construct the canonical projection map between jet spaces of a fiber bundle - Maple Programming Help

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JetCalculus[ProjectionTransformation] - construct the canonical projection map between jet spaces of a fiber bundle

Calling Sequences

ProjectionTransformation(n, m)

Parameters

n         - a non-negative integer, the order for the domain jet space

m         - a non-negative integer, the order for the range jet space,

Description

 • Let be a fiber bundle.Then ProjectionTransformation(n, m) defines the canonical projection map of the jet space to
 • The command ProjectionTransformation is part of the DifferentialGeometry:-JetCalculus package. It can be used in the form ProjectionTransformation(...) only after executing the commands with(DifferentialGeometry) and with(JetCalculus), but can always be used by executing DifferentialGeometry:-JetCalculus:-ProjectionTransformation(...).

Examples

 > $\mathrm{with}\left(\mathrm{DifferentialGeometry}\right):$$\mathrm{with}\left(\mathrm{JetCalculus}\right):$

Example 1.

Define the 6-th order jet bundle for with coordinates .

 > $\mathrm{DGsetup}\left(\left[x,y\right],\left[u\right],E,6\right):$

Define the canonical projection from ${J}^{6}(E$to ${J}^{3}\left(E\right)$.

 E > $\mathrm{Π1}≔\mathrm{ProjectionTransformation}\left(6,3\right)$
 ${\mathrm{Π1}}{:=}\left[{x}{=}{x}{,}{y}{=}{y}{,}{{u}}_{\left[\right]}{=}{{u}}_{\left[\right]}{,}{{u}}_{{1}}{=}{{u}}_{{1}}{,}{{u}}_{{2}}{=}{{u}}_{{2}}{,}{{u}}_{{1}{,}{1}}{=}{{u}}_{{1}{,}{1}}{,}{{u}}_{{1}{,}{2}}{=}{{u}}_{{1}{,}{2}}{,}{{u}}_{{2}{,}{2}}{=}{{u}}_{{2}{,}{2}}{,}{{u}}_{{1}{,}{1}{,}{1}}{=}{{u}}_{{1}{,}{1}{,}{1}}{,}{{u}}_{{1}{,}{1}{,}{2}}{=}{{u}}_{{1}{,}{1}{,}{2}}{,}{{u}}_{{1}{,}{2}{,}{2}}{=}{{u}}_{{1}{,}{2}{,}{2}}{,}{{u}}_{{2}{,}{2}{,}{2}}{=}{{u}}_{{2}{,}{2}{,}{2}}\right]$ (2.1)
 E > $\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{Π1},"DomainFrame"\right),\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{Π1},"DomainOrder"\right)$
 ${E}{,}{6}$ (2.2)
 E > $\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{Π1},"RangeFrame"\right),\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{Π1},"RangeOrder"\right)$
 ${E}{,}{3}$ (2.3)