The mass distribution of any rigid body can
be characterised as a measure on space, expressible as a
density function
({mass/volume}: ρ :{space}), for which integral(:ρ|V) is the mass within
any given volume V of space. The total mass of the body is then
integral(ρ).
In so far as space is flat enough that we can get away with describing the neighbourhood in which ρ is flat by a chart ({space}: :U), with U a vector space, whose representation of space-time's metric is close enough to unit-diagonalised that we don't notice the difference, we can reinterpret ρ as a density ({mass/volume}: ρ |U). As U is now a vector space, we get a quantity integral(: ρ(v).v ←v |U)/integral(ρ) which has the form of a unit-normalised weighted sum of the positions at which ρ is non-zero, which (because the total weight is one) is a well-formed formula for a position; this position is known as the centre of mass of the body. If we change our chart by a translation, or equivalently translate the body through space, without changing orientation, the displaced mass distribution is just (: ρ(v+k) ←v :) for some fixed vector k in U and the value of this unit-weighted sum also changes by k (if the weighted sum's total weight handn't been 1, this would not be true), so changing co-ordinates by the translation for which −k is the weighted sum, we can move the centre of mass of our body to the origin of our co-ordinates.
Written by Eddy.