calculate the horizontal exterior derivative of a bi-form on a jet space - Maple Programming Help

Home : Support : Online Help : Mathematics : DifferentialGeometry : JetCalculus : DifferentialGeometry/JetCalculus/HorizontalExteriorDerivative

JetCalculus[HorizontalExteriorDerivative] - calculate the horizontal exterior derivative of a bi-form on a jet space

Calling Sequences

HorizontalExteriorDerivative(${\mathbf{ω}}$)

Parameters

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 $E$. Let , ..., be a local system of jet coordinates. Every differential form on ${J}^{\infty }\left(E\right)$ can be expressed locally in terms of a sum of wedge products of 1-forms ${\mathrm{dx}}^{i}$ on and contact 1-forms,

.

Note that exterior derivatives of the contact 1-forms are

d

A differential $p-$form  is called a bi-form of degree $\left(r,s\right)$ if  it is a sum of wedge products of  -forms on  and contact 1-forms, that is,

where each ${C}^{{a}_{k}}$ is a contact 1-form.

The space of all $p$-forms then decomposes as a direct sum of bi-forms

The above formulas for the exterior derivative of the contact forms shows that and therefore   where

and

The differential operator is called the horizontal exterior derivative and the differential operator is called the vertical exterior derivative. One has that

and  .

The coordinate formulas for the horizontal exterior derivative are

.

The coordinate formulas for the vertical exterior derivative are

 • The command HorizontalExteriorDerivative(${\mathrm{ω}}$) returns the horizontal exterior derivative ${d}_{H}\left(\mathrm{ω}\right)$. The horizontal degree of must be less than the dimension of the base manifold $M$. The vertical exterior derivative is computed with the command VerticalExteriorDerivative.
 • The command HorizontalExteriorDerivative is part of the DifferentialGeometry:-JetCalculus package.  It can be used in the form HorizontalExteriorDerivative(...) only after executing the commands with(DifferentialGeometry) and with(JetCalculus), but can always be used by executing DifferentialGeometry:-JetCalculus:-HorizontalExteriorDerivative(...).

Examples

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

Example 1.

Create the jet space for the bundle $E$with coordinates .

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

Calculate the horizontal exterior derivative of a function.

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

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

 E > $\mathrm{ω1}≔A\left(x,y,{u}_{[]},{u}_{1},{u}_{2}\right)\mathrm{Dx}+B\left(x,y,{u}_{[]},{u}_{1},{u}_{2}\right)\mathrm{Dy}$
 ${\mathrm{ω1}}{:=}{A}{}\left({x}{,}{y}{,}{{u}}_{{[}{]}}{,}{{u}}_{{1}}{,}{{u}}_{{2}}\right){}{\mathrm{Dx}}{+}{B}{}\left({x}{,}{y}{,}{{u}}_{{[}{]}}{,}{{u}}_{{1}}{,}{{u}}_{{2}}\right){}{\mathrm{Dy}}$ (2.2)
 E > $\mathrm{HorizontalExteriorDerivative}\left(\mathrm{ω1}\right)$
 ${-}\left({{A}}_{{{u}}_{\left[\right]}}{}{{u}}_{{2}}{+}{{A}}_{{{u}}_{{1}}}{}{{u}}_{{1}{,}{2}}{+}{{A}}_{{{u}}_{{2}}}{}{{u}}_{{2}{,}{2}}{-}{{B}}_{{{u}}_{\left[\right]}}{}{{u}}_{{1}}{-}{{B}}_{{{u}}_{{1}}}{}{{u}}_{{1}{,}{1}}{-}{{B}}_{{{u}}_{{2}}}{}{{u}}_{{1}{,}{2}}{+}{{A}}_{{y}}{-}{{B}}_{{x}}\right){}{\mathrm{Dx}}{}{\bigwedge }{}{\mathrm{Dy}}$ (2.3)

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

 E > $\mathrm{ω2}≔\left({\mathrm{Cu}}_{2}\right)&wedge\left({\mathrm{Cv}}_{2}\right)$
 ${\mathrm{ω2}}{:=}{{\mathrm{Cu}}}_{{2}}{}{\bigwedge }{}{{\mathrm{Cv}}}_{{2}}$ (2.4)
 E > $\mathrm{HorizontalExteriorDerivative}\left(\mathrm{ω2}\right)$
 ${\mathrm{Dx}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{2}}{}{\bigwedge }{}{{\mathrm{Cv}}}_{{1}{,}{2}}{-}{\mathrm{Dx}}{}{\bigwedge }{}{{\mathrm{Cv}}}_{{2}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{1}{,}{2}}{+}{\mathrm{Dy}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{2}}{}{\bigwedge }{}{{\mathrm{Cv}}}_{{2}{,}{2}}{-}{\mathrm{Dy}}{}{\bigwedge }{}{{\mathrm{Cv}}}_{{2}}{}{\bigwedge }{}{{\mathrm{Cu}}}_{{2}{,}{2}}$ (2.5)