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Transformation of divergence theorem

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Transformation of divergence theorem in dynamical fields

Sergey B. Karavashkin, Special Laboratory for Fundamental Elaboration SELF

187 apt., 38 bldg., Prospect Gagarina, Kharkov, 61140, Ukraine

Phone: 38 (0572) 276624; e-mail: selflab@go.com

Abstract

Keywords: Theoretical physics, Mathematical physics, Wave physics, Vector algebra.

Classnames by MSC 2000: 76A02, 78A02, 78A25, 78A40

1.   Introduction

In the number of mentioned basic concepts, the finding of divergence of vector is an unalienable part of the EM field theory formalism. Using it, we express the conservation laws of charge, current, flux, energy, etc. Using the theorems basing on it, we develop the methods to study the distribution, propagation, attenuation of EM processes.

It is thought to be absolutely proved that in a charge-free region

where (, t)  is some vector whose parameters depend on co-ordinates and time. Rather, in the initial formulation of Poisson theorem, (, t) does not depend on time t, since “the operation (divergence of ) relates to the sources of the vector field ()” [2, p.29], not (, t).

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The more, the initial definition of the vector flux, on whose basis the divergence concept is formulated, also means the field being stationary. Particularly, in [1, p.91], when formulating its definition, it is supposed that “the vector flux depends on the size of surface, on the value of vector (P), and on the direction of this vector relatively to the perpendicular to the surface”. It means, in the initial definitions, the time dependence for the flux is absent also.

Necessity to take the divergence of flux in dynamical fields requires to broaden the domain, where the definition of divergence is true. Due to it, also for the vectors (, t), by default, its main definition was recognised valid in the following form: “The divergence of vector function of the point () is scalar function of the point, defined as

where V₁ is the region containing the point (); S₁ is the closed surface bounding the region V₁;   is the most distance from the point () to the point on the surface S₁ ” [3, p.166]. With it, also by default, all the theorems basing on the definition of divergence were kept unchanged, in that number the mentioned Poisson theorem.

As a result, accounting the electrical vector and magnetic vector   time-dependent, Poisson equation in the form (1) was included to the Maxwell system for dynamical fields free of charges and currents. Particularly, Landau writes: “The electromagnetic field in vacuum is defined by Maxwell equations in which we have to put = 0, j = 0 . Write them down again:

Solutions of these equations can be non-zero. It means that an electromagnetic field can exist even in absence of whatever charges” [4, p.143].

However, the force lines of stationary and dynamical fields in general case essentially differ. This is well-known not only in case of electrodynamics, but e.g. in hydrodynamics: “In general case, the force lines do not coincide with the trajectories. The family of lines of current xⁱ = xⁱ(c¹, c², c³, , t) (where  c¹, c², c³  are the generalised parameters and   (s, L)   is some function of the lines of current L  at the length of the arc s   along the line of current - S.K.) is time-dependent and different at different moments. However, the parameter  t  is included to the right parts of the differential equations of lines of current and of the differential equations determining the regularity of motion or trajectories of particles – only in the case of unsteady motions. In the case of steady motions, the difference between these equations disappears“ [5, p.41].