The Eulerian approach to continuum mechanics does not make use of a body manifold. Rather, all fields considered are defined on the space, or the space–time, manifolds. Sections of some vector bundle represent generalized velocities which need not be associated with the motion of material points. Using the theories of de Rham currents and generalized sections of vector bundles, we formulate a weak theory of forces and stresses represented by vector-valued currents. Considering generalized velocities represented by differential forms and interpreting such a form as a generalized potential field, we present a weak formulation of pre-metric, p -form electrodynamics as a natural example of the foregoing theory. Finally, it is shown that the assumptions leading to p -form electrodynamics may be replaced by the condition that the force functional is continuous with respect to the flat topology of forms.