Let E -> M and F -> N be two fiber bundles with dim(M) = m and dim(N) = n and let Phi: J^k(E) -> J^l(F) be a transformation.  Let (x^i), i = 1, ... m be a system of local coordinates on M and let (y^a), a = 1, ... n be a system of local coordinates on N.  Let F^a = y^a(Phi) be the y^a components of the map Phi--these are functions on J^k(E).  Then the total Jacobian of Phi is the m x n matrix J^a_i  = [D_i  F^a] where D_i denotes the total derivative with respect to x^i.  The pushforward of the total vector field X =X^iD_i on M is the  total vector Y=Y^aD_a , where  =Y^a =J^a_i X^i.

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