  We study in detail Hodge-Helmholtz decompositions in non-smooth exterior
domains filled with inhomogeneous and anisotropic media. We show decompositions
of alternating differential forms belonging to weighted Sobolev spaces into
irrotational and solenoidal forms. These decompositions are essential tools,
for example, in electro-magnetic theory for exterior domains. In the appendix
we translate our results to the classical framework of vector analysis.
