  We study here the relative cohomology and the Gauss-Manin connections
associated to an isolated singularity of a function on a manifold with
boundary, i.e. with a fixed hyperplane section. We prove several relative
analogs of classical theorems obtained mainly by E. Brieskorn and B. Malgrange,
concerning the properties of the Gauss-Manin connection as well as its
relations with the Picard-Lefschetz monodromy and the asymptotics of integrals
of holomorphic forms along the vanishing cycles. Finally, we give an
application in isochore deformation theory, i.e. the deformation theory of
boundary singularities with respect to a volume form. In particular we prove
the relative analog of J. Vey's isochore Morse lemma, J. -P. Fran\c{c}oise's
generalisation on the local normal forms of volume forms with respect to the
boundary singularity-preserving diffeomorphisms, as well as M. D. Garay's
theorem on the isochore version of Mather's versal unfolding theorem.
