The scalar product of two vectors
and is a scalar.
Its value is:
The scalar product is commutative:
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.
scalary applied to a vector is called divarication.
vectorially applied to a vector is called rotor.
Operations with :
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.
the scalar considered constant,
- the scalar considered constant,
- the vector considered constant,
- the vector considered constant.
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:
The whirl line is a curve tangent in each of its points to the whirl vector of the corresponding point .
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:
where - volume delimited by surface .
The circulation of velocity on a curve (C) is defined by:
represents the orientated element of the curve (- the versor of the tangent to the curve (C )).
The sense of circulation depends on the admitted sense in covering the curve.
in which represents the versor of the normal to the arbitrary surface bordered by the curve (C).
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