Let $\mathrm{\π}\:E\to M$be a fiber bundle, with base dimension $n$ and fiber dimension $m$ and let ${\mathrm{\π}}^k\:J^k\left(E\right) \to M$ be the $k$-th jet bundle with jet coordinates $\(x^i\, u^{\mathrm{\α}}\, u_{i_{}}^{\mathrm{\α}}\, u_{i_{}j}^{\mathrm{\α}}$, ..., $u_{\mathrm{ij} \cdot \cdot \cdot k}^{\mathrm{\α}}\)$. A total vector field on jet space is a vector field $Y$of the form $Y\= A^{\mathrm{\ℓ}}{\mathrm{D}}_{\mathrm{\ℓ}}$ , where the coefficients $A^{\mathrm{\ℓ}}$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,

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(...).

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

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

$\mathrm{X1}≔\mathrm{evalDG}\left(\mathrm{D\_x}\right)$

$\mathrm{X1}:=\mathrm{D\_x}$

$\mathrm{totX1}≔\mathrm{TotalVector}\left(\mathrm{X1}\right)$

$\mathrm{totX1}:=\mathrm{D\_x}\+u_1{\mathrm{D\_u}}_{\[\]}\+v_1{\mathrm{D\_v}}_{\[\]}$

$\mathrm{Prolong}\left(\mathrm{totX1}\,2\right)$

$\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}$

$\mathrm{X2}≔\mathrm{evalDG}\left(\mathrm{D\_u}\left[\right]\right)$

$\mathrm{X2}:={\mathrm{D\_u}}_{\[\]}$

$\mathrm{TotalVector}\left(\mathrm{X2}\right)$

$0\mathrm{D\_x}$

$\mathrm{X3}≔\mathrm{evalDG}\left(a\mathrm{D\_x}+b\mathrm{D\_y}+c\mathrm{D\_u}\left[\right]+d\mathrm{D\_v}\left[\right]\right)$

$\mathrm{X3}:=a\mathrm{D\_x}\+b\mathrm{D\_y}\+c{\mathrm{D\_u}}_{\[\]}\+d{\mathrm{D\_v}}_{\[\]}$

$\mathrm{totX3}≔\mathrm{TotalVector}\left(\mathrm{X3}\right)$

$\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}}_{\[\]}$

$\mathrm{DGsetup}\left(\left[x\,y\,z\right]\,\left[u\,v\,w\right]\,\mathrm{J33}\,3\right)\:$

$\mathrm{X4}≔w\left[1,2,3\right]\mathrm{D\_z}$

$\mathrm{X4}:=w_{1\,2\,3}\mathrm{D\_z}$

$\mathrm{totX4}≔\mathrm{TotalVector}\left(\mathrm{X4}\right)$

$\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}}_{\[\]}$

$\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}}_{\[\]}$

$\mathrm{Hook}\left(\mathrm{totX4}\,\mathrm{\ω1}\right),\mathrm{Hook}\left(\mathrm{totX4}\,\mathrm{\ω2}\right),\mathrm{Hook}\left(\mathrm{totX4}\,\mathrm{\ω3}\right)$

$0\,0\,0$

$\mathrm{evolX4}≔\mathrm{EvolutionaryVector}\left(\mathrm{X4}\right)$

$\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}}_{\[\]}$

$\mathrm{totX4}\&plus\mathrm{evolX4}$

$w_{1\,2\,3}\mathrm{D\_z}$