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 $\nabla\times \mathbf{H}=\mathbf{J}$ and $\nabla\cdot \mathrm{B}=0$ in fully anisotropic and inhomogeneous media characterized by the 3 × 3 positive-definite permeability matrix ${\mu}^{3\times 3}(x,\ y,\ z)$ . 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 $\mathcal{D}_{c}$ - and the associated supplementary $\mathcal{S}_{c}$ -forms. The existence of the $(\mathcal{D}_{c},\mathcal{S}_{c})$ —forms is demonstrated by construction. The accompanying paper (Part II) proves the internal consistency of the $(\mathcal{D}_{c},\mathcal{S}_{c})$ —forms rigorously by showing the sharp equivalence of the $(\mathcal{D}_{c},\ \mathcal{S}_{c})$ -forms with the originating Maxwell's and constitutive equations.