Magnetostatic problems in multiply connected regions: some properties of the curl operator

The two main versions of the boundary-value problem of magnetostatics (the one with h normal to the wall, the one with a tangential magnetic field) are studied in connection with a deep underlying mathematical structure: the existence of a decomposition of the space of square summable vector fields in five closed subspaces. This structure encompasses classical Helmholtz decomposition in its different guises. It is best conceived in terms of differential forms, but, for pedagogical reasons, this point of view has been left aside in the present text, which only makes use of classical notions of applied functional analysis.