As it was mentioned
in chapter 4, the mathematical representation of the *specific
elementary processes* (SEP) shall be made by means of vectors. As
for the concepts of *object *and* process*, which are basic
notions in the present paper, the *vector* definition is a
little bit different as compared to the one postulated by the classic
mathematics, and even comparatively to the one taught in the higher
education system. In this paper, the vectors are defined as even
directional quantitative variations of a single qualitative
attribute, deployed between two states: the *initial state*
(which corresponds with the application point) and the *final state*
(which corresponds with the vector’s apex). At the same time,
the vectors are processual objects which represent the total amount
of the invariant attributes (on their support domain) of specific SEP
(application point, direction, modulus etc.).

A special case when
the vector’s definition mentioned in this paper is different
from the currently-issued works is the case of carrier vectors, which
are components of the mathematic model for fluxes. According to the
existing works, a carrier (or slidable) vector is that vector with a
mobile application point. Under the meaning of the objectual
philosophy, this description is also completed by the transported
attribute because a carrier vector must “carry” (to be
attached) an amount which shall be moving once with the vector, just
as the flux density vector (FDV) is associated with the scalar
density *ρ* of the transporting amount.

Given the fact that
in case of SEP with spatial support, the vectors are the only
representative means for the *direction* attribute, there is a
class of vectors specialized in this field - the unit vectors - whose
modulus is always unitary and which are used as __direction
references__ (parts of the reference R systems), either for the
external RS or for the internal RS, or for the local RS (such as the
normal, tangent and bi-normal line into a point of a spatial curve).

Due to the specific
way of defining the vectors in this paper, some computing “artifices”
used in the geometric representation of the vectors from the classic
vector calculus must be regarded with discernment. For example, fig.
X.3.5.1 shows a classic operation of adding two vectors **V**_{1
}and **V**_{2}** **which have a common
application point S_{1,2}.

*Fig. X.3.5.1*

According to the
classic vector calculus (where the vectors are considered free), the
sum of the two vectors is the diagonal of the parallelogram which
have its sides in those vectors; in this case, it is no difference in
the result if the two vectors have the origin in the same point
(S_{1,2}), or the vector **V**_{2} is moved in
parallel with the origin into S_{12,} that is the apex of the
vector **V**_{1}. Processually speaking, if the two
vectors are SEP distributed on a temporal support (such as the real
processes), the fact that the vectors **V**_{1} and **V**_{2}
have an initial common state (S_{1,2}) means that the two
represented SEP are __simultaneous__, and if **V**_{2}**
**has the initial state equal with the final state of **V**_{1}**
**(S_{12}), then, the two SEP are __successive__. In
both situations, the resulting vector **V**_{R} has a
different temporal support (the amplitude, modulus of the vector is
the same in both situations but the temporal density of the resulting
SEP has another value).

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