With the first calling sequence Prolong(k) will prolong the current jet space to order k.  The vector fields, differential forms and contact forms to order k are all automatically assigned.

Let E -> M be a fiber bundle.  We say that a vector field Z on J^k(E) preserves the contact ideal on J^k(E) if for any contact form omega the Lie derivative of omega with respect to Z is again a contact form.  Let X be a projectable, point, contact, evolutionary, total,or generalized vector field with values in the tangent space E. (See AssignVectorType for the definitions of these types of vector fields.)  Then, for each k, there is a unique vector field Z on J^k(E) which preserves the contact ideal on J^k(E) and which projects pointwise to X.  This vector field Z is called the prolongation of X to order k.  The second calling sequence Prolong(X, k) computes the prolongation of X to order k.

Let E -> M and F -> N be fiber bundles.  We say that a transformation Psi: J^n(E) -> J^l(F) is a generalized contact transformation if for any contact form omega on J^l(F) the pullback of omega by Psi is a contact form on J^n(E).  Let Phi: J^m(E) -> F be a projectable transformation (n = 0), a point transformation (n = 0), a contact transformation, a differential substitution or a generalized differential substitution. (See AssignTransformType for the definitions of these different types of transformations.)  Then, for each k, there is a unique generalized contact transformation Psi: J^(n + k)(E) -> J^k(F) which covers Phi.  This transformation Psi is called the prolongation of Phi to order k.  The third calling sequence Prolong(Phi, k) computes the prolongation of Phi to order k.

Let E -> M be a fiber bundle and Delta be a system of n-th order differential equations defined as the zero set of a collection of functions F_a: J^n(E) -> R.  The k-th order prolongation of Delta is the system of (n + k)-th order differential equations defined as the zero set of the F_a and all their total derivatives to order k.  The third calling sequence Prolong(Delta, k) computes the prolongation of a system of differential equations Delta to order k.  Use the command DifferentialEquationData to convert a list of functions F_a: J^n(E) -> R into a differential equation data structure that can be passed to the Prolong command.  The result is a new differential equation (with DifferentialGeometry internal representation).

If a vector field, transformation or differential equation has been prolonged to a certain order using Prolong, then the prolonged objects may themselves be prolonged to a higher order using Prolong.

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