Chapter 17.8, Problem 40E

Gauss’ Law for gravitation The gravitational force due to a point mass M at the origin is proportional to F = GMr/|r|³, where r = 〈x, y, z〉 and G is the gravitational constant.

a.    Show that the flux of the force field across a sphere of radius a centered at the origin is ${\displaystyle {\iint }_{S}F\cdot ndS=4\pi GM}$

b.    Let S be the boundary of the region between two spheres centered at the origin of radius a and b with a < b. Use the Divergence Theorem to show that the net outward flux across S is zero.

c.    Suppose there is a distribution of mass within a region D Let ρ (x, y, z) be the mass density (mass per unit volume). Interpret the statement that

${\displaystyle \underset{S}{\iint }F\cdot ndS=4\pi G{\displaystyle \underset{D}{\iiint }\rho \left(x,y,z\right)dV.}}$

d.    Assuming F satisfies the conditions of the Divergence Theorem on D. conclude from part (c) that ▿ · F = 4pGρ.

e.    Because the gravitational force is conservative, it has a potential function ϕ. From part (d). conclude that ▿²ϕ = 4pGp.