Solenoidality of the Magnetic Induction Field, Electromagnetic Duality and Conservation of Momentum in Material Media

This monograph reconstructs, verifies and expands a thirty-five-page manuscript devoted to the dual form of Maxwell’s equations in the presence of electric and magnetic sources, the associated force laws, flux transport across moving surfaces, and the balance of electromagnetic momentum in material media. Assuming the momentum density D × B and a Minkowski-type stress tensor, the theoretical core leads to a global identity in which the defect k ρₘ - ∇·B appears. The reconstruction rigorously separates the branch in which Gauss’s law for magnetism is assumed from the non-circular branch in which it must be derived. The decisive logical point is the localisation principle: the vanishing of a single weighted integral does not imply the pointwise vanishing of the defect; instead, the conclusion ∇·B = k ρₘ follows if the balance holds over a localising family of subdomains, or in distributional form against all relevant test functions, and if H does not vanish on sets of positive measure. The general transport formula, interface conditions, force terms due to permittivity and permeability gradients, the distributional nature of point sources, the role of the unit convention represented by k, and the meaning of the choice D × B in the Abraham-Minkowski context are also reconstructed. Five appendices provide independent proofs, tensor identities, numerical checks, weak localisation, and a dimensional dictionary, making the treatment self-contained at specialist level. Global identity constituting the endpoint of the manuscript and the starting point of the localisation analysis: \( \int_{\Omega} \mathbf{H}(\mathbf{r},t) \left[ k\,\rho_m(\mathbf{r},t) - \nabla\!\cdot\!\mathbf{B}(\mathbf{r},t) \right] \,\mathrm{d}V =0 \) , (I.1).lobal identity constituting the endpoint of the manuscript and the starting point of the localisation analysis.