TotalVector - Maple Help

JetCalculus[TotalVector] - form the total part of a vector field

Calling Sequences

TotalVector(X)

Parameters

X    - a vector field or a generalized vector field on a fiber bundle

Description

 • Let be a fiber bundle, with base dimension $n$ and fiber dimension $m$ and let  be the $k$-th jet bundle with jet coordinates , ..., . A total vector field on jet space is a vector field of the form  , where the coefficients are functions on the jet space ${J}^{k}\left(E\right)$ and ${\mathrm{D}}_{\mathrm{ℓ}}$ is the total vector field for the coordinate ${x}^{\mathrm{ℓ}}$ , that is,

Total vector fields may be characterized intrinsically as generalized vector fields which annihilate all contact 1-forms. If is a generalized vector field on $E$, then the total part is

and the evolutionary part is

The prolongation of is the total vector field pr(.

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

Examples

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

Example 1.

Create the jet space ${J}^{2}\left(E\right)$ for the bundle with local coordinates. We calculate the total part of some vector fields.

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

Define a vector ${X}_{1}$ and compute its total part.

 E > $\mathrm{X1}≔\mathrm{evalDG}\left(\mathrm{D_x}\right)$
 ${\mathrm{X1}}{:=}{\mathrm{D_x}}$ (2.1)
 E > $\mathrm{totX1}≔\mathrm{TotalVector}\left(\mathrm{X1}\right)$
 ${\mathrm{totX1}}{:=}{\mathrm{D_x}}{+}{{u}}_{{1}}{}{{\mathrm{D_u}}}_{{[}{]}}{+}{{v}}_{{1}}{}{{\mathrm{D_v}}}_{{[}{]}}$ (2.2)

The prolongation of tot(is the total derivativewith respect to $x.$

 E > $\mathrm{Prolong}\left(\mathrm{totX1},2\right)$
 ${\mathrm{D_x}}{+}{{u}}_{{1}}{}{{\mathrm{D_u}}}_{{[}{]}}{+}{{v}}_{{1}}{}{{\mathrm{D_v}}}_{{[}{]}}{+}{{u}}_{{1}{,}{1}}{}{{\mathrm{D_u}}}_{{1}}{+}{{u}}_{{1}{,}{2}}{}{{\mathrm{D_u}}}_{{2}}{+}{{v}}_{{1}{,}{1}}{}{{\mathrm{D_v}}}_{{1}}{+}{{v}}_{{1}{,}{2}}{}{{\mathrm{D_v}}}_{{2}}{+}{{u}}_{{1}{,}{1}{,}{1}}{}{{\mathrm{D_u}}}_{{1}{,}{1}}{+}{{u}}_{{1}{,}{1}{,}{2}}{}{{\mathrm{D_u}}}_{{1}{,}{2}}{+}{{u}}_{{1}{,}{2}{,}{2}}{}{{\mathrm{D_u}}}_{{2}{,}{2}}{+}{{v}}_{{1}{,}{1}{,}{1}}{}{{\mathrm{D_v}}}_{{1}{,}{1}}{+}{{v}}_{{1}{,}{1}{,}{2}}{}{{\mathrm{D_v}}}_{{1}{,}{2}}{+}{{v}}_{{1}{,}{2}{,}{2}}{}{{\mathrm{D_v}}}_{{2}{,}{2}}$ (2.3)

Define a vector and compute its total part.

 E > $\mathrm{X2}≔\mathrm{evalDG}\left(\mathrm{D_u}\left[\right]\right)$
 ${\mathrm{X2}}{:=}{{\mathrm{D_u}}}_{{[}{]}}$ (2.4)
 E > $\mathrm{TotalVector}\left(\mathrm{X2}\right)$
 ${0}{}{\mathrm{D_x}}$ (2.5)

Define a vector and compute its total part.

 E > $\mathrm{X3}≔\mathrm{evalDG}\left(a\mathrm{D_x}+b\mathrm{D_y}+c\mathrm{D_u}\left[\right]+d\mathrm{D_v}\left[\right]\right)$
 ${\mathrm{X3}}{:=}{a}{}{\mathrm{D_x}}{+}{b}{}{\mathrm{D_y}}{+}{c}{}{{\mathrm{D_u}}}_{{[}{]}}{+}{d}{}{{\mathrm{D_v}}}_{{[}{]}}$ (2.6)
 E > $\mathrm{totX3}≔\mathrm{TotalVector}\left(\mathrm{X3}\right)$
 ${\mathrm{totX3}}{:=}{a}{}{\mathrm{D_x}}{+}{b}{}{\mathrm{D_y}}{+}\left({b}{}{{u}}_{{2}}{+}{a}{}{{u}}_{{1}}\right){}{{\mathrm{D_u}}}_{{[}{]}}{+}\left({b}{}{{v}}_{{2}}{+}{a}{}{{v}}_{{1}}\right){}{{\mathrm{D_v}}}_{{[}{]}}$ (2.7)

Example 2.

We show that the total part of a vector field annihilates the 1st order contact forms.

 E > $\mathrm{DGsetup}\left(\left[x,y,z\right],\left[u,v,w\right],\mathrm{J33},3\right):$
 J33 > $\mathrm{X4}≔w\left[1,2,3\right]\mathrm{D_z}$
 ${\mathrm{X4}}{:=}{{w}}_{{1}{,}{2}{,}{3}}{}{\mathrm{D_z}}$ (2.8)
 J33 > $\mathrm{totX4}≔\mathrm{TotalVector}\left(\mathrm{X4}\right)$
 ${\mathrm{totX4}}{:=}{{w}}_{{1}{,}{2}{,}{3}}{}{\mathrm{D_z}}{+}{{w}}_{{1}{,}{2}{,}{3}}{}{{u}}_{{3}}{}{{\mathrm{D_u}}}_{{[}{]}}{+}{{w}}_{{1}{,}{2}{,}{3}}{}{{v}}_{{3}}{}{{\mathrm{D_v}}}_{{[}{]}}{+}{{w}}_{{1}{,}{2}{,}{3}}{}{{w}}_{{3}}{}{{\mathrm{D_w}}}_{{[}{]}}$ (2.9)

A total vector field always annihilates the first order contact 1-forms.

 J33 > $\mathrm{ω1}≔\mathrm{convert}\left(\mathrm{Cu}\left[\right],\mathrm{DGform}\right);$$\mathrm{ω2}≔\mathrm{convert}\left(\mathrm{Cv}\left[\right],\mathrm{DGform}\right);$$\mathrm{ω3}≔\mathrm{convert}\left(\mathrm{Cw}\left[\right],\mathrm{DGform}\right)$
 ${\mathrm{ω1}}{:=}{-}{{u}}_{{1}}{}{\mathrm{dx}}{-}{{u}}_{{2}}{}{\mathrm{dy}}{-}{{u}}_{{3}}{}{\mathrm{dz}}{+}{{\mathrm{du}}}_{{[}{]}}$
 ${\mathrm{ω2}}{:=}{-}{{v}}_{{1}}{}{\mathrm{dx}}{-}{{v}}_{{2}}{}{\mathrm{dy}}{-}{{v}}_{{3}}{}{\mathrm{dz}}{+}{{\mathrm{dv}}}_{{[}{]}}$
 ${\mathrm{ω3}}{:=}{-}{{w}}_{{1}}{}{\mathrm{dx}}{-}{{w}}_{{2}}{}{\mathrm{dy}}{-}{{w}}_{{3}}{}{\mathrm{dz}}{+}{{\mathrm{dw}}}_{{[}{]}}$ (2.10)
 J33 > $\mathrm{Hook}\left(\mathrm{totX4},\mathrm{ω1}\right),\mathrm{Hook}\left(\mathrm{totX4},\mathrm{ω2}\right),\mathrm{Hook}\left(\mathrm{totX4},\mathrm{ω3}\right)$
 ${0}{,}{0}{,}{0}$ (2.11)

A vector field is always the sum of its total and evolutionary parts.

 J33 > $\mathrm{evolX4}≔\mathrm{EvolutionaryVector}\left(\mathrm{X4}\right)$
 ${\mathrm{evolX4}}{:=}{-}{{w}}_{{1}{,}{2}{,}{3}}{}{{u}}_{{3}}{}{{\mathrm{D_u}}}_{{[}{]}}{-}{{w}}_{{1}{,}{2}{,}{3}}{}{{v}}_{{3}}{}{{\mathrm{D_v}}}_{{[}{]}}{-}{{w}}_{{1}{,}{2}{,}{3}}{}{{w}}_{{3}}{}{{\mathrm{D_w}}}_{{[}{]}}$ (2.12)
 J33 > $\mathrm{totX4}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&plus\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathrm{evolX4}$
 ${{w}}_{{1}{,}{2}{,}{3}}{}{\mathrm{D_z}}$ (2.13)