Gauss's law for gravity

Gauss's law for gravity states that gravitational flux through a closed surface equals −4πG times the mass enclosed. It is the integral form of Newtonian gravity's field equation.

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Gravitational flux: The surface integral of a gravitational field over an oriented surface. Its integral over a closed surface is the quantity the law equates to enclosed mass.
Newton's shell theorem: The theorem that a spherically symmetric mass distribution attracts exterior bodies as though its mass were concentrated at its center. It follows from applying the flux law to spherical surfaces.
Newton's law of universal gravitation: The inverse-square law stating that two masses attract with force proportional to their product and inversely proportional to separation squared. Integrating its inverse-square field over a closed surface produces the flux law.
Gauss's law: The electrostatic law equating electric flux through a closed surface to enclosed charge divided by vacuum permittivity. Its positive source term contrasts with gravity's negative term for attractive fields.
Newtonian gravitational field: The force per unit mass experienced at each point in a Newtonian gravitational field. The law constrains this field's divergence and its flux through closed surfaces.
Uniform sphere: A sphere with constant mass density throughout its volume. The law gives its interior field by counting only the mass enclosed at each radius.
Gravitational constant: The proportionality constant G that sets the strength of Newtonian gravitational interactions. It scales the flux associated with each unit of enclosed mass.
Gauss's law for magnetism: The law stating that net magnetic flux through any closed surface is zero. It has no monopole source term, unlike gravitational flux around mass.
Divergence theorem: A theorem equating a vector field's flux through a closed surface with its divergence integrated over the enclosed volume. It converts the surface-flux law into a local differential equation.
Gravitational field of an infinite plane: The Newtonian gravitational field produced by an ideal infinite plane of uniform surface mass density. A pillbox Gaussian surface uses planar symmetry to determine its field.