Coordinate-Free Geometric Formulation for Anisotropic Conduction and Singular Boundaries

We present a coordinate-free geometric formulation for anisotropic electrical transport in two-dimensional media, integrated with Schwartz's theory of distributions for the rigorous treatment of singular boundaries. The material anisotropy is encoded in a direct conductivity ellipse, whose semi-axes are the principal conductivities σ_a and σ_b, leading to an intrinsic, basis-independent Ohm's law: J = σ_a (E·u_a)u_a + σ_b (E·u_b)u_b. Using distribution theory, we model two complementary types of geometric discontinuities: (i) insulating obstacles (holes), where the Heaviside step function H(g) yields the surface charge density ρ_s ∝ -[J·n]; and (ii) singular conducting channels (ideal nanowires), where the line delta distribution δ_{C_f} describes the one-dimensional channel with internal current I_f(s) determined by dI_f/ds = J_bulk · u_f. We reveal a profound duality between the two singularity types: the hole blocks the normal current component, generating surface charge accumulation; the nanowire is transparent to the normal component but captures and drains the parallel component, feeding its internal current. For the nanowire, we demonstrate the absence of lateral charge accumulation due to geometric transparency, while longitudinal accumulation λ(s) = -τ (J_bulk · u_f) emerges in the presence of field gradients. This is the first article of a trilogy on anisotropic transport in media with singular boundaries. It establishes the conceptual and mathematical foundation for the exact solution of the circular hole problem (Article II) and the analysis of numerical instabilities in truncated multipolar expansions (Article III). This work was supported by the Federal University of Ceará, Brazil, and is intended for submission to the European Journal of Physics. Suggested citation:C. R. Fernandes, Coordinate-Free Geometric Formulation for Anisotropic Conduction and Singular Boundaries, Zenodo (2026). DOI: 10.5281/zenodo.20721301