  We inquire into the relation between the curl operators, the Poisson
coboundary operators and contravariant derivatives on Poisson manifolds to
study the theory of differential operators in Poisson geometry. Given an
oriented Poisson manifold, we describe locally those two differential operators
in terms of Poisson connection whose torsion is vanishing. Moreover, we
introduce the notion of the modular operator for an oriented Poisson manifold.
For a symplectic manifold, we describe explicitly the modular operator in terms
of the curvature 2-section of Poisson connection, analogously to the
Weitzenb$\ddot{\rm o}$ck formula in Riemannian geometry.
