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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 noncommutative:
_{}
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)
GaussOstrogradskis 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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