  We consider the variational complex on infinite jet space and the complex of
variational derivatives for Lagrangians of multidimensional paths and study
relations between them. The discussion of the variational (bi)complex is set up
in terms of a flat connection in the jet bundle. We extend it to supercase
using a particular new class of forms. We establish relation of the complex of
variational derivatives and the variational complex. Certain calculus of
Lagrangians of multidimensional paths is developed. It is shown how covariant
Lagrangians of higher order can be used to represent characteristic classes.
