Let us assume that
the amount (property) of conveying, a scalar *M*, is presented
at a certain moment as a continuous 3D spatial distribution included
into *V* volume, with a certain position against a reference
system. It’s worth reminding that a spatial distribution of an
amount is also known as *field*, in mathematics; depending on
the distributed quantity (either scalar or vectorial), we shall have
a scalar or vectorial field.

In our case, the
scalar field of amount *M* is defined as a primary spatial
distribution, in which each element of volume *dV *on the
coordinates *x,y,z* is related to the scalar quantity:

* *
(5.2.1.1)

where
is the volume density of the continuous and even distribution of
amount *M*, on the volume element *dV*, which tends to zero
and “surrounds” a point with the position vector
.
As we may notice from the above-mentioned relation, the distribution
is static for the moment, the spatial coordinates of the distribution
do not depend on time. Under these static conditions, the total
quantity^{2}1
of amount *M* from* *volume *V* makes-up the *stockpile*
of this amount (see stockpile definition from chapter 2).

If the distribution
is moving, the position of each distribution element shall become
variable. In the overview of this chapter, it was mentioned that the
motion process of an object is characterized by *the* *motion
intensity*, as an existential attribute, which is also better
known as the velocity modulus, velocity which, according to the
classic theory^{2}2,
is given by the following relation:

(5.2.1.2)

The motion of a
continuous distribution is therefore represented by means of the
connection (association) of a vector displaying the motion intensity
and direction to each assigned value (more exactly, at each element
of the spatial primary distribution). However, in each point
(virtual, dimensionless point of the distribution), there is an
amount
which moves at the same time with it, therefore, we shall finally
have another amount which represents the transport process of the
attribute *M* in a certain point.

**Definition
5.2.1.1:** The** flux density vector** (FDV) of an amount M is
the following quantity:

(5.2.1.3)

Comment 5.2.1.1: According to the virtual approach (based on dimensionless points) of the flux, due to the infinite number of these points in any spatial interval, a nonsense occurs concerning the value of the distributed amount on each support element of primary distribution - a singular value from set {R}, namely an AAV. If a support interval, regardless its size, consists of an infinity of singular values, the logical conclusion is that a finite quantity of the assigned attribute, divided to an infinite number of singular values leads to a null value of the attribute associated to each singular value. But the question is how such a null value can lead, by means of addition (integration) to a result different from zero? This nonsense vanishes within the realizable objectual approach, where the support of any distribution contains a finite number of singular normal values.

This new amount - FDV - is a local
characteristic of the motion of a spatial distribution. One may
notice that through the attachment of a vector which represent the
transport velocity to each element of the primary distribution from
the volume *V* (occupied by the initial scalar distribution *M*),
the scalar field becomes a vectorial field, the distributed amount
being this time FDV of the amount *M*.

According to the vectors classification
made in chapter 4, FDV is a *carrier vector*, which conveys the
amount
attached to its application point, from the initial point, up to the
final point of the flux line (which shall be defined a little bit
later). The modulus of this vector represents the local
surface-density (see annex X.15) of the transport process, the
existential attribute of the abstract object *flux* in a virtual
point from the space.

In accordance with the virtual approach (and with the realizable one), there are two ways for the approach of this vector field, variable both in time and space, which is considered as a representation of a flux:

Analysis on the motion of

__a single object__(in this case, of a single volume element of the scalar distribution)__on a temporal interval__of the flux existence;Analysis on the spatial distribution of the

__ensemble__of objects belonging to the vector field (namely, of FDV set)__at a certain moment__, after the flux initiation.

The first method, also known as
Lagrange method, provides a *temporal distribution* of the
spatial position of a single object involved in the flux,
distribution which according to the current language is called
*trajectory* (pathway)*.* This pathway of a carrier vector
is, according to the classic theory, a continuous curve in space, at
which that particular vector remains permanently tangent, and which
is called a *flux line* (or a flow line).

The second method, also known as Euler
method, provides a snapshot at *t*_{k} moment of
the total or partly *spatial distribution* of the set of objects
which are in motion. This snapshot is a *global state* of the
flux, at *t*_{k} moment, a steady vector field in
2D or 3D space. If this vector field is crossed by a steady surface Σ
with a known equation (distribution relation) and area (reference
area), we shall get the *superficial* distribution of FDV of
that particular flux at the moment *t*_{k}, on
the surface Σ.

**Definition
5.2.1.2:** The area of a normal Σ surface in any point on
and which contains the set of __all__ the flux lines of the flux
**F**, is named **effective area** (synonym - **effective
section**) of this flux.

Considering the local elements of this distribution: the position vector of a point located on Σ, “surrounded” by an area element , , normal line on Σ in that point and , FDV in the same point (given by the relation 5.2.1.3). For simplifying these relations, we shall not mention their spatial coordinates, knowing that these coordinates make-up an invariant distribution for the points on Σ. According to the vector field theory, there is the following relation (see annex X.8):

(5.2.1.4)

which defines the *flux* of
vector
through the area Σ. According to this interpretation version,
the flux is a __scalar__ and represents the quantity from the
amount *M* which crosses the area Σ in the temporal
interval *dt.*

**Attention! **__This flux
definition is not valid according to the objectual philosophy__,
because the definition 5.2.1 is the valid one in this paper, which
states that the flux is a distributed *process*, that is a
vector field rather than a scalar one. In exchange, the relation
5.2.1.4 is also valid in this paper, but it defines the * global
flux intensity* through the area Σ.

Comment
5.2.1.2: As it is also stated in annex X.3, as compared to the
mathematic version of the flux definition of a vector, the concept of
flux cannot be approached in this paper if there is no displaceable
amount. Based on the mathematic version, the vector whose flux is
computed may be any free vector, such as the velocity
;
according to the objectual version, if the vector
is not attached to a displaceable scalar amount (
in the above-mentioned case) the flux concept does not make sense any
more. In accordance with the objectual version, the vector
,
whose flux must be computed, is always a __carrier vector__.

Under the above-mentioned conditions,
if the intersection of the volume *dV* with a calculus area Σ
is
,
knowing that the inner vector distribution is even, an * elementary
flux* results, a uniform vector distribution of FDV, with an
effective area
.
The vector distribution is uniform if both the modulus and the
vectors direction are invariant (field with parallel vectors
belonging to the same modulus).

21 We are dealing with quantity and stockpile only in case of the cumulative attributes (extensive), for which the addition, respectively, integration make sense.

22 Theory which allows the simultaneous existence of the position and velocity of an object at the same time.

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