The total part (TotalVector) of a generalized vector field X on the bundle E -> M is the generalized vector field Y on E -> M such that X - Y is a vertical vector and Hook(Y, omega) = 0 for any contact 1-form omega on J^1(E).

The vertical vector X - Y is called the evolutionary part of the vector field X.

The evolutionary part of a projectable vector field X has the following geometric interpretation.  Let Phi _t: E -> E be the flow of X.  Then Phi_t covers a map Psi_t: M -> M.  If sigma: M -> E is a section of E, then the induced flow in the space of sections is defined to be sigma_t(x) = Phi_t (sigma(Psi_( - t)(x))).  The derivative of sigma_t with respect to t, evaluated at t = 0, yields the components of the evolutionary part of X.

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