## X.3.5 Vectors

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 V1 and V2 which have a common application point S1,2.

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 (S1,2), or the vector V2 is moved in parallel with the origin into S12, that is the apex of the vector V1. Processually speaking, if the two vectors are SEP distributed on a temporal support (such as the real processes), the fact that the vectors V1 and V2 have an initial common state (S1,2) means that the two represented SEP are simultaneous, and if V2 has the initial state equal with the final state of V1 (S12), then, the two SEP are successive. In both situations, the resulting vector VR 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).