Complex Generated by Variational Derivatives. Lagrangian Formalism of Infinite Order and a Generalized Stokes' Formula

We prove that an analog of the exterior differential acts on the space of arbitrary Lagrangians of multidimensional paths on any manifold or supermanifold, thus making this space into a cochain complex. An analog of the Stokes' formula holds. The construction and the proofs are purely geometrical, in terms of the variation of corresponding actions.