divergence

Archivum mathematicum	239 - 240
S.B. Karavashkin

p. 239 The presence of this phase delay causes that the momentary values of (x, t) amplitude at the opposite surfaces are different. Because of it, the flux becomes dependent on the size of picked out region. It evidences that, when finding the divergence in dynamical fields, in general case, one should take into account the time characteristic of the flux. And when passing to the stationary fields, i.e. at 0 and/or c , the right part of (13) automatically vanishes, coming to the complete accord with conventional concept (1). 3. The complete proof of the divergence theorem in dynamical fields Generalising the above consideration of 1D flux, consider a general case of arbitrary flux of vector (, t). Let in some connective source-free space propagate some flux whose vector (, t) coincides with the direction of flux, as shown in Fig. 4; its time dependence is
	(14)
Fig. 4. General form of the time-variable field tube of the flux
To find the divergence of this vector, we will use, as before, a conventional definition (2) and the same technique to find the flux through the picked out region, but taking into account the presence of phase of the wave space-delay. Pick out of some region V, so that its ends coincide with the equiphase surfaces of flux, and its lateral surfaces – with the force tubes of current. Then, accounting that the wave propagation velocity is finite, we can write
	(15)
where l is the length of a picked out force tube; t is the wave phase delay. Thus, basing on (15) and taking into account the particular case considered above, we have set up a relation between the length of the picked out region and the phase delay. We will take this fact into account in the further studying. According to the statement of problem, the surface S consists of three components: S = S₁ + S₂ + S_l (where S₁, S₂ are the end surfaces, and S_l is the lateral surface of the picked out region). Taking into account the statement of problem and the results of previous investigation, the complete flux through the surface S is p. 240
	(16)
where (, t) = (, t) - (, t - t). In (16) we accounted at once the time shift of vector function, as well as the absence of flux through the lateral surface. The first integral of the right-part sum in (16) has not a phase shift t. In the source-free field it vanishes, since in this case the condition for the divergence of stationary vector function becomes true. The second right-hand integral in (16) in general case is non-zero and can be easy transformed into the integral over the space. For it, we will divide the picked out region into p small regions whose height (along the lines of current) is

After it, we can write the under-integral expression (, t) as
	(17)
where _i(, t) = (, t - (i - 1)t) - (, t- i t); in this case 1 i p. Taking the limit t 0 in (17), we come to the integral
	(18)
Substituting (18) into (16) and knowing that, according to Fig. 4, at the boundary S₂ the vector ₂, and ₂ coincides with the vector of flux , we obtain the required:
	(19)
Substituting (19) into (2), we come to the final expression for the divergence of vector:
	(20)