Every differential form on a jet space can be expressed as a sum of wedge products of 1-forms on M and contact 1-forms.  A differential form omega is called a bi-form of degree (r, s) if it is a sum of wedge products of r 1-forms on M and s contact 1-forms.  Alternatively, omega is of type (r, s) if omega(X_1, X_2, ... X_(r + s)) = 0 whenever more than r of the vector fields X_i are total vector fields or more than s of the vector fields X_i are vertical vector fields on J^k(E) -> M.  The non-negative integer r is called the horizontal degree of omega.  The non-negative integer s is called the vertical degree of omega.

If omega is called a bi-form of degree (r, s), then its exterior derivative d(omega) is a sum of two bi-forms, one of type (r + 1, s), the other of type (r, s + 1).  The type (r + 1, s) part is called the horizontal exterior derivative of omega and is denoted by dH(omega).  The type (r, s + 1) part is called the vertical exterior derivative of omega and is denoted by dV(omega).  Thus d(omega) = dH(omega) + dV(omega) from which it follows that dH^2 = 0, dH dV = - dV dH and dH^2 = 0.

If (x^i) are local coordinates on M and D_i denotes total differentiation with respect to x^i, then dH(omega) = D_i(omega) Dx^i.

The command VerticalExteriorDerivative is part of the DifferentialGeometry:-JetCalculus package.  It can be used in the form VerticalExteriorDerivative(...) only after executing the commands with(DifferentialGeometry) and with(JetCalculus), but can always be used by executing DifferentialGeometry:-JetCalculus:-VerticalExteriorDerivative(...).