  In this paper we give explicit formulas for higher order differential
operators on a finitely generated projective module $E$ on an arbitrary
commutative unital ring $A$. We use the differential operators constructed to
give a simple formula for the curvature of a classical connection and a
connection on a Lie-Rinehart algebra in terms of a "projective basis" $B$ for
$E$. A "projective basis" is sometimes referred to as a "dual basis". This
gives an explicit formula for the curvature $R_{\nabla_B}$ of a connection
$\nabla_B$ on $E$ defined in terms of a projective basis $B$ and an idempotent
$\phi$ for $E$. We also consider the notion of a stratification on the module
$E$ induced by a projective basis $B$. It turns out few stratifications are
induced by a projective basis.
