| dbp:proof
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- Start with the integral form of Gauss's law:
Apply this law to the situation where the volume V is a sphere of radius r centered on a point-mass M. It's reasonable to expect the gravitational field from a point mass to be spherically symmetric. By making this assumption, g takes the following form:
. Plugging this in, and using the fact that ∂V is a spherical surface with constant r and area ,
:
:
:
:
which is Newton's law. (en)
- g, the gravitational field at r, can be calculated by adding up the contribution to g due to every bit of mass in the universe . To do this, we integrate over every point s in space, adding up the contribution to g associated with the mass at s, where this contribution is calculated by Newton's law. The result is:
If we take the divergence of both sides of this equation with respect to r, and use the known theorem
where δ is the Dirac delta function, the result is
Using the "sifting property" of the Dirac delta function, we arrive at
which is the differential form of Gauss's law for gravity, as desired. (en)
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