  We compare the integration by parts of contact forms - leading to the
definition of the interior Euler operator - with the so-called canonical
splittings of variational morphisms. In particular, we discuss the possibility
of a generalization of the first method to contact forms of lower degree. We
define a suitable Residual operator for this case and, working out an original
conjecture by Olga Rossi, we recover the Krupka-Betounes equivalent for first
order field theories. A generalization to the second order case is discussed.
