Here are the explicit formulas for the vertical homotopy operators. Let , ..., be a local system of jet coordinates and let be the contact forms. The vertical radial vector field on is and its prolongation to jet space is
The flow of the vector field pr is the transformation given by = . The vertical homotopy operators are then defined in terms of pr and and the interior product operator (see Hook) by
As a concrete example, if is given by (, ...), then
Thus the formulas for the vertical homotopy operators are essentially the same as that for the standard de Rham homotopy operators.