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Theorem of curl of a potential vector

Sergey B. Karavashkin and Olga N. Karavashkina

Special Laboratory for Fundamental Elaboration SELF

e-mail: selftrans@yandex.ru , selflab@mail.ru

Abstract

Here we study the circulation and curl of potential vector in dynamical fields. We prove the theorem that in dynamical fields the curl of potential vector is proportional to the vector product of unit vector of flux 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 finite velocity of wave space-propagation. We consider the applications of this theorem to acoustic and electromagnetic fields. We describe the results of experimental studying the transversal acoustic waves in gas medium, which corroborates the possibility to form the transversal wave by way of linear superposition of potential dynamical fields.

Keywords: theoretical physics, mathematical physics, wave physics, vector algebra, acoustics, electromagnetic theory, dynamical potential fields.

Classification by MSC 2000: 76A02, 76B47, 76N15, 76Q05, 78A02, 78A25, 78A40.

Classification by PASC 2001: 03.50.-z; 03.50.De; 41.20.Jb; 43.20.+g; 43.90.+v; 46.25.Cc; 46.40.Cd

1. Introduction

One of principal peculiarities of vector fields is that ''if the divergence and curl of field ()  have been determined at each point  ()  of region V, then everywhere in V  the function ()  can be presented as the sum of vortex-free field ₁()  and of solenoidal field ₂() :

(the Helmholtz decomposition theorem)'' [1, p.173].

We can say that the equation

''is the sufficient condition for the conservative field, i.e., for description of the field by the gradient of some potential function'' [2, p. 89]. In other words, if ()  is potential, then in the studied domain the equality (2) is true unconditionally.

Note that the definition and the conventional theorem of curl of vector have been formulated for stationary fields in which the vector depends only on co-ordinates, not on time. We can easy check it, seeing the argument on which ()  depends in the conventional definition and theorem. Particularly, in the Stokes theorem: ''If the vector function  ()  was one-valued and had the continuous particular derivatives everywhere in the finite surface one-connective region V ₁, and if the surface S  belonging to the region V ₁   is one-connective, continuous and bounded by a regular closed curve C, then

i.e., the curvilinear integral of  ()  about a closed path C is equal to the flux of vector () through the surface tightened on the path C'' [2, p.171].

Usually this potentiality definition is extended also to dynamical fields, in that number to complex acoustic fields in gas medium. As is known, ''in liquids and gases the elastic resistance to the transversal displacement of particles does not exist, only to the volume variation, i.e. to the compression or rarefaction. So in such substances only longitudinal waves can propagate'' [3, p.114]. ''Thus we have in the vector form

(where   is the displacement velocity of molecules of elementary gas region, and is the acoustic potential), which means the Stokes theorem: the curl of gradient is zero identically. Hence, the velocity potential technique is applicable only to the vortex-free motions. But acoustic motion in liquids and gases always stays vortexless, even if taking into account the viscosity effect. We can show that near the solid bound the vortex layer exists always, but this boundary layer is extremely thin'' [4, p.392].