Let omega be a bi-form of degree (r, s) on J^k(R^n, R^m).  Then omega is called dV closed if dV(omega) = 0 where dV denotes the vertical exterior derivative and dV exact if there exists a bi-form eta of degree (r, s - 1) on J^l(R^n, R^m) such that omega = dV(eta).

If dV(omega) = 0 then there are numerous algorithms for finding a bi-form eta such that omega = dV(eta).  One approach is to use the vertical homotopy operators.  These homotopy operators are very similar to the usual homotopy operators for the deRham complex.

The optional arguments available to DeRhamHomotopy can be invoked with VerticalHomotopy.

If omega is a bi-form of degree (r, s) with s > 1, then VerticalHomotopy(omega) returns a bi-form of degree (r, s - 1).

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

with(DifferentialGeometry): with(JetCalculus):

DGsetup([x, y], [u], E, 1):

omega1 := evalDG(Cu[] &w Cu[1] &w Cu[2]);

VerticalExteriorDerivative(omega1);

eta1a := VerticalHomotopy(omega1);

omega1 &minus VerticalExteriorDerivative(eta1a);

eta1b := VerticalHomotopy(omega1, path = "zigzag");

eta1b := VerticalHomotopy(omega1, path = "zigzag", variableorder = [u[1], u[2], u[], u[1, 1], u[1, 2], u[2, 2]]);