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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).
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