A sum rule for the magnetic permeability of arbitrary textures 1 Andr´ e Thiaville Laboratoire de Physique des Solides, Univ. Paris-Sud, CNRS UMR8502, 91405 Orsay Cedex, France Nicolas Vukadinovic Dassault Aviation, 92552 Saint-Cloud, France Olivier Acher CEA-DAM, BP 16, 37260 Monts, France ∗ (Dated: November 12, 2012) Abstract The f-sum rule for the magnetic permeability, derived previously for an assembly of isolated macrospins, is generalized for an arbitrary non-uniform three-dimensional magnetization texture, in which the magnetizations at different points are coupled by exchange and magnetostatic inter- actions. The sum value depends only on the magnetic texture at rest. It has no direct contribution from the exchange energy, but depends on the anisotropy, applied field and demagnetizing energies. The derived formula is tested against numerical calculations for several complex and very different magnetization structures. This generalized sum rule should be useful for experiments, numerical simulations, and metrology. 1