This annex shows as
a remember the most frequent relations from the theory of vector
fields which are used throughout the entire paper, with the
specification that the notion’s denominations are the ones __from
the mathematics field__, some of them being redefined in the
present paper.

**The gradient**of a scalar field***(x,y,z)*is a vector given by the following relation:

(X.8.1)

where
are
the versors of the
axes **X,Y,Z**.

**The elementary flux**of the vector is named the product , where is the oriented area element (). If the area element surrounds a point*P(x, y, z)*, then, the elementary flux will be in point*P*.

**Total**(global)**flux**of the vector through any surface is:

(X.8.2)

The total flux

**productivity**of the volume . The ratio is the**average productivity**of the volume unit and the limit of this ratio when all the points of the surface tend to an internal point*P,*it is named the**divergence**of the vector fieldin the point**V***P*.

(X.8.3)

Under the assumption that the partial derivatives of are continuous in P, there is a limit which can be expressed by means of:

(X.8.4)

**The curl***C**curl*of the field in the point*P*. Therefore,

(X.8.5)

where

(X.8.6)

Circulation
on the curve *C*
goes directly against
(rule of the right screw). The curl
may be also written as a symbolic
determinant:

(X.8.7)

**The divergence’s integral formula**(Gauss-Ostrogradski):

(X.8.8)

where
is the confined area of the volume
.
The sense of the normal line on the oriented surface is considered to
be positive __to the outside__.
Other two relations are coming from the relation X.8.8, that is the
**curl’s****
integral formula**:

(X.8.9)

which may be also written as:

(X.8.10)

and **the gradient’s integral
formula**:

(X.8.11)

**Stokes’s formula**:

(X.8.12)

where
is any surface limited by the confined curve *C*.

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