Tensors are taught mainly in a context of tensor fields and in fact usually for curved spaces where they have no competition. In the context of tensor fields, the reals that as a matrix compose the tensor, depend on which point in the field is examined, and also the orientation of the coordinate system at that point. In our tensor moments there is no field and thus no dependency on which point the value is for. The reals in the tensor moments do depend on the coordinate system however. They also depend on the vantage point about which moments are taken and in these notes that point is always the origin of the coordinate system.
In a coordinate free description of moments it is still necessary to nominate a vantage point about which the moments are taken. The center of gravity is a coordinate free concept which, together with the total mass, captures the information of first moments. I do not now know how to capture coordinate free concepts for 2nd moments but perhaps the ellipsoid of inertia does this.
I do not know how to define moments except in flat spaces. Tensors moments are not fields which depend on what point in space you are at, but depend instead on properties of bodies, or mass distributions. The values mjk depend on the location of the origin of coordinates as well as the basis vectors for the coordinate system.
If we translate a coordinate system leaving the mass distribution fixed (x'j = xj + dj), then our moments change as follows:
m' = m
m'j = ∫ (xj + dj)dm = mj + mdj.
This is unlike any relation in tensor fields.
m'jk = ∫ (xj + dj)(xk + dk)dm = mjk + djmk + dkmj + djdkm.
An oblique coordinate system has a metric tensor that is not a function of the coordinates.
x'i = ΣjAijxj
m' = det(A)m (See Jacobian)
m'j = ∫ x'jdm = ∫Aijxjdm
At this point we must unpack our dm as ρ∏dxj where ρ is the density expressed in the current coordinate system. ρ is a scalar field but it depends on the coordinate system thus: the number of grams per cubic inch is different from the number of grams per cm3. ρ gets an optional prime as well. This contravenes a venerable physics tradition that variables such as ρ are to be taken as denoting lengths rather than number of standard length units. In one of my physics classes it was a flunking offense to write ρ = 1.6 instead of ρ = 1.6 g cm–3. But in tensor analysis this tradition causes great difficulties as in ‘tensor densities’. A component of a tensor is a real number!