 form the total part of a vector field - Maple Programming Help

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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

 > with(DifferentialGeometry): with(JetCalculus):

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.

 > DGsetup([x, y], [u, v], E, 2):

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

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

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

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

Define a vector and compute its total part.

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

Define a vector and compute its total part.

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

Example 2.

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

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

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

 J33 > omega1 := convert(Cu[], DGform); omega2 := convert(Cv[], DGform); omega3 := convert(Cw[], DGform);
 ${\mathrm{ω1}}{:=}{-}{{u}}_{{1}}{}{\mathrm{dx}}{-}{{u}}_{{2}}{}{\mathrm{dy}}{-}{{u}}_{{3}}{}{\mathrm{dz}}{+}{{\mathrm{du}}}_{{[}{]}}$
 ${\mathrm{ω2}}{:=}{-}{{v}}_{{1}}{}{\mathrm{dx}}{-}{{v}}_{{2}}{}{\mathrm{dy}}{-}{{v}}_{{3}}{}{\mathrm{dz}}{+}{{\mathrm{dv}}}_{{[}{]}}$
 ${\mathrm{ω3}}{:=}{-}{{w}}_{{1}}{}{\mathrm{dx}}{-}{{w}}_{{2}}{}{\mathrm{dy}}{-}{{w}}_{{3}}{}{\mathrm{dz}}{+}{{\mathrm{dw}}}_{{[}{]}}$ (2.10)
 J33 > Hook(totX4, omega1), Hook(totX4, omega2), Hook(totX4, omega3);
 ${0}{,}{0}{,}{0}$ (2.11)

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

 J33 > evolX4 := EvolutionaryVector(X4);
 ${\mathrm{evolX4}}{:=}{-}{{w}}_{{1}{,}{2}{,}{3}}{}{{u}}_{{3}}{}{{\mathrm{D_u}}}_{{[}{]}}{-}{{w}}_{{1}{,}{2}{,}{3}}{}{{v}}_{{3}}{}{{\mathrm{D_v}}}_{{[}{]}}{-}{{w}}_{{1}{,}{2}{,}{3}}{}{{w}}_{{3}}{}{{\mathrm{D_w}}}_{{[}{]}}$ (2.12)
 J33 > totX4 &plus evolX4;
 ${{w}}_{{1}{,}{2}{,}{3}}{}{\mathrm{D_z}}$ (2.13)

 See Also