Sum rule for the magnetic permeability of arbitrary textures

The $f$-sum rule for the magnetic permeability, derived previously for an assembly of isolated macrospins, is generalized for an arbitrary nonuniform three-dimensional magnetization texture, in which the magnetizations at different points are coupled by exchange and magnetostatic interactions. 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.