This article is cited in 3 papers

Differential Equations and Mathematical Physics

On a class of vector fields

G. G. Islamov

Udmurt State University, Izhevsk, 426034, Russian Federation

Abstract: It is shown that a simple postulate “The displacement field of the vacuum is a normalized electric field”, is equivalent to three parametric representation of the displacement field of the vacuum:
$$ u(x;t) = P(x) \cos k(x)t + Q(x) \sin k(x)t. $$
Here $t$ — time; $k(x)$ — frequency vibrations at the point of three-dimensional Euclidean space; $P(x), Q(x)$ — a pair of stationary orthonormal vector fields; $(k,P, Q)$ — parameter list of the displacement field. In this case, the normalization factor has dimension $T^{-2}$. The speed of the displacement field
$$ v(x;t) = \frac{\partial u(x;t)}{\partial t} = k(x)(Q(x) \cos k(x)t - P(x) \sin k(x)t). $$
The electric field corresponding to this distribution of the displacement field of vacuum, is given by the formula
$$ E(x;t) = -\frac{\partial v(x;t)}{\partial t} = k^2(x)u(x;t). $$
Moreover, the magnetic induction
$$ B(x;t) = \mathop{\mathrm{rot }} v(x; t). $$
These constructions are used in the determination of local and global solutions of Maxwell's equations describing the dynamics of electromagnetic fields.

Keywords: local and global solutions of Maxwell's equations, spectral problem for rotor operator, the small flow of the displacement field.

UDC: 517.958:[535+537.812]

MSC: 78A25, 83C50

Original article submitted 19/XII/2014
revision submitted – 19/II/2015

DOI: 10.14498/vsgtu1382

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