9

Theorem of curl of a potential vector

6. Corollaries of the curl theorem

The main corollary of the proved theorem is that when transiting to the stationary fields, the right part of (25) automatically vanishes. With it the curl of vector becomes time-independent and takes the standard form. Hence, the proved theorem is generalising for that known and applicable both to dynamical and stationary fields.

The same transformation of (25) takes place in case of vector being longitudinal to the flux direction. Irrespectively of dynamical pattern of field, the right part of this expression vanishes also. Therefore the revealed peculiarities of circulation and curl of the vector are true only and exceptionally for the perpendicular component of the vector of dynamical flux.

We should mark that the non-zero value of circulation and curl of vector does not mean that in dynamical fields the potential vectors transform into solenoidal. As we said above, the non-zero value of curl is caused by the finite velocity of wave processes propagation in space that does not require the transformation of the vector per se. The vector remains potential and has the gradient being the principal peculiarity of vector potentiality. True, we should define more exactly that the gradient of given vector has all features inherent in dynamical fields.

For solenoidal vectors the proved theorem is also true. Actually, according to (20), in dynamical fields the circulation of vector can be presented in the form of two summands. The first of them describes the stationary vortex process disregarding the finite velocity of wave space-propagation. For a potential vector this summand is equal to zero, but for a solenoidal one it has some finite value. The second summand notes that the propagation velocity is finite. If the solenoidal vector of field was a delaying function, then the second summand also does not vanish. Thus, for solenoidal vectors, the right part of (20) - and consequently of (25) - consists of two summands. The first of them describes the solenoidal pattern of vector, and the second determines its dynamical pattern.

And the last important corollary of the proved theorem is that the resulting expression, describing the curl of potential vector being normal to the flux propagation direction, has acquired the four-dimensional form. The solution of given differential equation is a delaying function. Consequently, for transversal fields the wave properties are determined by the differential equation not of the second order, as conventional, but of the first order. And the solutions naturally satisfy the wave equation also.

7. Conclusions

As the result of carried out investigation, we have proved that in dynamical fields the curl of potential vector is proportional to the vector product of an unit vector of flux propagation direction by the particular derivative of flux of vector with respect to time. With it, the vector remains its potential pattern, since the circulation is conditioned by the finiteness of wave space-propagation velocity.

For solenoidal vectors, the curl of vector depends additionally also on the vortical properties of vector.

The proved theorem is generalising for that known for stationary fields.

The wave pattern of obtained regularity retains only for the vector component being transversal to the direction of flux propagation. For the longitudinal component, the curl of vector is zero, irrespectively of the dynamical properties of field.

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