  A deformed differential calculus is developed based on an associative
star-product. In two dimensions the Hamiltonian vector fields model the algebra
of pseudo-differential operator, as used in the theory of integrable systems.
Thus one obtains a geometric description of the operators. A dual theory is
also possible, based on a deformation of differential forms. This calculus is
applied to a number of multidimensional integrable systems, such as the KP
hierarchy, thus obtaining a geometrical description of these systems. The limit
in which the deformation disappears corresponds to taking the dispersionless
limit in these hierarchies.
