Natural Science Forum / Physics / Research / December 2004

One imagines summing the gravitational interaction with each point
within the extended mass will give you an overall total identical to
the classical expectation that (less tidal forces) all the mass could
have been cocentrated at the center of mass to give identical results.

Sounds like job for Green's function.

If I understand what you are to show... it's certainly not true, which
makes me wonder if the problem is rather odd or if I misunderstand or
if it's been lost in translation:
"Immagine a an extended (regular or irregular) body immersed in a
gravitational force field (not necessarily that produced by a
pointmass, but more generally of any kind). I have to show that the
medium gravitational force exerted on the body turns out to be the same
force we find at its center of mass."

The force exerted by a (Newtonian) gravity field on an object is not
the force that would be exerted on a point mass at the center of mass,
which is what I think you're saying you have to show. Rather, there is
a point called the "center of gravity" where the force exerted on the
object is the same as the force on that point.  To dispel the
center-of-mass idea, consider a very long dumbbell with heavy weights
on the end and a light rod connecting them held vertically in the
gravity field of a spherical mass. Let the rod be several times longer
than the distance from the center of the sphere to the center of the
lower weight on the dumbbell.  The force is a bit over the weight of
that dumbbell; if both weights were moved to the center of mass it
would clearly be much less.

Now, if you want to prove the existence of the center of gravity,
which is what I suspect you really want to do, then what you're
probably looking for is the (integral) Mean Value Theorem for
3-vectors.  "A continuous function attains its average at some point".
You can look that up at http://www.mathreference.com/ca-int,mvt.html.
Note that gravity fields are assumed to be continuous while density may
not be.

Note that in a uniform gravity field, the center of mass and the
center of gravity coincide, but in general, they do not.