According to
chapter 4, a *vector* is a literal or graphical representation
of a *specific elementary process* (SEP), which means an
elementary variation of a single attribute, distributed on an
elementary variation of a support attribute, that is an element of a
derived distribution of a primary distribution. The variation takes
place between two S_{0}-type states (two elements of primary
distribution)^{1}07,
*initial state *and *final state*, states which represent
the __asymptotic__ boundaries of the vector object, but at the
same time, they are also its internal references.

The density and
direction of the derived distribution element, which are invariant in
the elementary support interval, are states S_{x} (where *x*
is the rank of the derived distribution) of SEP. As we have seen in
chapter 3, an object is characterized by a set made-up from six basic
properties which form the *general object model*:

P1 Set of the distributed properties;

P2 Type of the support attribute;

P3 Distribution type;

P4 Amount of the support domain;

P5 Internal reference system;

P6 Simultaneous
existence of all these properties, at the same time *t*.

As for the vectors,
they belong to SEP-type, the set of the distributed properties
consists of only a single property (variable attribute). The support
attribute type is mostly the temporal attribute, but it can be other
type as well, such as the spatial one (for example, in case of the
gradient), frequential (applicable in case of the phonetic or musical
processes) etc. As regards the vectors, the distribution type is
always the same, that is the *linear distribution*. The amount
of the support domain is always the same, the domain on which the
distribution linearity is maintained. The internal reference system
(RS) is made-up from the two above-mentioned states S_{x},
which are the marks between which the SEP takes place (asymptotic
boundaries of the internal domain), and these states are defined
against an external reference system (the reference system against
which the internal reference of the primary distribution is being
determined).

As for the actual case of the velocity vectors, the set of the basic properties is:

P1 Variation of the spatial position;

P2 Temporal attribute;

P3 Linear distribution;

P4 Interval on which the density (velocity) is kept even;

P5 Initial or final state (position) between which the position variation takes place;

P6 The
moment *t*_{f}
of the final state, the right-side internal reference of the temporal
interval
,
which represents the SEP support (the left-side
internal reference* t*_{i}
can be also used for the abstract SEP).

All
the abstract objects which have the same model make-up a *class*
of objects. When we are dealing with a set of objects from the same
class, all these objects shall have a set of common properties, that
is even the set which makes-up the class model. Due to distinction
reasons, the concrete objects of the same class (instances) must also
contain specific attributes, besides the common attributes, between
which there are non-zero differences (contrast).

In case of the vectors (but also of other objects), the quantitative contrast is determined by means of a difference between the values of a specific attribute; but, we saw that the relations deployed between two or more objects are relations developed between their internal reference systems, systems evaluated against an external reference system, common for all the set vectors.

The
internal reference system of a vector *k*
(where *k*
is the running number of the vector from the arranged set of *n*
vectors which make-up the vector field) is made-up from the
properties P5, out of which the reference may be either the final
state *S*_{0k}*(t*_{f}*)*,
or the initial state *S*_{0k}*(t*_{i}*)*.
In case of the velocity
fields, defined against an external
RS, the two states (elements of a primary spatial distribution) are
the positions:

(X.17.1)

and:

(X.17.2)

The
state *S*_{1k}*(t*_{f}*)*,
the velocity, is the density of the first rank derived distribution
element of the primary spatial distribution:

(X.17.3)

The velocity field is therefore made-up
from *n* vectors, which are simultaneously found in the interval
(interval
with a right-side internal reference at the moment *t*_{f}),
with disjoint spatial positions, given by the sets
(set
of the application points) and
(the
set of the peak values of the velocity vectors). If we are taking
into account that the spatial velocity is a property of some MS, the
spatial-temporal exclusion principle of MS postulates that both the
set
and
are strict sets, because the spatial positions of more MS at the same
moment *t *cannot
be identical. Otherwise speaking, the spatial positions of the
vectors from the velocity field are external
specific attributes for these objects. The set of the positions
is
a complex abstract object, for which a *natural
internal reference* can be defined, that
is a reference which shall be a common component (since it is a
reference) for all the objects from the set.

The natural internal
reference of a set made-up from *n*
values
is
the arithmetic mean**
**of the set
values, given by the following relation:

(X.17.4)

so that each property from the set is made-up from two components:

(X.17.5)

where
is the common component of all the objects at the
moment *t*_{i}
(the internal natural reference), and
is the *natural
specific property* of the object k
(deviation,
dispersion) at the same moment
(against the internal natural reference).

The natural specific properties have the basic property of having a null internal natural reference (specific properties do not have a common component). Therefore, we shall have:

(X.17.6)

The abstract object
,
given by the relation X.17.4 represents the *global* (general)*
position* of the objects set (of the vector field) at the moment
*t*_{i},
against the external position reference.

Similarly,
at the moment *t*_{f},
we shall have:

(X.17.7)

If , this means that there is an overall motion of the objects set with the following common (global) velocity:

(X.17.8)

which is also a natural reference of the velocity field, against which a set of natural specific velocity rates shall result. All of these are coming through the derivation of the relation X.17.5 in relation with the time, resulting that:

(X.17.9)

The global motion can be also time
variable, consequently, a global acceleration
could be deployed, a global variation of the acceleration, so on. The
most relevant fact is that all the global amounts, because they are
considered as natural references, are also common properties to all
the vectors from the vectorial field.

107
This assertion is applicable for the first rank derived
distributions. In general, a P_{n}-type
SEP has two
-type as its boundaries.

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