Let E -> M and F -> N be two fiber bundles.  [i] A map Phi : E -> F which sends the fibers of E to fibers of N (and hence covers a map Phi0: M -> N) is called a projectable transformation.  [ii] A map Phi: E -> F is called a point transformation.  [iii] A transformation Phi: J^1(E) -> J^1(F) is called a contact transformation if the fiber dimensions of E and F are 1 and Phi pulls back the contact form on J^1(F) to a multiple of the contact form on J^1(E).  [iv] If Phi: J^k(E) -> F and the total Jacobian of Phi is the identity matrix, then Phi is called a differential substitution. [v] A map Phi: J^k(E) -> F is called a generalized differential substitution.  [vi] A transformation not of one the types [i]--[v] is called generic.

Explicit coordinate formulas for these various types of maps are given in Example 1.

Any transformation of type [i]--[v] can be prolonged to higher order jet spaces.  See Prolong for further information.

The type of a transformation and its prolongation order can be determined by the command DGinfo with keyword "TransformationType".

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