3D Diagonalization and Supplementation of Maxwell's Magneto-Static Field Equations in Fully Anisotropic and Inhomogeneous Media - Part I: Proof of Existence by Construction

Consider the Maxwell's magneto-static equations ∇×H=J and ∇·B=0 in fully anisotropic and inhomogeneous media characterized by the 3 × 3 positive-definite permeability matrix μ
^3×3
(x, y, z). This paper establishes - for the first time - that the aforementioned system of the governing and constitutive equations can be diagonalized with respect to the arbitrarily chosen z-axis leading to the Dc - and the associated supplementary Sc -forms. The existence of the (Dc,Sc) -forms is demonstrated by construction. The accompanying paper (Part II) proves the internal consistency of the (Dc,Sc) -forms rigorously by showing the sharp equivalence of the (Dc, Sc) -forms with the originating Maxwell's and constitutive equations.