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Basic mathematics



The scalar product of two vectors

 and    is a scalar.

Its value is:

.              (1.1)

 .                (1.2)

The scalar product is commutative:

.                            (1.3)

The vectorial product of two vectors and is a vector perpendicular on the plane determined by those vectors, directed in such a manner that the trihedral ,and should be rectangular.

.                     (1.4)

The modulus of the vectorial product is given by the relation:


.                    (1.5)

The vectorial product is non-commutative:

                          (1.6)

The mixed product of three vectors , and  is a scalar.

.                   (1.7)

The double vectorial product of three vectors , and  is a vector situated in the plane .

The formula of the double vectorial product:

.                (1.8)

The operator is defined by:

.                   (1.9)

 applied to a scalar is called gradient.

.                 (1.10)

 scalary applied to a vector is called divarication.


.                   (1.11)

 vectorially applied to a vector is called rotor.

.                   (1.12)

Operations with :

.                    (1.13)

.                     (1.14)

.                 (1.15)

When  acts upon a product:

-         in the first place has differential and only then vectorial proprieties;

-         all the vectors or the scalars upon which it doesn’t act must, in the end, be placed in front of the operator;

-         it mustn’t be placed alone at the end.




.     (1.16)

.        (1.17)

. (1.18)

,                (1.19)

,              (1.20)

,                 (1.21)

,                (1.22)

.    (1.23)

- the scalar  considered constant,

- the scalar  considered constant,

- the vector  considered constant,

- the vector  considered constant.

If:

                            (1.24)

then:

.                 (1.25)

The streamline is a curve tangent in each of its points to the velocity vector of the corresponding point .

The equation of the streamline is obtained by writing that the tangent to streamline is parallel to the vector velocity in its corresponding point:

.                         (1.26)

The whirl line is a curve tangent in each of its points to the whirl vector of the corresponding point .

.                           (1.27)

The equation of the whirl line is obtained by writing that the tangent to whirl line is parallel with the vector whirl in its corresponding point:

.                        (1.28)

Gauss-Ostrogradski’s relation:

,                      (1.29)

where - volume  delimited by surface .

The circulation of velocity on a curve (C) is defined by:

                            (1.30)

in which

                        (1.31)

represents the orientated element of the curve (- the versor of the tangent to the curve (C )).

Fig.1.1

                  (1.32)

The sense of circulation depends on the admitted sense in covering the curve.

.                   (1.33)

Also:

.                 (1.34)

Stokes’ relation:

              (1.35)

in which  represents the versor of the normal to the arbitrary surface  bordered by the curve (C).








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