  We prove that the space of vector fields on the boundary of a bounded domain
in three dimensions is decomposed into three subspaces orthogonal to each
other: elements of the first one extend to the inside of the domain as gradient
fields of harmonic functions, the second one to the outside of the domain as
gradient fields of harmonic functions, and the third one to both the inside and
the outside as divergence-free harmonic vector fields. We also characterize by
the first Betti numbers the codimensions of inclusions of various subspaces
related to the decomposition.
