Chapter 14.1, Problem 50E

Gravitational force due to a mass The gravitational force on a point mass m due to a point mass M at the origin is a gradient field with potential $U\left(r\right)=\frac{GMm}{r}$, where G is the gravitational constant and $r=\sqrt{x^2+y^2+z^2}$ is the distance between the masses.

a.    Find the components of the gravitational force in the x-,y-, and z-directions, where F(x, y, z) = –▿U(x, y z).

b.    Show that the gravitational force points in the radial direction (outward from point mass M) and the radial component is $F\left(r\right)=\frac{GMm}{r^2}.$

c.    Show that the vector field is orthogonal to the equipotential surfaces at all points in the domain of U.