On the geometry of variational calculus on some functional bundles
  We first generalize the operation of formal exterior differential in the case of finite dimensional fibered manifolds and then we extend it to certain bundles of smooth maps. In order to characterize the operator order of some morphisms between our bundles of smooth maps, we introduce the concept of fiberwise $(k,r)$-jet. The relations to the Euler-Lagrange morphism of the variational calculus are described. 
Antonella Cabras, Josef Jany\v{s}ka, Ivan Kol\'a\v{r}