Jet spaces play a fundamental role in the geometric approach to the calculus of variations and to differential equations.  The JetCalculus package is a specialized package for symbolic computations on jet spaces and is fully compatible with the other packages and commands in DifferentialGeometry.

This package contains commands for prolonging both vector fields and transformations to jet spaces and for calculating Euler-Lagrange equations for variational problems with any number of independent and dependent variables and any number of derivatives.

Jet spaces admit a very important generalization of the de Rham complex which is called the variational bicomplex -- so named because one of the differentials in the variational bicomplex  can be identified with the Euler-Lagrange operator for the calculus of variations.  The JetCalculus packages provides the full functionality needed for computations within the theoretical framework of the variational bicomplex, including the horizontal and vertical exterior derivative operators, the associated homotopy operators and the integration by parts operator.

This package can be used in conjunction with the PDEtools package for the systematic analysis of symmetries of differential equations, for the study of conservation laws and integrable evolution equations, for the study of invariant variational  problems, and for the inverse problem to the calculus of variations.  It will also be of interest to those working in the areas of integrable systems and exterior differential systems.

The JetCalculus package is a sub-package of DifferentialGeometry.  Each command in the JetCalculus package can be accessed by using either the long form or the short form of the command name in the command calling sequence.

AssignTransformationType: assign a type (projectable, point, contact, ...) to a transformation.

AssignVectorType: assign a type (projectable, point, contact, ...) to a vector.

DifferentialEquationData: create a data structure for a system of differential equations.

EulerLagrange: calculate the Euler-Lagrange equations for a Lagrangian.

EvolutionaryVector: find the evolutionary part of a vector field.

GeneralizedLieBracket: find the Lie bracket of two generalized vector fields.

GeneratingFunctionToContactVector: find the contact vector field defined by a generating function.

HigherEulerOperators: apply the higher Euler operators to a function or a differential bi-form.

HorizontalExteriorDerivative: calculate the horizontal exterior derivative of a bi-form on a jet space.

HorizontalHomotopy: apply the horizontal homotopy operator to a bi-form on a jet space.

IntegrationByParts: apply the integration by parts operator to a differential bi-form.

Noether: find the conservation law for the  Euler-Lagrange equations from a given symmetry of the Lagrangian

ProjectedPullback: pullback a differential bi-form of type (r, s) by a transformation to a differential bi-form of type (r, s).

ProjectionTransformation: construct the canonical projection map between jet spaces of a fiber bundle.

Prolong: prolong a jet space, vector field, transformation, or differential equation to a higher order jet space.

PushforwardTotalVector: pushforward a total vector field by a transformation.

TotalDiff: take the total derivative of an expression, a differential form or a contact form.

TotalJacobian: find the Jacobian of a transformation using total derivatives.

TotalVector: find the total part of a vector field.

VerticalExteriorDerivative: calculate the vertical exterior derivative of a bi-form on a jet space.

VerticalHomotopy: apply the vertical homotopy operator to a bi-form on a jet space.

ZigZag: lift a dH-closed form on a jet space to a d-closed form.