Some new results on higher order analogs ${\bf dR}_\sigma $ , $\sigma $ being a sequence of positive integers, of the standard de Rham complex ${\bf % dR}\equiv {\bf dR}_{\left( 1,...,1,..\right) }$ are presented. First we sketch an algebraic machinery which allows us to prove an analog of the infinitesimal Stoke's formula (also called the Cartan homotopy formula) for $% {\bf dR}_\sigma $'s with non-decreasing $\sigma $'s. Second, we outline the key points of the proof that the cohomology of ${\bf dR}_\sigma $ does not depend on $\sigma $, under some smoothness assumptions.
16 pages