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 idea conjectured by Olga Rossi, we recover the Krupka-Betounes equivalent for first order field theories. A generalization to the second order case is discussed.