 ProjectedPullback - Maple Help

JetCalculus[ProjectedPullback] - pullback a differential bi-form of type (r, s) by a transformation to a differential bi-form of type (r, s)

Calling Sequences

ProjectedPullback(${\mathbf{φ}}$, ${\mathbf{ω}}$)

Parameters

$\mathrm{φ}$    - a transformation between two jet spaces

- a differential bi-form of type defined on the range jet space of $\mathrm{φ}$ Description

 • Let be a fiber bundle, with base dimension $n$ and fiber dimension $m$ and let be the $\mathrm{∞}$-th jet bundle of $E$. The space of -forms ${\mathrm{Ω}}^{p}\left({J}^{\mathrm{∞}}\left(E\right)\right)$ decomposes into a direct sum, where is the space of bi-forms of horizontal degree $r$ and vertical degree 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 , then let denote the type component of in the decomposition (*). The command convert/DGbifom calculates the various bi-graded components of a form .Let $F\to N$be another fiber bundle and let . Let be a differential bi-form of type on . Then the projected pullback of is denote by and defined by .
 • Two special cases of this general definition should be noted.

[i]  If $\mathrm{φ}$ is the prolongation of a projectable transformation from to, then the pullback ${\mathrm{φ}}^{*}$ is a bi-degree preserving transformation, that is, if be a differential bi-form of type on ${J}^{\ell }\left(F\right)$, then ${\mathrm{φ}}^{*}\left(\mathrm{η}\right)$ is a differential bi-form of type on Hence ${\mathrm{\phi }}^{†}\left(\mathrm{\eta }\right)$.

[ii] Suppose that  is the prolongation of a point transformation, a contact transformation, a differential substitution or a generalized differential substitution. (See AssignTransformationType for the definitions of these different types of transformations.) Then if is a differential bi-form of type on , where ${\mathrm{\omega }}_{i}$ is a bi-form of degree  on ${J}^{k}\left(E\right)$. In these cases the command ProjectedPullback(${\mathbf{φ}}$$,$ ${\mathbf{ω}}$) returns the type bi-form ${\mathrm{\omega }}_{0}$.

 • 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(...). Examples

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

First initialize several different jet spaces over bundles ${E}_{1}\to {M}_{1}$, . The dimension of the base spaces are dim(${M}_{1}$) =2, dim(${M}_{2}$) =1, dim(${M}_{3}$) =3.

 > $\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):$

Example 1.

Define a transformation ${\mathrm{φ}}_{1}:{E}_{1}\to {E}_{2}$. This transformation is a projectable transformation and therefore pullbacks by the prolongation of ${\mathrm{φ}}_{1}$can be calculated directly using the Pullback command.

 E3 > $\mathrm{Φ1}≔\mathrm{Transformation}\left(\mathrm{E1},\mathrm{E2},\left[t=x,{v}_{[]}={x}^{2}{u}_{[]}\right]\right)$
 ${\mathrm{Φ1}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{\mathrm{E1}}{,}{0}\right]{,}\left[{\mathrm{E2}}{,}{0}\right]\right]{,}\left[\right]{,}\left[\left[\begin{array}{ccc}{1}& {0}& {0}\\ {2}{}{x}{}{{u}}_{\left[\right]}& {0}& {{x}}^{{2}}\end{array}\right]\right]\right]{,}\left[\left[{x}{,}{t}\right]{,}\left[{{x}}^{{2}}{}{{u}}_{\left[\right]}{,}{{v}}_{\left[\right]}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{\mathrm{E1}}{,}{0}\right]{,}\left[{\mathrm{E2}}{,}{0}\right]\right]{,}\left[\right]{,}\left[\left[\begin{array}{ccc}{1}& {0}& {0}\\ {2}{}{x}{}{{u}}_{\left[\right]}& {0}& {{x}}^{{2}}\end{array}\right]\right]\right]{,}\left[\left[{x}{,}{t}\right]{,}\left[{{x}}^{{2}}{}{{u}}_{\left[\right]}{,}{{v}}_{\left[\right]}\right]\right]\right]\right)$ (2.1)
 E1 > $\mathrm{prPhi1}≔\mathrm{Prolong}\left(\mathrm{Φ1},2\right)$
 ${\mathrm{prPhi1}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{\mathrm{E1}}{,}{2}\right]{,}\left[{\mathrm{E2}}{,}{2}\right]\right]{,}\left[{"projectable"}{,}{2}\right]{,}\left[\left[\begin{array}{cccccccc}{1}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {2}{}{x}{}{{u}}_{\left[\right]}& {0}& {{x}}^{{2}}& {0}& {0}& {0}& {0}& {0}\\ {2}{}{x}{}{{u}}_{{1}}{+}{2}{}{{u}}_{\left[\right]}& {0}& {2}{}{x}& {{x}}^{{2}}& {0}& {0}& {0}& {0}\\ {2}{}{x}{}{{u}}_{{1}{,}{1}}{+}{4}{}{{u}}_{{1}}& {0}& {2}& {4}{}{x}& {0}& {{x}}^{{2}}& {0}& {0}\end{array}\right]\right]\right]{,}\left[\left[{x}{,}{t}\right]{,}\left[{{x}}^{{2}}{}{{u}}_{\left[\right]}{,}{{v}}_{\left[\right]}\right]{,}\left[{{x}}^{{2}}{}{{u}}_{{1}}{+}{2}{}{x}{}{{u}}_{\left[\right]}{,}{{v}}_{{1}}\right]{,}\left[{{x}}^{{2}}{}{{u}}_{{1}{,}{1}}{+}{4}{}{x}{}{{u}}_{{1}}{+}{2}{}{{u}}_{\left[\right]}{,}{{v}}_{{1}{,}{1}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{\mathrm{E1}}{,}{2}\right]{,}\left[{\mathrm{E2}}{,}{2}\right]\right]{,}\left[{"projectable"}{,}{2}\right]{,}\left[\left[\begin{array}{cccccccc}{1}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {2}{}{x}{}{{u}}_{\left[\right]}& {0}& {{x}}^{{2}}& {0}& {0}& {0}& {0}& {0}\\ {2}{}{x}{}{{u}}_{{1}}{+}{2}{}{{u}}_{\left[\right]}& {0}& {2}{}{x}& {{x}}^{{2}}& {0}& {0}& {0}& {0}\\ {2}{}{x}{}{{u}}_{{1}{,}{1}}{+}{4}{}{{u}}_{{1}}& {0}& {2}& {4}{}{x}& {0}& {{x}}^{{2}}& {0}& {0}\end{array}\right]\right]\right]{,}\left[\left[{x}{,}{t}\right]{,}\left[{{x}}^{{2}}{}{{u}}_{\left[\right]}{,}{{v}}_{\left[\right]}\right]{,}\left[{{x}}^{{2}}{}{{u}}_{{1}}{+}{2}{}{x}{}{{u}}_{\left[\right]}{,}{{v}}_{{1}}\right]{,}\left[{{x}}^{{2}}{}{{u}}_{{1}{,}{1}}{+}{4}{}{x}{}{{u}}_{{1}}{+}{2}{}{{u}}_{\left[\right]}{,}{{v}}_{{1}{,}{1}}\right]\right]\right]\right)$ (2.2)
 E1 > $\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{prPhi1},"TransformationType"\right)$
 $\left[{"projectable"}{,}{2}\right]$ (2.3)

Pullback the contact 1-form Cv on to a contact form on -- this can be done with either the Pullback command or the ProjectedPullback command.

 E1 > $\mathrm{Pullback}\left(\mathrm{prPhi1},{\mathrm{Cv}}_{1}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{\mathrm{E1}}{,}\left[{0}{,}{1}\right]\right]{,}\left[\left[\left[{3}\right]{,}{2}{}{x}\right]{,}\left[\left[{4}\right]{,}{{x}}^{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{\mathrm{E1}}{,}\left[{0}{,}{1}\right]\right]{,}\left[\left[\left[{3}\right]{,}{2}{}{x}\right]{,}\left[\left[{4}\right]{,}{{x}}^{{2}}\right]\right]\right]\right)$ (2.4)
 E1 > $\mathrm{ProjectedPullback}\left(\mathrm{prPhi1},{\mathrm{Cv}}_{1}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{\mathrm{E1}}{,}\left[{0}{,}{1}\right]\right]{,}\left[\left[\left[{3}\right]{,}{2}{}{x}\right]{,}\left[\left[{4}\right]{,}{{x}}^{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{\mathrm{E1}}{,}\left[{0}{,}{1}\right]\right]{,}\left[\left[\left[{3}\right]{,}{2}{}{x}\right]{,}\left[\left[{4}\right]{,}{{x}}^{{2}}\right]\right]\right]\right)$ (2.5)

Example 2

Define a point transformation ${\mathrm{\phi }}_{1}:{E}_{1}\to {E}_{3}$ and prolong it to a transformation .

 E1 > $\mathrm{Φ2}≔\mathrm{Transformation}\left(\mathrm{E1},\mathrm{E3},\left[p={u}_{[]},q=y,r=1,{w}_{[]}=x\right]\right)$
 ${\mathrm{Φ2}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{\mathrm{E1}}{,}{0}\right]{,}\left[{\mathrm{E3}}{,}{0}\right]\right]{,}\left[\right]{,}\left[\left[\begin{array}{ccc}{0}& {0}& {1}\\ {0}& {1}& {0}\\ {0}& {0}& {0}\\ {1}& {0}& {0}\end{array}\right]\right]\right]{,}\left[\left[{{u}}_{\left[\right]}{,}{p}\right]{,}\left[{y}{,}{q}\right]{,}\left[{1}{,}{r}\right]{,}\left[{x}{,}{{w}}_{\left[\right]}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{\mathrm{E1}}{,}{0}\right]{,}\left[{\mathrm{E3}}{,}{0}\right]\right]{,}\left[\right]{,}\left[\left[\begin{array}{ccc}{0}& {0}& {1}\\ {0}& {1}& {0}\\ {0}& {0}& {0}\\ {1}& {0}& {0}\end{array}\right]\right]\right]{,}\left[\left[{{u}}_{\left[\right]}{,}{p}\right]{,}\left[{y}{,}{q}\right]{,}\left[{1}{,}{r}\right]{,}\left[{x}{,}{{w}}_{\left[\right]}\right]\right]\right]\right)$ (2.6)
 E1 > $\mathrm{prPhi2}≔\mathrm{Prolong}\left(\mathrm{Φ2},1\right)$
 ${\mathrm{prPhi2}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{\mathrm{E1}}{,}{1}\right]{,}\left[{\mathrm{E3}}{,}{1}\right]\right]{,}\left[{"point"}{,}{1}\right]{,}\left[\left[\begin{array}{ccccc}{0}& {0}& {1}& {0}& {0}\\ {0}& {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}\\ {1}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-}\frac{{1}}{{{u}}_{{1}}^{{2}}}& {0}\\ {0}& {0}& {0}& \frac{{{u}}_{{2}}}{{{u}}_{{1}}^{{2}}}& {-}\frac{{1}}{{{u}}_{{1}}}\\ {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]\right]{,}\left[\left[{{u}}_{\left[\right]}{,}{p}\right]{,}\left[{y}{,}{q}\right]{,}\left[{1}{,}{r}\right]{,}\left[{x}{,}{{w}}_{\left[\right]}\right]{,}\left[\frac{{1}}{{{u}}_{{1}}}{,}{{w}}_{{1}}\right]{,}\left[{-}\frac{{{u}}_{{2}}}{{{u}}_{{1}}}{,}{{w}}_{{2}}\right]{,}\left[{0}{,}{{w}}_{{3}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{\mathrm{E1}}{,}{1}\right]{,}\left[{\mathrm{E3}}{,}{1}\right]\right]{,}\left[{"point"}{,}{1}\right]{,}\left[\left[\begin{array}{ccccc}{0}& {0}& {1}& {0}& {0}\\ {0}& {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}\\ {1}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-}\frac{{1}}{{{u}}_{{1}}^{{2}}}& {0}\\ {0}& {0}& {0}& \frac{{{u}}_{{2}}}{{{u}}_{{1}}^{{2}}}& {-}\frac{{1}}{{{u}}_{{1}}}\\ {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]\right]{,}\left[\left[{{u}}_{\left[\right]}{,}{p}\right]{,}\left[{y}{,}{q}\right]{,}\left[{1}{,}{r}\right]{,}\left[{x}{,}{{w}}_{\left[\right]}\right]{,}\left[\frac{{1}}{{{u}}_{{1}}}{,}{{w}}_{{1}}\right]{,}\left[{-}\frac{{{u}}_{{2}}}{{{u}}_{{1}}}{,}{{w}}_{{2}}\right]{,}\left[{0}{,}{{w}}_{{3}}\right]\right]\right]\right)$ (2.7)

Calculate the projected pullback of the type (1, 0) form $\mathrm{Dp}$.

 E1 > $\mathrm{ProjectedPullback}\left(\mathrm{prPhi2},\mathrm{Dp}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{\mathrm{E1}}{,}\left[{1}{,}{0}\right]\right]{,}\left[\left[\left[{1}\right]{,}{{u}}_{{1}}\right]{,}\left[\left[{2}\right]{,}{{u}}_{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{\mathrm{E1}}{,}\left[{1}{,}{0}\right]\right]{,}\left[\left[\left[{1}\right]{,}{{u}}_{{1}}\right]{,}\left[\left[{2}\right]{,}{{u}}_{{2}}\right]\right]\right]\right)$ (2.8)

Calculate the projected pullback of the type (1, 1) form .

 E1 > $\mathrm{ω}≔\mathrm{Dp}&wedge\left({\mathrm{Cw}}_{[]}\right)$
 ${\mathrm{ω}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{\mathrm{E3}}{,}\left[{1}{,}{1}\right]\right]{,}\left[\left[\left[{1}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{\mathrm{E3}}{,}\left[{1}{,}{1}\right]\right]{,}\left[\left[\left[{1}{,}{4}\right]{,}{1}\right]\right]\right]\right)$ (2.9)
 E3 > $\mathrm{ProjectedPullback}\left(\mathrm{prPhi2},\mathrm{ω}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{\mathrm{E1}}{,}\left[{1}{,}{1}\right]\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{3}\right]{,}{-}\frac{{{u}}_{{2}}}{{{u}}_{{1}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{\mathrm{E1}}{,}\left[{1}{,}{1}\right]\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{3}\right]{,}{-}\frac{{{u}}_{{2}}}{{{u}}_{{1}}}\right]\right]\right]\right)$ (2.10)

To illustrate the definition of the projected pullback we re-derive this result using the usual Pullback command. First convert $\mathrm{ω}$ from a bi-form to a form ${\mathrm{θ}}_{1}$.

 E1 > $\mathrm{θ1}≔\mathrm{convert}\left(\mathrm{ω},\mathrm{DGform}\right)$
 ${\mathrm{θ1}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{E3}}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{-}{{w}}_{{2}}\right]{,}\left[\left[{1}{,}{3}\right]{,}{-}{{w}}_{{3}}\right]{,}\left[\left[{1}{,}{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{E3}}{,}{2}\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{-}{{w}}_{{2}}\right]{,}\left[\left[{1}{,}{3}\right]{,}{-}{{w}}_{{3}}\right]{,}\left[\left[{1}{,}{4}\right]{,}{1}\right]\right]\right]\right)$ (2.11)

Then pullback using

 E3 > $\mathrm{θ2}≔\mathrm{Pullback}\left(\mathrm{prPhi2},\mathrm{θ1}\right)$
 ${\mathrm{θ2}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{E1}}{,}{2}\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{3}\right]{,}{-}\frac{{{u}}_{{2}}}{{{u}}_{{1}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"form"}{,}{\mathrm{E1}}{,}{2}\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{3}\right]{,}{-}\frac{{{u}}_{{2}}}{{{u}}_{{1}}}\right]\right]\right]\right)$ (2.12)

Then convert back to a bi-form and take the type [1, 1] part.

 E1 > $\mathrm{θ3}≔\mathrm{convert}\left(\mathrm{θ2},\mathrm{DGbiform},\left[1,1\right]\right)$
 ${\mathrm{θ3}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{\mathrm{E1}}{,}\left[{1}{,}{1}\right]\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{3}\right]{,}{-}\frac{{{u}}_{{2}}}{{{u}}_{{1}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{\mathrm{E1}}{,}\left[{1}{,}{1}\right]\right]{,}\left[\left[\left[{1}{,}{3}\right]{,}{-1}\right]{,}\left[\left[{2}{,}{3}\right]{,}{-}\frac{{{u}}_{{2}}}{{{u}}_{{1}}}\right]\right]\right]\right)$ (2.13)

Example 3

Define a differential substitution and prolong it to a transformation .

 E1 > $\mathrm{Φ3}≔\mathrm{Transformation}\left(\mathrm{E2},\mathrm{E1},\left[x={v}_{[]},y={v}_{1},{u}_{[]}={v}_{2}\right]\right)$
 ${\mathrm{Φ3}}{:=}{\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{\mathrm{E2}}{,}{1}\right]{,}\left[{\mathrm{E1}}{,}{0}\right]\right]{,}\left[\right]{,}\left[\left[\begin{array}{ccc}{0}& {1}& {0}\\ {0}& {0}& {1}\\ {0}& {0}& {0}\end{array}\right]\right]\right]{,}\left[\left[{{v}}_{\left[\right]}{,}{x}\right]{,}\left[{{v}}_{{1}}{,}{y}\right]{,}\left[{{v}}_{{2}}{,}{{u}}_{\left[\right]}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{\mathrm{E2}}{,}{1}\right]{,}\left[{\mathrm{E1}}{,}{0}\right]\right]{,}\left[\right]{,}\left[\left[\begin{array}{ccc}{0}& {1}& {0}\\ {0}& {0}& {1}\\ {0}& {0}& {0}\end{array}\right]\right]\right]{,}\left[\left[{{v}}_{\left[\right]}{,}{x}\right]{,}\left[{{v}}_{{1}}{,}{y}\right]{,}\left[{{v}}_{{2}}{,}{{u}}_{\left[\right]}\right]\right]\right]\right)$ (2.14)
 E2 > $\mathrm{prPhi3}≔\mathrm{Prolong}\left(\mathrm{Φ3},1\right)$
 ${\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{{2}{}{{v}}_{{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{{2}{}{{v}}_{{1}}{}{{v}}_{{1}{,}{2}}{}{{v}}_{{1}{,}{1}}}{{\left({{v}}_{{1}}^{{2}}{+}{{v}}_{{1}{,}{1}}^{{2}}\right)}^{{2}}}\\ {0}& {0}& {-}\frac{{2}{}{{v}}_{{1}}{}{{v}}_{{1}{,}{2}}{}{{v}}_{{1}{,}{1}}}{{\left({{v}}_{{1}}^{{2}}{+}{{v}}_{{1}{,}{1}}^{{2}}\right)}^{{2}}}& {-}\frac{{2}{}{{v}}_{{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{{2}{}{{v}}_{{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{{2}{}{{v}}_{{1}}{}{{v}}_{{1}{,}{2}}{}{{v}}_{{1}{,}{1}}}{{\left({{v}}_{{1}}^{{2}}{+}{{v}}_{{1}{,}{1}}^{{2}}\right)}^{{2}}}\\ {0}& {0}& {-}\frac{{2}{}{{v}}_{{1}}{}{{v}}_{{1}{,}{2}}{}{{v}}_{{1}{,}{1}}}{{\left({{v}}_{{1}}^{{2}}{+}{{v}}_{{1}{,}{1}}^{{2}}\right)}^{{2}}}& {-}\frac{{2}{}{{v}}_{{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)$ (2.15)

Calculate the projected pullback of the type (1, 0) form

 E2 > $\mathrm{ProjectedPullback}\left(\mathrm{prPhi3},2\mathrm{Dx}+3\mathrm{Dy}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{\mathrm{E2}}{,}\left[{1}{,}{0}\right]\right]{,}\left[\left[\left[{1}\right]{,}{3}{}{{v}}_{{1}{,}{1}}{+}{2}{}{{v}}_{{1}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{\mathrm{E2}}{,}\left[{1}{,}{0}\right]\right]{,}\left[\left[\left[{1}\right]{,}{3}{}{{v}}_{{1}{,}{1}}{+}{2}{}{{v}}_{{1}}\right]\right]\right]\right)$ (2.16)

Calculate the projected pullback of the type (1, 0) form ${\mathrm{Cu}}_{\left[\right]}$

 E2 > $\mathrm{ProjectedPullback}\left(\mathrm{prPhi3},{u}_{1}{\mathrm{Cu}}_{[]}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{\mathrm{E2}}{,}\left[{0}{,}{1}\right]\right]{,}\left[\left[\left[{2}\right]{,}{-}\frac{{{v}}_{{1}}^{{2}}{}{{v}}_{{1}{,}{2}}^{{2}}}{{\left({{v}}_{{1}}^{{2}}{+}{{v}}_{{1}{,}{1}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{{v}}_{{1}}{}{{v}}_{{1}{,}{2}}^{{2}}{}{{v}}_{{1}{,}{1}}}{{\left({{v}}_{{1}}^{{2}}{+}{{v}}_{{1}{,}{1}}^{{2}}\right)}^{{2}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{\mathrm{E2}}{,}\left[{0}{,}{1}\right]\right]{,}\left[\left[\left[{2}\right]{,}{-}\frac{{{v}}_{{1}}^{{2}}{}{{v}}_{{1}{,}{2}}^{{2}}}{{\left({{v}}_{{1}}^{{2}}{+}{{v}}_{{1}{,}{1}}^{{2}}\right)}^{{2}}}\right]{,}\left[\left[{3}\right]{,}{-}\frac{{{v}}_{{1}}{}{{v}}_{{1}{,}{2}}^{{2}}{}{{v}}_{{1}{,}{1}}}{{\left({{v}}_{{1}}^{{2}}{+}{{v}}_{{1}{,}{1}}^{{2}}\right)}^{{2}}}\right]\right]\right]\right)$ (2.17)