Let $\mathrm{\π}\:E\to M$be a fiber bundle, with base dimension $n$ and fiber dimension $m$ and let ${\mathrm{\π}}^{\mathrm{\∞}} \:J^{\mathrm{\∞}}\left(E\right) \to M$be the $\mathrm{\∞}$-th jet bundle of $E$. The space of $p$-forms ${\mathrm{\Ω}}^p\left(J^{\mathrm{\∞}}\left(E\right)\right)$ decomposes into a direct sum${\mathrm{\Ω}}^p\left(J^{\infty } \right)\= \underset{r\+s \=p}{\overset{}{⨁}} {\mathrm{\Omega }}^{\left(r\,s\right)}\left(J^{\infty }\left(E\right)\right)$, where ${\mathrm{\Omega }}^{\left(r\,s\right)} \left(J^{\infty }\left(E\right)\right)$is the space of bi-forms of horizontal degree $r$ and vertical degree $s\.$The precise definition of the space ${\mathrm{\Ω}}^{\left(r\,s\right)}\left(J^{\infty }\left(E\right)\right)$is given in the help page for the horizontal exterior derivative. If $\mathrm{\ω} \in {\mathrm{\Ω}}^p\left(J^{\infty } \right)$, then let ${\mathrm{\ω}}^{\left[r\,s\right]}$denote the type $\left(r\,s\right)$component of $\mathrm{\ω}$in the decomposition (*). The command convert/DGbifom calculates the various bi-graded components of a form $\mathrm{\ω} \in {\mathrm{\Ω}}^p\left(J^{\infty } \right)$.Let $F\to N$$$be another fiber bundle and let $\mathrm{\φ}\: J^k\left(E\right) \to j^{\mathrm{\ℓ}}\left(F\right)$. Let $\mathrm{\η}$be a differential bi-form of type $\left(r\,s\right)$on $J^{\ell }\left(F\right)$. Then the projected pullback of $\mathrm{\η}$is denote by ${\mathrm{\φ}}^{†}\left(\mathrm{\ω}\right)$and defined by ${\mathrm{\φ}}^{†}\left(\mathrm{\η}\right) \= {\left({\mathrm{\Φ}}^{\*}\left(\mathrm{\η}\right)\right)}^{\left[r\,s\right]}$.

 Two special cases of this general definition should be noted.

Use ProjectedPullback to transform a Lagrangian bi-form to a new Lagrangian bi-form using any of the above transformations.$$

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

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

$\mathrm{DGsetup}\left(\left[x\,y\right]\,\left[u\right]\,\mathrm{E1}\,2\right)\:$$\mathrm{DGsetup}\left(\left[t\right]\,\left[v\right]\,\mathrm{E2}\,2\right)\:$$\mathrm{DGsetup}\left(\left[p\,q\,r\right]\,\left[w\right]\,\mathrm{E3}\,2\right)\:$

$\mathrm{\Φ1}≔\mathrm{Transformation}\left(\mathrm{E1}\,\mathrm{E2}\,\left[t\=x\,v_{\[\]}\=x^2{u}_{\[\]}\right]\right)$

$\mathrm{\Φ1}:=\mathrm{\_DG}\left(\left[\left[''transformation''\,\left[\left[\mathrm{E1}\,0\right]\,\left[\mathrm{E2}\,0\right]\right]\,\left[\right]\,\left[\left[\begin{array}{ccc}1& 0& 0\\ 2x{u}_{\left[\right]}& 0& x^2\end{array}\right]\right]\right]\,\left[\left[x\,t\right]\,\left[x^2{u}_{\left[\right]}\,v_{\left[\right]}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''transformation''\,\left[\left[\mathrm{E1}\,0\right]\,\left[\mathrm{E2}\,0\right]\right]\,\left[\right]\,\left[\left[\begin{array}{ccc}1& 0& 0\\ 2x{u}_{\left[\right]}& 0& x^2\end{array}\right]\right]\right]\,\left[\left[x\,t\right]\,\left[x^2{u}_{\left[\right]}\,v_{\left[\right]}\right]\right]\right]\right)$

$\mathrm{prPhi1}≔\mathrm{Prolong}\left(\mathrm{\Φ1}\,2\right)$

$\mathrm{prPhi1}:=\mathrm{\_DG}\left(\left[\left[''transformation''\,\left[\left[\mathrm{E1}\,2\right]\,\left[\mathrm{E2}\,2\right]\right]\,\left[''projectable''\,2\right]\,\left[\left[\begin{array}{cccccccc}1& 0& 0& 0& 0& 0& 0& 0\\ 2x{u}_{\left[\right]}& 0& x^2& 0& 0& 0& 0& 0\\ 2x{u}_1+2u_{\left[\right]}& 0& 2x& x^2& 0& 0& 0& 0\\ 2x{u}_{1,1}+4u_1& 0& 2& 4x& 0& x^2& 0& 0\end{array}\right]\right]\right]\,\left[\left[x\,t\right]\,\left[x^2{u}_{\left[\right]}\,v_{\left[\right]}\right]\,\left[x^2{u}_1+2x{u}_{\left[\right]}\,v_1\right]\,\left[x^2{u}_{1,1}+4x{u}_1+2u_{\left[\right]}\,v_{1,1}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''transformation''\,\left[\left[\mathrm{E1}\,2\right]\,\left[\mathrm{E2}\,2\right]\right]\,\left[''projectable''\,2\right]\,\left[\left[\begin{array}{cccccccc}1& 0& 0& 0& 0& 0& 0& 0\\ 2x{u}_{\left[\right]}& 0& x^2& 0& 0& 0& 0& 0\\ 2x{u}_1+2u_{\left[\right]}& 0& 2x& x^2& 0& 0& 0& 0\\ 2x{u}_{1,1}+4u_1& 0& 2& 4x& 0& x^2& 0& 0\end{array}\right]\right]\right]\,\left[\left[x\,t\right]\,\left[x^2{u}_{\left[\right]}\,v_{\left[\right]}\right]\,\left[x^2{u}_1+2x{u}_{\left[\right]}\,v_1\right]\,\left[x^2{u}_{1,1}+4x{u}_1+2u_{\left[\right]}\,v_{1,1}\right]\right]\right]\right)$

$\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{prPhi1}\,''TransformationType''\right)$

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

$\mathrm{Pullback}\left(\mathrm{prPhi1}\,{\mathrm{Cv}}_1\right)$

$\mathrm{\_DG}\left(\left[\left[''biform''\,\mathrm{E1}\,\left[0\,1\right]\right]\,\left[\left[\left[3\right]\,2x\right]\,\left[\left[4\right]\,x^2\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''biform''\,\mathrm{E1}\,\left[0\,1\right]\right]\,\left[\left[\left[3\right]\,2x\right]\,\left[\left[4\right]\,x^2\right]\right]\right]\right)$

$\mathrm{ProjectedPullback}\left(\mathrm{prPhi1}\,{\mathrm{Cv}}_1\right)$

$\mathrm{\_DG}\left(\left[\left[''biform''\,\mathrm{E1}\,\left[0\,1\right]\right]\,\left[\left[\left[3\right]\,2x\right]\,\left[\left[4\right]\,x^2\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''biform''\,\mathrm{E1}\,\left[0\,1\right]\right]\,\left[\left[\left[3\right]\,2x\right]\,\left[\left[4\right]\,x^2\right]\right]\right]\right)$

$\mathrm{\Φ2}≔\mathrm{Transformation}\left(\mathrm{E1}\,\mathrm{E3}\,\left[p\=u_{\[\]}\,q\=y\,r\=1\,w_{\[\]}\=x\right]\right)$

$\mathrm{\Φ2}:=\mathrm{\_DG}\left(\left[\left[''transformation''\,\left[\left[\mathrm{E1}\,0\right]\,\left[\mathrm{E3}\,0\right]\right]\,\left[\right]\,\left[\left[\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 0& 0& 0\\ 1& 0& 0\end{array}\right]\right]\right]\,\left[\left[u_{\left[\right]}\,p\right]\,\left[y\,q\right]\,\left[1\,r\right]\,\left[x\,w_{\left[\right]}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''transformation''\,\left[\left[\mathrm{E1}\,0\right]\,\left[\mathrm{E3}\,0\right]\right]\,\left[\right]\,\left[\left[\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 0& 0& 0\\ 1& 0& 0\end{array}\right]\right]\right]\,\left[\left[u_{\left[\right]}\,p\right]\,\left[y\,q\right]\,\left[1\,r\right]\,\left[x\,w_{\left[\right]}\right]\right]\right]\right)$

$\mathrm{prPhi2}≔\mathrm{Prolong}\left(\mathrm{\Φ2}\,1\right)$

$\mathrm{prPhi2}:=\mathrm{\_DG}\left(\left[\left[''transformation''\,\left[\left[\mathrm{E1}\,1\right]\,\left[\mathrm{E3}\,1\right]\right]\,\left[''point''\,1\right]\,\left[\left[\begin{array}{ccccc}0& 0& 1& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 1& 0& 0& 0& 0\\ 0& 0& 0& -\frac{1}{u_1^2}& 0\\ 0& 0& 0& \frac{u_2}{u_1^2}& -\frac{1}{u_1}\\ 0& 0& 0& 0& 0\end{array}\right]\right]\right]\,\left[\left[u_{\left[\right]}\,p\right]\,\left[y\,q\right]\,\left[1\,r\right]\,\left[x\,w_{\left[\right]}\right]\,\left[\frac{1}{u_1}\,w_1\right]\,\left[-\frac{u_2}{u_1}\,w_2\right]\,\left[0\,w_3\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''transformation''\,\left[\left[\mathrm{E1}\,1\right]\,\left[\mathrm{E3}\,1\right]\right]\,\left[''point''\,1\right]\,\left[\left[\begin{array}{ccccc}0& 0& 1& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 1& 0& 0& 0& 0\\ 0& 0& 0& -\frac{1}{u_1^2}& 0\\ 0& 0& 0& \frac{u_2}{u_1^2}& -\frac{1}{u_1}\\ 0& 0& 0& 0& 0\end{array}\right]\right]\right]\,\left[\left[u_{\left[\right]}\,p\right]\,\left[y\,q\right]\,\left[1\,r\right]\,\left[x\,w_{\left[\right]}\right]\,\left[\frac{1}{u_1}\,w_1\right]\,\left[-\frac{u_2}{u_1}\,w_2\right]\,\left[0\,w_3\right]\right]\right]\right)$

$\mathrm{ProjectedPullback}\left(\mathrm{prPhi2}\,\mathrm{Dp}\right)$

$\mathrm{\_DG}\left(\left[\left[''biform''\,\mathrm{E1}\,\left[1\,0\right]\right]\,\left[\left[\left[1\right]\,u_1\right]\,\left[\left[2\right]\,u_2\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''biform''\,\mathrm{E1}\,\left[1\,0\right]\right]\,\left[\left[\left[1\right]\,u_1\right]\,\left[\left[2\right]\,u_2\right]\right]\right]\right)$

$\mathrm{\ω}≔\mathrm{Dp}\&wedge\left({\mathrm{Cw}}_{\[\]}\right)$

$\mathrm{\ω}:=\mathrm{\_DG}\left(\left[\left[''biform''\,\mathrm{E3}\,\left[1\,1\right]\right]\,\left[\left[\left[1\,4\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''biform''\,\mathrm{E3}\,\left[1\,1\right]\right]\,\left[\left[\left[1\,4\right]\,1\right]\right]\right]\right)$

$\mathrm{ProjectedPullback}\left(\mathrm{prPhi2}\,\mathrm{\ω}\right)$

$\mathrm{\_DG}\left(\left[\left[''biform''\,\mathrm{E1}\,\left[1\,1\right]\right]\,\left[\left[\left[1\,3\right]\,\mathrm{-1}\right]\,\left[\left[2\,3\right]\,-\frac{u_2}{u_1}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''biform''\,\mathrm{E1}\,\left[1\,1\right]\right]\,\left[\left[\left[1\,3\right]\,\mathrm{-1}\right]\,\left[\left[2\,3\right]\,-\frac{u_2}{u_1}\right]\right]\right]\right)$

$\mathrm{\θ1}≔\mathrm{convert}\left(\mathrm{\ω}\,\mathrm{DGform}\right)$

$\mathrm{\θ1}:=\mathrm{\_DG}\left(\left[\left[''form''\,\mathrm{E3}\,2\right]\,\left[\left[\left[1\,2\right]\,-w_2\right]\,\left[\left[1\,3\right]\,-w_3\right]\,\left[\left[1\,4\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,\mathrm{E3}\,2\right]\,\left[\left[\left[1\,2\right]\,-w_2\right]\,\left[\left[1\,3\right]\,-w_3\right]\,\left[\left[1\,4\right]\,1\right]\right]\right]\right)$

$\mathrm{\θ2}≔\mathrm{Pullback}\left(\mathrm{prPhi2}\,\mathrm{\θ1}\right)$

$\mathrm{\θ2}:=\mathrm{\_DG}\left(\left[\left[''form''\,\mathrm{E1}\,2\right]\,\left[\left[\left[1\,3\right]\,\mathrm{-1}\right]\,\left[\left[2\,3\right]\,-\frac{u_2}{u_1}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''form''\,\mathrm{E1}\,2\right]\,\left[\left[\left[1\,3\right]\,\mathrm{-1}\right]\,\left[\left[2\,3\right]\,-\frac{u_2}{u_1}\right]\right]\right]\right)$

$\mathrm{\θ3}≔\mathrm{convert}\left(\mathrm{\θ2}\,\mathrm{DGbiform}\,\left[1\,1\right]\right)$

$\mathrm{\θ3}:=\mathrm{\_DG}\left(\left[\left[''biform''\,\mathrm{E1}\,\left[1\,1\right]\right]\,\left[\left[\left[1\,3\right]\,\mathrm{-1}\right]\,\left[\left[2\,3\right]\,-\frac{u_2}{u_1}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''biform''\,\mathrm{E1}\,\left[1\,1\right]\right]\,\left[\left[\left[1\,3\right]\,\mathrm{-1}\right]\,\left[\left[2\,3\right]\,-\frac{u_2}{u_1}\right]\right]\right]\right)$

$\mathrm{\Φ3}≔\mathrm{Transformation}\left(\mathrm{E2}\,\mathrm{E1}\,\left[x\=v_{\[\]}\,y\=v_1\,u_{\[\]}\=v_2\right]\right)$

$\mathrm{\Φ3}:=\mathrm{\_DG}\left(\left[\left[''transformation''\,\left[\left[\mathrm{E2}\,1\right]\,\left[\mathrm{E1}\,0\right]\right]\,\left[\right]\,\left[\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right]\right]\right]\,\left[\left[v_{\left[\right]}\,x\right]\,\left[v_1\,y\right]\,\left[v_2\,u_{\left[\right]}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''transformation''\,\left[\left[\mathrm{E2}\,1\right]\,\left[\mathrm{E1}\,0\right]\right]\,\left[\right]\,\left[\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right]\right]\right]\,\left[\left[v_{\left[\right]}\,x\right]\,\left[v_1\,y\right]\,\left[v_2\,u_{\left[\right]}\right]\right]\right]\right)$

$\mathrm{prPhi3}≔\mathrm{Prolong}\left(\mathrm{\Φ3}\,1\right)$

$\mathrm{prPhi3}:=\mathrm{\_DG}\left(\left[\left[''transformation''\,\left[\left[\mathrm{E2}\,2\right]\,\left[\mathrm{E1}\,1\right]\right]\,\left[''generalizedDifferentialSubstitution''\,1\right]\,\left[\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 0\\ 0& 0& -\frac{2v_1^2{v}_{1,2}}{{\left(v_1^2+v_{1,1}^2\right)}^2}+\frac{v_{1,2}}{v_1^2+v_{1,1}^2}& -\frac{2v_1{v}_{1,2}{v}_{1,1}}{{\left(v_1^2+v_{1,1}^2\right)}^2}\\ 0& 0& -\frac{2v_1{v}_{1,2}{v}_{1,1}}{{\left(v_1^2+v_{1,1}^2\right)}^2}& -\frac{2v_{1,1}^2{v}_{1,2}}{{\left(v_1^2+v_{1,1}^2\right)}^2}+\frac{v_{1,2}}{v_1^2+v_{1,1}^2}\end{array}\right]\right]\right]\,\left[\left[v_{\left[\right]}\,x\right]\,\left[v_1\,y\right]\,\left[v_2\,u_{\left[\right]}\right]\,\left[\frac{v_1{v}_{1,2}}{v_1^2+v_{1,1}^2}\,u_1\right]\,\left[\frac{v_{1,1}{v}_{1,2}}{v_1^2+v_{1,1}^2}\,u_2\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''transformation''\,\left[\left[\mathrm{E2}\,2\right]\,\left[\mathrm{E1}\,1\right]\right]\,\left[''generalizedDifferentialSubstitution''\,1\right]\,\left[\left[\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 0\\ 0& 0& -\frac{2v_1^2{v}_{1,2}}{{\left(v_1^2+v_{1,1}^2\right)}^2}+\frac{v_{1,2}}{v_1^2+v_{1,1}^2}& -\frac{2v_1{v}_{1,2}{v}_{1,1}}{{\left(v_1^2+v_{1,1}^2\right)}^2}\\ 0& 0& -\frac{2v_1{v}_{1,2}{v}_{1,1}}{{\left(v_1^2+v_{1,1}^2\right)}^2}& -\frac{2v_{1,1}^2{v}_{1,2}}{{\left(v_1^2+v_{1,1}^2\right)}^2}+\frac{v_{1,2}}{v_1^2+v_{1,1}^2}\end{array}\right]\right]\right]\,\left[\left[v_{\left[\right]}\,x\right]\,\left[v_1\,y\right]\,\left[v_2\,u_{\left[\right]}\right]\,\left[\frac{v_1{v}_{1,2}}{v_1^2+v_{1,1}^2}\,u_1\right]\,\left[\frac{v_{1,1}{v}_{1,2}}{v_1^2+v_{1,1}^2}\,u_2\right]\right]\right]\right)$

$\mathrm{ProjectedPullback}\left(\mathrm{prPhi3}\,2\mathrm{Dx}\+3\mathrm{Dy}\right)$

$\mathrm{\_DG}\left(\left[\left[''biform''\,\mathrm{E2}\,\left[1\,0\right]\right]\,\left[\left[\left[1\right]\,3v_{1,1}+2v_1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''biform''\,\mathrm{E2}\,\left[1\,0\right]\right]\,\left[\left[\left[1\right]\,3v_{1,1}+2v_1\right]\right]\right]\right)$

$\mathrm{ProjectedPullback}\left(\mathrm{prPhi3}\,u_1{\mathrm{Cu}}_{\[\]}\right)$

$\mathrm{\_DG}\left(\left[\left[''biform''\,\mathrm{E2}\,\left[0\,1\right]\right]\,\left[\left[\left[2\right]\,-\frac{v_1^2{v}_{1,2}^2}{{\left(v_1^2+v_{1,1}^2\right)}^2}\right]\,\left[\left[3\right]\,-\frac{v_1{v}_{1,2}^2{v}_{1,1}}{{\left(v_1^2+v_{1,1}^2\right)}^2}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''biform''\,\mathrm{E2}\,\left[0\,1\right]\right]\,\left[\left[\left[2\right]\,-\frac{v_1^2{v}_{1,2}^2}{{\left(v_1^2+v_{1,1}^2\right)}^2}\right]\,\left[\left[3\right]\,-\frac{v_1{v}_{1,2}^2{v}_{1,1}}{{\left(v_1^2+v_{1,1}^2\right)}^2}\right]\right]\right]\right)$