So far, we have discussed only about
distributions which had a single type of support attribute (a single
independent variable) because it was the most simple case of support
and only the clear understanding of the *distribution *__concept__
was important. If the distributed attribute *y* simultaneously
depends on many variables
(independent
both in relation to *y* and each other), relation 2.2.1 (based
on the simplifying assumption that it is a function) may be written
as it follows:

(2.6.1)

In this case, we
are dealing with the classic situation of a continuous function with
many variables. As for the multiple support distributions, it is
important to understand that this support is made-up from the union
of *n *individual ranges of each variable, each combination of
distinct singular values which may be assigned to the *n*
variables being related to a single value of *y*. Otherwise
speaking, an element belonging to this distribution (in case of the
primary realizable distribution) is made-up from a normal value *y*
associated through relation *f* to *n* normal
__simultaneously-existent__ values of the multiple support. In
such case of a multiple support (also known as multidimensional, such
is, for instance, the Euclidean 3D space), the relations 2.2.5 and
2.2.8 are also multiplied by *n* folds. Therefore, we shall
have:

, , … (2.6.2)

relations which display the *partial*
density values of the primary distribution elements, where
are
linked by means of the following relation:

(2.6.3)

are the fractions belonging to the
value of the distributed attribute *y*_{k} which
correspond to each support variable. The same expression shall be
written for the density of the first rank derivative distribution:

, ... (2.6.4)

where are the specific variations of the distributed amount due to the corresponding variations of the support elements (variables), the specific variations being the elements of the total variation:

(2.6.5)

It must be
mentioned that the density values given by the relations 2.6.4 are
achieved under the conditions of a total invariance of the other *n-1*
support variables. These density values are the equivalent of the
first rank partial derivatives from the differential calculus and the
relation 2.6.5 is the equivalent of the total differential of the
function *f *of variables *n.* The high-rank density values
of the distributions and the variations of the same rank of the
distributed attribute shall be approached again in the following
chapters.

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