Basic mathematics

The scalar product of two vectors

and is a scalar.

Its value is:

.

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.

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

The vectorial product is non-commutative:

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

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

The formula of the double vectorial product:

The operator is defined by:

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.

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 doesnt act must, in the end, be placed in front of the operator;

it mustnt 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-Ostrogradskis 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).