g = − ∇ ϕ. {\displaystyle \mathbf {g} =-\nabla \phi.}

Substituting this into Gauss's law, ∇ ⋅ (− ∇ ϕ) = − 4 π G ρ, {\displaystyle \nabla \cdot (-\nabla \phi)=-4\pi G\rho,} yields Poisson's equation for gravity: ∇ 2 ϕ = 4 π G ρ. {\displaystyle \nabla ^{2}\phi =4\pi G\rho.}

If the mass density is zero, Poisson's equation reduces to Laplace's equation. The corresponding Green's function can be used to calculate the potential at distance r from a central point mass m (i.e., the fundamental solution). In three dimensions the potential is ϕ (r) = − G m r, {\displaystyle \phi (r)={\frac {-Gm}{r}},} which is equivalent to Newton's law of universal gravitation.