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)\:$

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

$\mathrm{\Φ1}:=\left[t=x\,v_{\left[\right]}=x^2{u}_{\left[\right]}\right]$

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

$\mathrm{prPhi1}:=\left[t=x\,v_{\left[\right]}=x^2{u}_{\left[\right]}\,v_1=x^2{u}_1+2x{u}_{\left[\right]}\,v_{1,1}=x^2{u}_{1,1}+4x{u}_1+2u_{\left[\right]}\right]$

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

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

$\mathrm{Pullback}\left(\mathrm{prPhi1}\,\mathrm{Cv}\left[1\right]\right)$

$2x{\mathrm{Cu}}_{\left[\right]}+x^2{\mathrm{Cu}}_1$

$\mathrm{ProjectedPullback}\left(\mathrm{prPhi1}\,\mathrm{Cv}\left[1\right]\right)$

$2x{\mathrm{Cu}}_{\left[\right]}+x^2{\mathrm{Cu}}_1$

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

$\mathrm{\Φ2}:=\left[p=u_{\left[\right]}\,q=y\,r=1\,w_{\left[\right]}=x\right]$

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

$\mathrm{prPhi2}:=\left[p=u_{\left[\right]}\,q=y\,r=1\,w_{\left[\right]}=x\,w_1=\frac{1}{u_1}\,w_2=-\frac{u_2}{u_1}\,w_3=0\right]$

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

$u_1\mathrm{Dx}+u_2\mathrm{Dy}$

$\mathrm{\omega }≔\mathrm{Dp}\&wedge\mathrm{Cw}\left[\right]$

$\mathrm{\ω}:=\mathrm{Dp}\bigwedge {\mathrm{Cw}}_{\left[\right]}$

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

$-\mathrm{Dx}\bigwedge {\mathrm{Cu}}_{\left[\right]}-\frac{u_2}{u_1}\mathrm{Dy}\bigwedge {\mathrm{Cu}}_{\left[\right]}$

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

$\mathrm{\θ1}:=-w_2\mathrm{dp}\bigwedge \mathrm{dq}-w_3\mathrm{dp}\bigwedge \mathrm{dr}+\mathrm{dp}\bigwedge {\mathrm{dw}}_{\left[\right]}$

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

$\mathrm{\θ2}:=-\mathrm{dx}\bigwedge {\mathrm{du}}_{\left[\right]}-\frac{u_2}{u_1}\mathrm{dy}\bigwedge {\mathrm{du}}_{\left[\right]}$

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

$\mathrm{\θ3}:=-\mathrm{Dx}\bigwedge {\mathrm{Cu}}_{\left[\right]}-\frac{u_2}{u_1}\mathrm{Dy}\bigwedge {\mathrm{Cu}}_{\left[\right]}$

$\mathrm{\Φ3}≔\mathrm{Transformation}\left(\mathrm{E2}\,\mathrm{E1}\,\left[x=v\left[\right]\,y=v\left[1\right]\,u\left[\right]=v\left[2\right]\right]\right)$

$\mathrm{\Φ3}:=\left[x=v_{\left[\right]}\,y=v_1\,u_{\left[\right]}=v_2\right]$

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

$\mathrm{prPhi3}:=\left[x=v_{\left[\right]}\,y=v_1\,u_{\left[\right]}=v_2\,u_1=\frac{v_1{v}_{1,2}}{v_1^2+v_{1,1}^2}\,u_2=\frac{v_{1,1}{v}_{1,2}}{v_1^2+v_{1,1}^2}\right]$

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

$\left(3v_{1,1}+2v_1\right)\mathrm{Dt}$

$\mathrm{ProjectedPullback}\left(\mathrm{prPhi3}\,u\left[1\right]\mathrm{Cu}\left[\right]\right)$

$-\frac{v_1^2{v}_{1,2}^2}{{\left(v_1^2+v_{1,1}^2\right)}^2}{\mathrm{Cv}}_{\left[\right]}-\frac{v_1{v}_{1,2}^2{v}_{1,1}}{{\left(v_1^2+v_{1,1}^2\right)}^2}{\mathrm{Cv}}_1$