  We develop a formula (Theorem 5.1) which allows to compute top Chern classes
of vector bundles on the vanishing locus $V(s)$ of a section of this bundle.
This formula particularly applies in the case when $V(s)$ is the union of
locally complete intersections giving the individual contribution of each
component and their mutual intersections. We conclude with applications to the
enumeration of rational curves in complete intersections in projective space.
