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 Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]: Year: Volume: Issue: Page: Find

 Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2015, Volume 19, Number 4, Pages 680–696 (Mi vsgtu1382)

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

DOI: https://doi.org/10.14498/vsgtu1382

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Bibliographic databases:

UDC: 517.958:[535+537.812]
MSC: 78A25, 83C50
Original article submitted 19/XII/2014
revision submitted – 19/II/2015

Citation: G. G. Islamov, “On a class of vector fields”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 19:4 (2015), 680–696

Citation in format AMSBIB
\Bibitem{Isl15} \by G.~G.~Islamov \paper On a class of vector fields \jour Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.] \yr 2015 \vol 19 \issue 4 \pages 680--696 \mathnet{http://mi.mathnet.ru/vsgtu1382} \crossref{https://doi.org/10.14498/vsgtu1382} \zmath{https://zbmath.org/?q=an:06969187} \elib{http://elibrary.ru/item.asp?id=25687496}