3D Diagonalization and Supplementation of Electrostatic Field Equations in Fully Anisotropic and Inhomogeneous Media Proof of Existence and Consistency

Abstract

Consider Maxwell’s homogeneous curl equation divergence × E = 0 for the electric field vector E and the inhomogeneous divergence equation divergence · D = rho for the dielectric displacement vector D and the charge density function rho in the static limit. Assume an (x, y, z)–Cartesian coordinate system. Consider the constitutive equation D = epsilon E, with the 3 × 3 position-dependent positive-definite permittivity matrix epsilon (x, y, z) modeling fully anisotropic and inhomogeneous dielectric media. This paper proves that divergence × E = 0 and divergence · D = rho along with D = epsilon E are diagonalizable with respect to the arbitrarily chosen z–axis leading to the Dc–form. The existence of an associated supplementary equation, the Sc–form, has also been demonstrated. Finally, it is shown that the constructed (Dc, Sc)–forms are sharply equivalent with the originating set of equations divergence × E = 0, divergence · D = rho, and D = epsilon E, and, thus, internally consistent. The proof scheme is relative in the sense that its validity hinges on the consistency of Maxwell’s equations in the static limit and the material realizability conditions.