Let $\mathrm{\π}\:E\to M$ be a fiber bundle and let ${\mathrm{\π}}^k\:J^k\left(E\right)\to M$be the associated bundle of $k-$jets.

 The command AssignVectorType(X) returns the vector field $X$, but with internal representation of $X$ changed to encode its vector type. The type of a vector and its prolongation order can be determined by the command DGinfo with the keyword "VectorType".

Any of the vector fields [i]--[vi] admit a unique lift to a vector field with values in $J^p\left(E\right)$by the Lie prolongation formula. We call, for example, a vector field on $J^p\left(E\right)$obtained from the prolongation of a point vector field a prolonged point vector field. See Prolong for further information.

Explicit coordinate formulas for each of these types of vector fields in the case $E \={\mathrm{\ℝ}}^2\times \mathrm{\ℝ} \to {\mathrm{\ℝ}}^2$ are presented in Example 1.

The type of a vector field and its prolongation order can be determined by the command DGinfo with keyword "VectorType".

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

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

$\mathrm{DGsetup}\left(\left[x\,y\right]\,\left[u\right]\,\mathrm{J22}\,4\right)\:$

$\mathrm{X1}≔\mathrm{evalDG}\left(A\left(x\,y\,u\left[\right]\right)\mathrm{D\_x}+B\left(x\,y\,u\left[\right]\right)\mathrm{D\_y}+C\left(x\,y\,u\left[\right]\right)\mathrm{D\_u}\left[\right]\right)$

$\mathrm{X1}:=A\left(x\,y\,u_{\left[\right]}\right)\mathrm{D\_x}\+B\left(x\,y\,u_{\left[\right]}\right)\mathrm{D\_y}\+C\left(x\,y\,u_{\left[\right]}\right){\mathrm{D\_u}}_{\left[\right]}$

$\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{X1}\,''VectorType''\right)$

$\mathrm{newX1}≔\mathrm{AssignVectorType}\left(\mathrm{X1}\right)$

$\mathrm{newX1}:=A\left(x\,y\,u_{\left[\right]}\right)\mathrm{D\_x}\+B\left(x\,y\,u_{\left[\right]}\right)\mathrm{D\_y}\+C\left(x\,y\,u_{\left[\right]}\right){\mathrm{D\_u}}_{\left[\right]}$

$\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{newX1}\,''VectorType''\right)$

$\left[''point''\,0\right]$

$\mathrm{X2}≔\mathrm{evalDG}\left(A\left(x\,y\right)\mathrm{D\_x}+B\left(x\,y\right)\mathrm{D\_y}+C\left(x\,y\,u\left[\right]\right)\mathrm{D\_u}\left[\right]\right)$

$\mathrm{X2}:=A\left(x\,y\right)\mathrm{D\_x}\+B\left(x\,y\right)\mathrm{D\_y}\+C\left(x\,y\,u_{\left[\right]}\right){\mathrm{D\_u}}_{\left[\right]}$

$\mathrm{newX2}≔\mathrm{AssignVectorType}\left(\mathrm{X2}\right)$

$\mathrm{newX2}:=A\left(x\,y\right)\mathrm{D\_x}\+B\left(x\,y\right)\mathrm{D\_y}\+C\left(x\,y\,u_{\left[\right]}\right){\mathrm{D\_u}}_{\left[\right]}$

$\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{newX2}\,''VectorType''\right)$

$\left[''projectable''\,0\right]$

$\mathrm{PDEtools}\left[\mathrm{declare}\right]\left(S\left(x\,y\,u\left[\right]\,u\left[1\right]\,u\left[2\right]\right)\,\mathrm{quiet}\right)$

$\mathrm{X3}≔\mathrm{GeneratingFunctionToContactVector}\left(S\left(x\,y\,u\left[\right]\,u\left[1\right]\,u\left[2\right]\right)\right)$

$\mathrm{X3}:=-S_{u_1}\mathrm{D\_x}-S_{u_2}\mathrm{D\_y}\+\left(-u_1{S}_{u_1}-u_2{S}_{u_2}+S\right){\mathrm{D\_u}}_{\left[\right]}\+\left(u_1{S}_{u_{\left[\right]}}+S_x\right){\mathrm{D\_u}}_1\+\left(u_2{S}_{u_{\left[\right]}}+S_y\right){\mathrm{D\_u}}_2$

$\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{X3}\,''VectorType''\right)$

$\mathrm{newX3}≔\mathrm{AssignVectorType}\left(\mathrm{X3}\right)\:$

$\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{newX3}\,''VectorType''\right)$

$\left[''contact''\,1\right]$

$\mathrm{vars}≔x,y,u\left[\right],u\left[1\right],u\left[2\right]\:$

$\mathrm{X4}≔\mathrm{evalDG}\left(A\left(\mathrm{vars}\right)\mathrm{D\_x}+B\left(\mathrm{vars}\right)\mathrm{D\_y}+C\left(\mathrm{vars}\right)\mathrm{D\_u}\left[\right]\right)$

$\mathrm{X4}:=A\left(x\,y\,u_{\left[\right]}\,u_1\,u_2\right)\mathrm{D\_x}\+B\left(x\,y\,u_{\left[\right]}\,u_1\,u_2\right)\mathrm{D\_y}\+C\left(x\,y\,u_{\left[\right]}\,u_1\,u_2\right){\mathrm{D\_u}}_{\left[\right]}$

$\mathrm{newX4}≔\mathrm{AssignVectorType}\left(\mathrm{X4}\right)\:$

$\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{newX4}\,''VectorType''\right)$

$\left[''generalized''\,0\right]$

$\mathrm{vars}≔x,y,u\left[\right],u\left[1\right],u\left[2\right],u\left[1,1\right],u\left[1,1,1\right],u\left[1,1,1,1\right]\:$

$\mathrm{X4}≔\mathrm{evalDG}\left(A\left(\mathrm{vars}\right)\mathrm{D\_x}\right)$

$\mathrm{X4}:=A\left(x\,y\,u_{\left[\right]}\,u_1\,u_2\,u_{1,1}\,u_{1,1,1}\,u_{1,1,1,1}\right)\mathrm{D\_x}$

$\mathrm{newX4}≔\mathrm{AssignVectorType}\left(\mathrm{X4}\right)\:$

$\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{newX4}\,''VectorType''\right)$

$\left[''generalized''\,0\right]$

$\mathrm{vars}≔x,y,u\left[\right],u\left[1\right],u\left[2\right],u\left[1,1\right],u\left[1,2\right],u\left[2,2\right]\:$

$\mathrm{X5}≔\mathrm{evalDG}\left(C\left(\mathrm{vars}\right)\mathrm{D\_u}\left[\right]\right)$

$\mathrm{X5}:=C\left(x\,y\,u_{\left[\right]}\,u_1\,u_2\,u_{1,1}\,u_{1,2}\,u_{2,2}\right){\mathrm{D\_u}}_{\left[\right]}$

$\mathrm{newX5}≔\mathrm{AssignVectorType}\left(\mathrm{X5}\right)\:$

$\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{newX5}\,''VectorType''\right)$

$\left[''evolutionary''\,0\right]$

$\mathrm{X6}≔\mathrm{evalDG}\left(A\mathrm{D\_x}+B\mathrm{D\_y}+\left(Au\left[1\right]+Bu\left[2\right]\right)\mathrm{D\_u}\left[\right]\right)$

$\mathrm{X6}:=A\mathrm{D\_x}\+B\mathrm{D\_y}\+\left(A{u}_1+B{u}_2\right){\mathrm{D\_u}}_{\left[\right]}$

$\mathrm{newX6}≔\mathrm{AssignVectorType}\left(\mathrm{X6}\right)\:$

$\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{newX6}\,''VectorType''\right)$

$\left[''total''\,0\right]$

$\mathrm{X7}≔\mathrm{evalDG}\left(\mathrm{D\_u}\left[1\right]\right)$

$\mathrm{X7}≔{\mathrm{D\_u}}_1$

$\mathrm{newX7}≔\mathrm{AssignVectorType}\left(\mathrm{X7}\right)\:$

$\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{newX7}\,''VectorType''\right)$

$\left[''generic''\,''NA''\right]$

$\mathrm{DGsetup}\left(\left[x\right]\,\left[u\right]\,E\,7\right)\:$

$\mathrm{X1}≔\mathrm{evalDG}\left(u\left[\right]\mathrm{D\_x}-x\mathrm{D\_u}\left[\right]+\left(-1-{u\left[1\right]}^2\right)\mathrm{D\_u}\left[1\right]-3u\left[1\right]u\left[1,1\right]\mathrm{D\_u}\left[1,1\right]\right)$

$\mathrm{X1}≔u_{\left[\right]}\mathrm{D\_x}-x{\mathrm{D\_u}}_{\left[\right]}-\left(u_1^2+1\right){\mathrm{D\_u}}_1-3u_1{u}_{1,1}{\mathrm{D\_u}}_{1,1}$

$\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{X1}\,''VectorType''\right)$

$\mathrm{newX1}≔\mathrm{AssignVectorType}\left(\mathrm{X1}\right)$

$\mathrm{newX1}≔u_{\left[\right]}\mathrm{D\_x}-x{\mathrm{D\_u}}_{\left[\right]}-\left(u_1^2+1\right){\mathrm{D\_u}}_1-3u_1{u}_{1,1}{\mathrm{D\_u}}_{1,1}$

$\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{newX1}\,''VectorType''\right)$

$\left[''point''\,2\right]$

$\mathrm{X2}≔\mathrm{evalDG}\left(x\mathrm{D\_x}+u\left[0\right]\mathrm{D\_u}\left[\right]-u\left[2\right]\mathrm{D\_u}\left[1\right]-2u\left[1,1,1\right]\mathrm{D\_u}\left[1,1,1\right]-3u\left[1,1,1,1\right]\mathrm{D\_u}\left[1,1,1,1\right]\right)$

$\mathrm{X2}≔x\mathrm{D\_x}+u_0{\mathrm{D\_u}}_{\left[\right]}-u_2{\mathrm{D\_u}}_1-2u_{1,1,1}{\mathrm{D\_u}}_{1,1,1}-3u_{1,1,1,1}{\mathrm{D\_u}}_{1,1,1,1}$

$\mathrm{newX2}≔\mathrm{AssignVectorType}\left(\mathrm{X2}\right)$

$\mathrm{newX2}≔x\mathrm{D\_x}+u_0{\mathrm{D\_u}}_{\left[\right]}-u_2{\mathrm{D\_u}}_1-2u_{1,1,1}{\mathrm{D\_u}}_{1,1,1}-3u_{1,1,1,1}{\mathrm{D\_u}}_{1,1,1,1}$

$\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{newX2}\,''VectorType''\right)$

$\left[''generic''\,''NA''\right]$

$\mathrm{X3}≔\mathrm{evalDG}\left(-2u\left[1\right]\mathrm{D\_x}+\left(-{u\left[1\right]}^2+x^2\right)\mathrm{D\_u}\left[\right]+2x\mathrm{D\_u}\left[1\right]+\left(2+2{u\left[1,1\right]}^2\right)\mathrm{D\_u}\left[1,1\right]+6u\left[2\right]u\left[1,1,1\right]\mathrm{D\_u}\left[1,1,1\right]\right)$

$\mathrm{X3}≔-2u_1\mathrm{D\_x}+\left(x^2-u_1^2\right){\mathrm{D\_u}}_{\left[\right]}+2x{\mathrm{D\_u}}_1+\left(2u_{1,1}^2+2\right){\mathrm{D\_u}}_{1,1}+6u_2{u}_{1,1,1}{\mathrm{D\_u}}_{1,1,1}$

$\mathrm{newX3}≔\mathrm{AssignVectorType}\left(\mathrm{X3}\right)$

$\mathrm{newX3}≔-2u_1\mathrm{D\_x}+\left(x^2-u_1^2\right){\mathrm{D\_u}}_{\left[\right]}+2x{\mathrm{D\_u}}_1+\left(2u_{1,1}^2+2\right){\mathrm{D\_u}}_{1,1}+6u_2{u}_{1,1,1}{\mathrm{D\_u}}_{1,1,1}$

$\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{newX3}\,''VectorType''\right)$

$\left[''generic''\,''NA''\right]$

$\mathrm{X4}≔\mathrm{evalDG}\left(u\left[1,1\right]\mathrm{D\_x}-u\left[1,1,1\right]\mathrm{D\_u}\left[\right]+\left(-u\left[1,1,1,1\right]-u\left[1\right]u\left[1,1,1\right]\right)\mathrm{D\_u}\left[1\right]\right)$

$\mathrm{X4}≔u_{1,1}\mathrm{D\_x}-u_{1,1,1}{\mathrm{D\_u}}_{\left[\right]}-\left(u_1{u}_{1,1,1}+u_{1,1,1,1}\right){\mathrm{D\_u}}_1$

$\mathrm{newX4}≔\mathrm{AssignVectorType}\left(\mathrm{X4}\right)\:$

$\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{newX4}\,''VectorType''\right)$

$\left[''generalized''\,1\right]$

$\mathrm{X5}≔\mathrm{evalDG}\left(u\left[1,1\right]\mathrm{D\_u}\left[\right]+u\left[1,1,1\right]\mathrm{D\_u}\left[1\right]+u\left[1,1,1,1\right]\mathrm{D\_u}\left[1,1\right]+u\left[1,1,1,1,1\right]\mathrm{D\_u}\left[1,1,1\right]+u\left[1,1,1,1,1,1\right]\mathrm{D\_u}\left[1,1,1,1\right]+u\left[1,1,1,1,1,1,1\right]\mathrm{D\_u}\left[1,1,1,1,1,1\right]\right)$

$\mathrm{X5}:=u_{1,1}{\mathrm{D\_u}}_{\left[\right]}\+u_{1,1,1}{\mathrm{D\_u}}_1\+u_{1,1,1,1}{\mathrm{D\_u}}_{1,1}\+u_{1,1,1,1,1}{\mathrm{D\_u}}_{1,1,1}\+u_{1,1,1,1,1,1}{\mathrm{D\_u}}_{1,1,1,1}\+u_{1,1,1,1,1,1,1}{\mathrm{D\_u}}_{1,1,1,1,1,1}$

$\mathrm{newX5}≔\mathrm{AssignVectorType}\left(\mathrm{X5}\right)\:$

$\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{newX5}\,''VectorType''\right)$

$\left[''generic''\,''NA''\right]$

$\mathrm{X6}≔\mathrm{Prolong}\left(\mathrm{D\_x}+u\left[1\right]\mathrm{D\_u}\left[\right]\,3\right)$

$\mathrm{X6}:=\mathrm{D\_x}\+u_1{\mathrm{D\_u}}_{\left[\right]}\+u_{1,1}{\mathrm{D\_u}}_1\+u_{1,1,1}{\mathrm{D\_u}}_{1,1}\+u_{1,1,1,1}{\mathrm{D\_u}}_{1,1,1}$

$\mathrm{newX6}≔\mathrm{AssignVectorType}\left(\mathrm{X5}\right)\:$

$\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{newX6}\,''VectorType''\right)$

$\left[''generic''\,''NA''\right]$