Let E -> M be a fiber bundle and J^k(E) -> M the associate bundle of k-jets.  [i] A vector field X on E is called a point vector field.  [ii] If X is a point vector field which projects to a vector field on M, then X is called a projectable vector field.  [iii] If the fiber dimension of E is 1 and X is a vector field on J^1(E) which preserves the contact 1-form on J^1(E) (under Lie differentiation), then X is a contact vector field.  [iv] If X is a vector field on J^k(E) which takes values in the tangent space of E, then X is called a generalized vector field on E (also called a Lie-Backlund vector field or a dynamical vector field).  [v] If X is a generalized vector field on E which projects to the zero vector field on M, then X is called a vertical generalized vector field or an evolutionary vector on E.  [vi] A generalized vector field which annihilates the contact ideal on J^1(E) is called a total vector field.

Any of the vector fields [i]--[vi] admit a unique lift to a vector field with values in J^p(E) by the Lie prolongation formula.  We call, for example, a vector field on J^p(E) obtained from the prolongation of a point vector field a prolonged point vector field.  See Prolong for further information.

A vector field not of the types [i]--[vi] or a prolongation of such is labeled generic.

Explicit coordinate formulas for these types of vector fields in the case R^2 x R -> R^2 are presented in Example 1.

The type of a vector field and its prolongation order can be determined by the command DGinfo with keyword "VectorType".

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