Math - Gravity Sphere

Using vector calculus show that the mass of a planet exerts a force on an object as if all the planet's mass were at the center of the planet. 

The calculation shows that gravitational simplifications can be applied to objects inside or outside a spherically symmetrical body. This theorem has particular application to astronomy.   Isaac Newton performed the calculation and summarized the results as: 

A spherically symmetric body affects external objects gravitationally as though all of its mass were concentrated at a point at its center.  

If the body is a spherically symmetric shell (i.e., a hollow ball), no net gravitational force is exerted by the shell on any object inside, regardless of the object's location within the shell.

The mass m0 is located at xyz coordinates: (0, 0, Z0)

u is a unit vector from the mass m0 to an arbitrary point on the hollow sphere. 

Find the formula to compute a small area on the sphere:  delta-A