VerticalExteriorDerivative - Maple Help

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JetCalculus[VerticalExteriorDerivative] - calculate the vertical exterior derivative of a bi-form on a jet space

Calling Sequences

VerticalExteriorDerivative(${\mathrm{ω}}$)

Parameters

$\mathrm{\omega }$     - a differential bi-form on the jet space of a fiber bundle

Description

 • Let be a fiber bundle, with base dimension $n$ and fiber dimension $m$ and let  be the infinite jet bundle of $E$. The  -forms ${\mathrm{Ω}}^{p}\left({J}^{\mathrm{∞}}\left(E\right)\right)$ can be graded by horizontal and vertical (or contact) degree and, with respect to this grading, the exterior derivative operator can be decomposed as The horizontal exterior derivative raises the horizontal degree by 1 and the vertical exterior derivative raises the vertical degree by 1. For details, see HorizontalExteriorDerivative.
 • The command VerticalExteriorDerivative($\mathrm{ω}$) returns the vertical exterior derivative ${d}_{V}\left(\mathrm{ω}\right)$.
 • The command VerticalExteriorDerivative is part of the DifferentialGeometry:-JetCalculus package.  It can be used in the form VerticalExteriorDerivative(...) only after executing the commands with(DifferentialGeometry) and with(JetCalculus), but can always be used by executing DifferentialGeometry:-JetCalculus:-VerticalExteriorDerivative(...).

Examples

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

Example 1.

Create the jet space ${J}^{2}\left(E\right)$with coordinates

 > $\mathrm{DGsetup}\left(\left[x,y\right],\left[u,v\right],E,2\right):$

Calculate the vertical exterior derivative of a function.

 E > $F≔f\left(x,y,u\left[\right],v\left[\right],u\left[1\right],u\left[2\right],v\left[1\right],v\left[2\right]\right):$
 E > $\mathrm{PDEtools}\left[\mathrm{declare}\right]\left(F,\mathrm{quiet}\right):$
 E > $\mathrm{VerticalExteriorDerivative}\left(F\right)$
 ${{f}}_{{{u}}_{\left[\right]}}{}{{\mathrm{Cu}}}_{\left[\right]}{+}{{f}}_{{{v}}_{\left[\right]}}{}{{\mathrm{Cv}}}_{\left[\right]}{+}{{f}}_{{{u}}_{{1}}}{}{{\mathrm{Cu}}}_{{1}}{+}{{f}}_{{{u}}_{{2}}}{}{{\mathrm{Cu}}}_{{2}}{+}{{f}}_{{{v}}_{{1}}}{}{{\mathrm{Cv}}}_{{1}}{+}{{f}}_{{{v}}_{{2}}}{}{{\mathrm{Cv}}}_{{2}}$ (2.1)

Calculate the vertical exterior derivative of a type (1, 0) bi-form.

 E > $\mathrm{ω1}≔\mathrm{evalDG}\left(u\left[2,2\right]\mathrm{Dx}+v\left[1,1,1\right]\mathrm{Dy}\right)$
 ${\mathrm{ω1}}{≔}{{u}}_{{2}{,}{2}}{}{\mathrm{Dx}}{+}{{v}}_{{1}{,}{1}{,}{1}}{}{\mathrm{Dy}}$ (2.2)
 E > $\mathrm{VerticalExteriorDerivative}\left(\mathrm{ω1}\right)$
 ${-}{\mathrm{Dx}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{2}{,}{2}}{-}{\mathrm{Dy}}{}{\bigwedge }{}{{\mathrm{Cv}}}_{{1}{,}{1}{,}{1}}$ (2.3)

Calculate the vertical exterior derivative of a type (0, 2) bi-form.

 E > $\mathrm{ω2}≔\mathrm{evalDG}\left(v\left[1,1\right]u\left[2,2,2\right]\mathrm{Cu}\left[2\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&w\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{Cv}\left[2\right]\right)$
 ${\mathrm{ω2}}{≔}{{v}}_{{1}{,}{1}}{}{{u}}_{{2}{,}{2}{,}{2}}{}{{\mathrm{Cu}}}_{{2}}{}{\bigwedge }{}{{\mathrm{Cv}}}_{{2}}$ (2.4)
 E > $\mathrm{VerticalExteriorDerivative}\left(\mathrm{ω2}\right)$
 ${{u}}_{{2}{,}{2}{,}{2}}{}{{\mathrm{Cu}}}_{{2}}{}{\bigwedge }{}{{\mathrm{Cv}}}_{{2}}{}{\bigwedge }{}{{\mathrm{Cv}}}_{{1}{,}{1}}{+}{{v}}_{{1}{,}{1}}{}{{\mathrm{Cu}}}_{{2}}{}{\bigwedge }{}{{\mathrm{Cv}}}_{{2}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{2}{,}{2}{,}{2}}$ (2.5)