Let us consider a 3D even, finite and
invariant spatial distribution of an attribute, such as, for
instance, the mass density
.
In relation to an external reference **X,Y,Z**,
each DP_{i}
of the distribution (the primary distribution element) has the
position vector
,
according to the figure X.20.1.

*Fig. X.20.1*

The common component of the set of
position vectors is also a position vector which shall be written as
.
In relation to this unique and invariant (as internal position)
vector of the distribution, each position vector
has a differential (specific) component
.
If by convention, the following rule is settled for the set of the
distribution elements: *the common
component of the set of specific components is null*
(nonexistent), then, this condition transposed in the example shown
in the figure X.20.1 becomes:

(X.20.1)

which if it is written by components, it means:

, , (X.20.2)

If
we presume that 3D distribution contains *n*
elements, namely, *i=[1,n]*,
the relations X.20.2 may be also written as:

, , (X.20.3)

hence the result is:

, , (X.20.4)

the notorious relations which define
the coordinates of the mass-center (or the weight-center) of a
material object with __even distribution__ of the mass density.

This means that the
center’s position represents the common component of the
spatial positions of the distribution elements set, that is an
abstract object which represents at the same time an *internal
natural reference* T of that distribution (against which the
specific components of each element are being estimated). The
relations X.20.4 are applicable for any discrete distribution, such
as, for instance, the finite sets of numerical values; in this case,
the common component (the natural internal reference) of these sets
is the *mean arithmetical value*. Also, the relations X.20.4
justify the relation X.20.1 (which was used before its
justification), fact which shows that the value of the common
component is null, only if the sum of its individual (specific)
values is null (assertion applicable for the quantitative
attributes).

Comment X.20.1: It must be emphasized that both the common component of a set of objects and the other reference values, either natural or artificial, are abstract objects without any correspondence to the objects of the set used for their determination (they are external references to these objects). If we are talking about the set of vectors from the figure X.20.1, is a position vector belonging to an imaginary point (there is no element of the distribution placed on that position, unless it was randomly arranged like that).

Copyright © 2006-2011 Aurel Rusu. All rights reserved.