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

 The command AssignTransformationType($\mathbf{\φ}$ ) returns the transformation $\mathrm{\φ}$, but with internal representation $\mathrm{\phi }$ of  changed to encode its transformation type. The type of a transformation and its prolongation order can be determined by the command DGinfo with the keyword "TransformationType".

Any transformation of type [i]--[v] can be prolonged to higher order jet spaces. See Prolong for further information.

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

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

$\mathrm{DGsetup}\left(\left[x\,y\right]\,\left[u\right]\,E\,4\right)\:$$\mathrm{DGsetup}\left(\left[z\right]\,\left[v\right]\,F\,4\right)\:$$\mathrm{DGsetup}\left(\left[p\,q\right]\,\left[w\right]\,K\,4\right)\:$

$\mathrm{\Φ1}≔\mathrm{Transformation}\left(E\,F\,\left[z=A\left(x\,y\right)\,v\left[\right]=B\left(x\,y\,u\left[\right]\right)\right]\right)$

$\mathrm{\Φ1}≔\left[z=A\left(x\,y\right)\,v_{\left[\right]}=B\left(x\,y\,u_{\left[\right]}\right)\right]$

$\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{\Φ1}\,''TransformationType''\right)$

$\mathrm{newPhi1}≔\mathrm{AssignTransformationType}\left(\mathrm{\Φ1}\right)$

$\mathrm{newPhi1}≔\left[z=A\left(x\,y\right)\,v_{\left[\right]}=B\left(x\,y\,u_{\left[\right]}\right)\right]$

$\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{newPhi1}\,''TransformationType''\right)$

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

$\mathrm{\Φ2}≔\mathrm{Transformation}\left(E\,F\,\left[z=A\left(x\,y\,u\left[\right]\right)\,v\left[\right]=B\left(x\,y\,u\left[\right]\right)\right]\right)$

$\mathrm{\Φ2}≔\left[z=A\left(x\,y\,u_{\left[\right]}\right)\,v_{\left[\right]}=B\left(x\,y\,u_{\left[\right]}\right)\right]$

$\mathrm{newPhi2}≔\mathrm{AssignTransformationType}\left(\mathrm{\Φ2}\right)$

$\mathrm{newPhi2}≔\left[z=A\left(x\,y\,u_{\left[\right]}\right)\,v_{\left[\right]}=B\left(x\,y\,u_{\left[\right]}\right)\right]$

$\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{newPhi2}\,''TransformationType''\right)$

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

$\mathrm{\Φ3}≔\mathrm{Transformation}\left(E\,K\,\left[p=-u\left[1\right]\,q=y\,w\left[\right]=-u\left[1\right]x+u\left[\right]\,w\left[1\right]=x\,w\left[2\right]=u\left[2\right]\right]\right)$

$\mathrm{\Φ3}≔\left[p=-u_1\,q=y\,w_{\left[\right]}=-u_{1}x+u_{\left[\right]}\,w_1=x\,w_2=u_2\right]$

$\mathrm{newPhi3}≔\mathrm{AssignTransformationType}\left(\mathrm{\Φ3}\right)$

$\mathrm{newPhi3}≔\left[p=-u_1\,q=y\,w_{\left[\right]}=-u_{1}x+u_{\left[\right]}\,w_1=x\,w_2=u_2\right]$

$\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{newPhi3}\,''TransformationType''\right)$

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

$\mathrm{\Φ4}≔\mathrm{Transformation}\left(F\,E\,\left[x=z\,y=1\,u\left[\right]=v\left[\right]\,u\left[1\right]=v\left[1\right]\,u\left[2\right]=0\right]\right)$

$\mathrm{\Φ4}≔\left[x=z\,y=1\,u_{\left[\right]}=v_{\left[\right]}\,u_1=v_1\,u_2=0\right]$

$\mathrm{newPhi4}≔\mathrm{AssignTransformationType}\left(\mathrm{\Φ4}\right)$

$\mathrm{newPhi4}≔\left[x=z\,y=1\,u_{\left[\right]}=v_{\left[\right]}\,u_1=v_1\,u_2=0\right]$

$\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{newPhi3}\,''TransformationType''\right)$

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

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

$\mathrm{vars}≔x,y,u_{\left[\right]},u_1,u_2,u_{1,1},u_{1,2},u_{2,2}$

$\mathrm{\Φ5}≔\mathrm{Transformation}\left(E\,K\,\left[p=x\,q=y\,w\left[\right]=A\left(\mathrm{vars}\right)\right]\right)$

$\mathrm{\Φ5}≔\left[p=x\,q=y\,w_{\left[\right]}=A\left(x\,y\,u_{\left[\right]}\,u_1\,u_2\,u_{1,1}\,u_{1,2}\,u_{2,2}\right)\right]$

$\mathrm{newPhi5}≔\mathrm{AssignTransformationType}\left(\mathrm{\Φ5}\right)\:$

$\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{newPhi5}\,''TransformationType''\right)$

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

$\mathrm{\Φ5}≔\mathrm{Transformation}\left(E\,F\,\left[z=A\left(\mathrm{vars}\right)\,v\left[\right]=B\left(\mathrm{vars}\right)\right]\right)$

$\mathrm{\Φ5}≔\left[z=A\left(x\,y\,u_{\left[\right]}\,u_1\,u_2\,u_{1,1}\,u_{1,2}\,u_{2,2}\right)\,v_{\left[\right]}=B\left(x\,y\,u_{\left[\right]}\,u_1\,u_2\,u_{1,1}\,u_{1,2}\,u_{2,2}\right)\right]$

$\mathrm{newPhi5}≔\mathrm{AssignTransformationType}\left(\mathrm{\Φ5}\right)\:$

$\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{newPhi5}\,''TransformationType''\right)$

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

$\mathrm{\Φ6}≔\mathrm{Transformation}\left(E\,F\,\left[z=u\left[1\right]y\,v\left[\right]=u\left[2\right]+xu\left[\right]\,v\left[1\right]=y\right]\right)$

$\mathrm{\Φ6}≔\left[z=u_{1}y\,v_{\left[\right]}=x{u}_{\left[\right]}+u_2\,v_1=y\right]$

$\mathrm{newPhi6}≔\mathrm{AssignTransformationType}\left(\mathrm{\Φ6}\right)$

$\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{newPhi6}\,''TransformationType''\right)$

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