  Let $X=\{X_t: t\ge 0\}$ be a stationary piecewise continuous $\R^d$-valued
process that moves between jumps along the integral curves of a given
continuous vector field, and let $S\subset\R^d$ be a smooth surface. The aim of
this paper is to derive a multivariate version of Rice's formula, relating the
intensity of the point process of (localized) continuous crossings of $S$ by
$X$ to the distribution of $X_0$. Our result is illustrated by examples
relating to queueing networks and stress release network models.
