Stokes' formulae on classical symbol valued forms and applications

The Wodzicki residue and the cut-off integral extend to classical symbol-valued forms. We show that they obey a Stokes' type property and that the extended Wodzicki residue can be interpreted as a complex residue like the ordinary one. In the case of cut-off integrals, Stokes' property (i.e. vanishing on exact forms) only holds for non-integer order symbol-valued forms and leads to an integration by parts formula and translation invariance for cut-off integrals on non-integer order classical symbols. The extended Wodzicki residue yields an even residue cycle on classical symbols and an odd cochain (the cosphere cochain) which measures an obstruction to Stokes' property of the cut-off integral on integer order symbol-valued forms.