Chapter 17.3, Problem 67E

Gravitational potential The gravitational force between two point masses M and m is

     $F=GMm\frac{r}{{\left|r\right|}^3}=GMm\frac{\langle x,y,z\rangle }{{(x^2+y^2+z^2)}^{3/2}},$

  where G is the gravitational constant.

a.    Verify that this force field is conservative on any region excluding the origin.

b.    Find a potential function φ for this force field such that F = –▿φ.

c.    Suppose the object with mass m is moved from a point A to a point B, where A is a distance r₁ from M and B is a distance r₂, from M. Show that the work done in moving the object is $GMm\left(\frac{1}{r_2}-\frac{1}{r_1}\right)$.

d.    Does the work depend on the path between A and B? Explain.