3

Theorem of curl of a potential vector

2. Preliminary analysis

As a preliminary analysis, consider a simplified model of 1D potential flux of vector (, t)  whose general form is shown at the top of Fig. 2.

Let in some one-connective domain , free of vortexes and currents, in direction of the axis x propagate 1D homogeneous flux, whose vector (, t) is normal to the direction of flux and the axis z. In the meanwhile we will not determine the flux as potential or vortex. For the present it will be important only that the force vector of flux is normal to the propagation direction. The item of vector solenoidality we will consider after having worked out the necessary basis.

We will suppose also that the disturbance propagation velocity is finite, and the function F(x, t) has a form

We will use again the technique used in [10] to study the divergence of vector in dynamical fields. Pick out of the studied region   three paths  ABD₁E₁ , ABD₂E₂ , ABD₃E₃  having the common side AB, as this is shown in Fig. 2. To visualise, choose the distance between the sides  D₁E₁ , D₂E₂ , D₃E₃   and the side AB equal to  2/3k ; /k ;  4/3k  relatively.

At these conditions, take the standard definition of curl of vector which, being general, must be true both for stationary and dynamical fields. This definition is the following (see, e.g., [1, p.83], [11, p.116]):

where   is the flux vector; P   is some point of the surface ;   is the perpendicular to the surface ; L   is the closed path bounding the surface ; d is some infinitesimal vector element of the length of path L.

For the considered problem, (9) can be written as

where  S_i   is the cross sectional area of the ith path picked out;  i = 1, 2, 3  for ABD₁E₁ , ABD₂E₂ , ABD₃E₃   relatively.

As the paths picked out are finite, we first will consider the reduced circulation R_i :

is the circulation of vector (, t)  about the path L_i .