PushforwardTotalVector - Maple Help
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JetCalculus[PushforwardTotalVector] - push forward a total vector field by a transformation

Calling Sequences

PushforwardTotalVector(${\mathbf{φ}}$)

Parameters

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

Description

 • Let and $F\to N$ be two fiber bundles with associated jet spaces and  and with jet coordinates , ..., and , ..., respectively. Let be a transformation and let , ..., be the components of $\mathrm{φ}$. Then the total Jacobian of $\mathrm{φ}$is the matrix $\left[{\mathrm{D}}_{i}{\mathrm{φ}}^{a}\right]$, where ${\mathrm{D}}_{i}$ denotes the total derivative with respect to ${x}^{i}$. The push forward of the total vector field on is the total vector , where .
 • The command PushforwardTotalVector is part of the DifferentialGeometry:-JetCalculus package. It can be used in the form PushforwardTotalVector(...) only after executing the commands with(DifferentialGeometry) and with(JetCalculus), but can always be used by executing DifferentialGeometry:-JetCalculus:-PushforwardTotalVector(...).

Examples

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

Example 1.

First initialize two different jet spaces over bundles ${E}_{1}\to {M}_{1}$, ${E}_{2}\to {M}_{2}$. The dimension of the base spaces are dimdim.

 > $\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):$

Define a transformation  and compute its total Jacobian (a matrix).

 E3 > $\mathrm{φ}≔\mathrm{Transformation}\left(\mathrm{E1},\mathrm{E2},\left[t={u}_{2,0},{v}_{[]}=xy\right]\right)$
 ${\mathrm{\phi }}{≔}{\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{\mathrm{E1}}{,}{2}\right]{,}\left[{\mathrm{E2}}{,}{0}\right]\right]{,}\left[\right]{,}\left[\left[\begin{array}{cccccccc}{0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {y}& {x}& {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]\right]{,}\left[\left[{{u}}_{{2}{,}{0}}{,}{t}\right]{,}\left[{x}{}{y}{,}{{v}}_{\left[\right]}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{\mathrm{E1}}{,}{2}\right]{,}\left[{\mathrm{E2}}{,}{0}\right]\right]{,}\left[\right]{,}\left[\left[\begin{array}{cccccccc}{0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {y}& {x}& {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]\right]{,}\left[\left[{{u}}_{{2}{,}{0}}{,}{t}\right]{,}\left[{x}{}{y}{,}{{v}}_{\left[\right]}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{\mathrm{E1}}{,}{2}\right]{,}\left[{\mathrm{E2}}{,}{0}\right]\right]{,}\left[\right]{,}\left[\left[\begin{array}{cccccccc}{0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {y}& {x}& {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]\right]{,}\left[\left[{{u}}_{{2}{,}{0}}{,}{t}\right]{,}\left[{x}{}{y}{,}{{v}}_{\left[\right]}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{\mathrm{E1}}{,}{2}\right]{,}\left[{\mathrm{E2}}{,}{0}\right]\right]{,}\left[\right]{,}\left[\left[\begin{array}{cccccccc}{0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {y}& {x}& {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]\right]{,}\left[\left[{{u}}_{{2}{,}{0}}{,}{t}\right]{,}\left[{x}{}{y}{,}{{v}}_{\left[\right]}\right]\right]\right]\right)$ (2.1)
 E1 > $\mathrm{J1}≔\mathrm{TotalJacobian}\left(\mathrm{φ}\right)$
 ${\mathrm{J1}}{≔}\left[\begin{array}{cc}{{u}}_{{0}{,}{1}{,}{2}}& {{u}}_{{0}{,}{2}{,}{2}}\end{array}\right]$ (2.2)

Define a vector field on ${M}_{1}$ and its total part on ${J}^{4}\left({E}_{1}\right)$.

 E1 > $X≔a\mathrm{D_x}+b\mathrm{D_y}$
 ${X}{≔}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{E1}}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{E1}}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{E1}}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{E1}}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){}{a}{+}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{E1}}{,}\left[\right]\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{E1}}{,}\left[\right]\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{E1}}{,}\left[\right]\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{E1}}{,}\left[\right]\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){}{b}$ (2.3)
 E1 > $\mathrm{totX}≔\mathrm{Prolong}\left(\mathrm{TotalVector}\left(X\right),3\right)$
 ${\mathrm{totX}}{≔}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{E1}}{,}\left[{"total"}{,}{3}\right]\right]{,}\left[\left[\left[{1}\right]{,}{a}\right]{,}\left[\left[{2}\right]{,}{b}\right]{,}\left[\left[{3}\right]{,}{{u}}_{{1}}{}{a}{+}{{u}}_{{2}}{}{b}\right]{,}\left[\left[{4}\right]{,}{a}{}{{u}}_{{1}{,}{1}}{+}{b}{}{{u}}_{{1}{,}{2}}\right]{,}\left[\left[{5}\right]{,}{a}{}{{u}}_{{1}{,}{2}}{+}{b}{}{{u}}_{{2}{,}{2}}\right]{,}\left[\left[{6}\right]{,}{a}{}{{u}}_{{1}{,}{1}{,}{1}}{+}{b}{}{{u}}_{{1}{,}{1}{,}{2}}\right]{,}\left[\left[{7}\right]{,}{a}{}{{u}}_{{1}{,}{1}{,}{2}}{+}{b}{}{{u}}_{{1}{,}{2}{,}{2}}\right]{,}\left[\left[{8}\right]{,}{a}{}{{u}}_{{1}{,}{2}{,}{2}}{+}{b}{}{{u}}_{{2}{,}{2}{,}{2}}\right]{,}\left[\left[{9}\right]{,}{a}{}{{u}}_{{1}{,}{1}{,}{1}{,}{1}}{+}{b}{}{{u}}_{{1}{,}{1}{,}{1}{,}{2}}\right]{,}\left[\left[{10}\right]{,}{a}{}{{u}}_{{1}{,}{1}{,}{1}{,}{2}}{+}{b}{}{{u}}_{{1}{,}{1}{,}{2}{,}{2}}\right]{,}\left[\left[{11}\right]{,}{a}{}{{u}}_{{1}{,}{1}{,}{2}{,}{2}}{+}{b}{}{{u}}_{{1}{,}{2}{,}{2}{,}{2}}\right]{,}\left[\left[{12}\right]{,}{a}{}{{u}}_{{1}{,}{2}{,}{2}{,}{2}}{+}{b}{}{{u}}_{{2}{,}{2}{,}{2}{,}{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{E1}}{,}\left[{"total"}{,}{3}\right]\right]{,}\left[\left[\left[{1}\right]{,}{a}\right]{,}\left[\left[{2}\right]{,}{b}\right]{,}\left[\left[{3}\right]{,}{{u}}_{{1}}{}{a}{+}{{u}}_{{2}}{}{b}\right]{,}\left[\left[{4}\right]{,}{a}{}{{u}}_{{1}{,}{1}}{+}{b}{}{{u}}_{{1}{,}{2}}\right]{,}\left[\left[{5}\right]{,}{a}{}{{u}}_{{1}{,}{2}}{+}{b}{}{{u}}_{{2}{,}{2}}\right]{,}\left[\left[{6}\right]{,}{a}{}{{u}}_{{1}{,}{1}{,}{1}}{+}{b}{}{{u}}_{{1}{,}{1}{,}{2}}\right]{,}\left[\left[{7}\right]{,}{a}{}{{u}}_{{1}{,}{1}{,}{2}}{+}{b}{}{{u}}_{{1}{,}{2}{,}{2}}\right]{,}\left[\left[{8}\right]{,}{a}{}{{u}}_{{1}{,}{2}{,}{2}}{+}{b}{}{{u}}_{{2}{,}{2}{,}{2}}\right]{,}\left[\left[{9}\right]{,}{a}{}{{u}}_{{1}{,}{1}{,}{1}{,}{1}}{+}{b}{}{{u}}_{{1}{,}{1}{,}{1}{,}{2}}\right]{,}\left[\left[{10}\right]{,}{a}{}{{u}}_{{1}{,}{1}{,}{1}{,}{2}}{+}{b}{}{{u}}_{{1}{,}{1}{,}{2}{,}{2}}\right]{,}\left[\left[{11}\right]{,}{a}{}{{u}}_{{1}{,}{1}{,}{2}{,}{2}}{+}{b}{}{{u}}_{{1}{,}{2}{,}{2}{,}{2}}\right]{,}\left[\left[{12}\right]{,}{a}{}{{u}}_{{1}{,}{2}{,}{2}{,}{2}}{+}{b}{}{{u}}_{{2}{,}{2}{,}{2}{,}{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{E1}}{,}\left[{"total"}{,}{3}\right]\right]{,}\left[\left[\left[{1}\right]{,}{a}\right]{,}\left[\left[{2}\right]{,}{b}\right]{,}\left[\left[{3}\right]{,}{{u}}_{{1}}{}{a}{+}{{u}}_{{2}}{}{b}\right]{,}\left[\left[{4}\right]{,}{a}{}{{u}}_{{1}{,}{1}}{+}{b}{}{{u}}_{{1}{,}{2}}\right]{,}\left[\left[{5}\right]{,}{a}{}{{u}}_{{1}{,}{2}}{+}{b}{}{{u}}_{{2}{,}{2}}\right]{,}\left[\left[{6}\right]{,}{a}{}{{u}}_{{1}{,}{1}{,}{1}}{+}{b}{}{{u}}_{{1}{,}{1}{,}{2}}\right]{,}\left[\left[{7}\right]{,}{a}{}{{u}}_{{1}{,}{1}{,}{2}}{+}{b}{}{{u}}_{{1}{,}{2}{,}{2}}\right]{,}\left[\left[{8}\right]{,}{a}{}{{u}}_{{1}{,}{2}{,}{2}}{+}{b}{}{{u}}_{{2}{,}{2}{,}{2}}\right]{,}\left[\left[{9}\right]{,}{a}{}{{u}}_{{1}{,}{1}{,}{1}{,}{1}}{+}{b}{}{{u}}_{{1}{,}{1}{,}{1}{,}{2}}\right]{,}\left[\left[{10}\right]{,}{a}{}{{u}}_{{1}{,}{1}{,}{1}{,}{2}}{+}{b}{}{{u}}_{{1}{,}{1}{,}{2}{,}{2}}\right]{,}\left[\left[{11}\right]{,}{a}{}{{u}}_{{1}{,}{1}{,}{2}{,}{2}}{+}{b}{}{{u}}_{{1}{,}{2}{,}{2}{,}{2}}\right]{,}\left[\left[{12}\right]{,}{a}{}{{u}}_{{1}{,}{2}{,}{2}{,}{2}}{+}{b}{}{{u}}_{{2}{,}{2}{,}{2}{,}{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{E1}}{,}\left[{"total"}{,}{3}\right]\right]{,}\left[\left[\left[{1}\right]{,}{a}\right]{,}\left[\left[{2}\right]{,}{b}\right]{,}\left[\left[{3}\right]{,}{{u}}_{{1}}{}{a}{+}{{u}}_{{2}}{}{b}\right]{,}\left[\left[{4}\right]{,}{a}{}{{u}}_{{1}{,}{1}}{+}{b}{}{{u}}_{{1}{,}{2}}\right]{,}\left[\left[{5}\right]{,}{a}{}{{u}}_{{1}{,}{2}}{+}{b}{}{{u}}_{{2}{,}{2}}\right]{,}\left[\left[{6}\right]{,}{a}{}{{u}}_{{1}{,}{1}{,}{1}}{+}{b}{}{{u}}_{{1}{,}{1}{,}{2}}\right]{,}\left[\left[{7}\right]{,}{a}{}{{u}}_{{1}{,}{1}{,}{2}}{+}{b}{}{{u}}_{{1}{,}{2}{,}{2}}\right]{,}\left[\left[{8}\right]{,}{a}{}{{u}}_{{1}{,}{2}{,}{2}}{+}{b}{}{{u}}_{{2}{,}{2}{,}{2}}\right]{,}\left[\left[{9}\right]{,}{a}{}{{u}}_{{1}{,}{1}{,}{1}{,}{1}}{+}{b}{}{{u}}_{{1}{,}{1}{,}{1}{,}{2}}\right]{,}\left[\left[{10}\right]{,}{a}{}{{u}}_{{1}{,}{1}{,}{1}{,}{2}}{+}{b}{}{{u}}_{{1}{,}{1}{,}{2}{,}{2}}\right]{,}\left[\left[{11}\right]{,}{a}{}{{u}}_{{1}{,}{1}{,}{2}{,}{2}}{+}{b}{}{{u}}_{{1}{,}{2}{,}{2}{,}{2}}\right]{,}\left[\left[{12}\right]{,}{a}{}{{u}}_{{1}{,}{2}{,}{2}{,}{2}}{+}{b}{}{{u}}_{{2}{,}{2}{,}{2}{,}{2}}\right]\right]\right]\right)$ (2.4)

Push forward totX to a vector field on ${J}^{4}\left({E}_{2}\right)$

 E1 > $\mathrm{PushforwardTotalVector}\left(\mathrm{φ},\mathrm{totX}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{E2}}{,}\left[{"total"}{,}{3}\right]\right]{,}\left[\left[\left[{1}\right]{,}{{u}}_{{0}{,}{1}{,}{2}}{}{a}{+}{{u}}_{{0}{,}{2}{,}{2}}{}{b}\right]{,}\left[\left[{2}\right]{,}{{v}}_{{1}}{}\left({{u}}_{{0}{,}{1}{,}{2}}{}{a}{+}{{u}}_{{0}{,}{2}{,}{2}}{}{b}\right)\right]{,}\left[\left[{3}\right]{,}\left({{u}}_{{0}{,}{1}{,}{2}}{}{a}{+}{{u}}_{{0}{,}{2}{,}{2}}{}{b}\right){}{{v}}_{{1}{,}{1}}\right]{,}\left[\left[{4}\right]{,}\left({{u}}_{{0}{,}{1}{,}{2}}{}{a}{+}{{u}}_{{0}{,}{2}{,}{2}}{}{b}\right){}{{v}}_{{1}{,}{1}{,}{1}}\right]{,}\left[\left[{5}\right]{,}\left({{u}}_{{0}{,}{1}{,}{2}}{}{a}{+}{{u}}_{{0}{,}{2}{,}{2}}{}{b}\right){}{{v}}_{{1}{,}{1}{,}{1}{,}{1}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{E2}}{,}\left[{"total"}{,}{3}\right]\right]{,}\left[\left[\left[{1}\right]{,}{{u}}_{{0}{,}{1}{,}{2}}{}{a}{+}{{u}}_{{0}{,}{2}{,}{2}}{}{b}\right]{,}\left[\left[{2}\right]{,}{{v}}_{{1}}{}\left({{u}}_{{0}{,}{1}{,}{2}}{}{a}{+}{{u}}_{{0}{,}{2}{,}{2}}{}{b}\right)\right]{,}\left[\left[{3}\right]{,}\left({{u}}_{{0}{,}{1}{,}{2}}{}{a}{+}{{u}}_{{0}{,}{2}{,}{2}}{}{b}\right){}{{v}}_{{1}{,}{1}}\right]{,}\left[\left[{4}\right]{,}\left({{u}}_{{0}{,}{1}{,}{2}}{}{a}{+}{{u}}_{{0}{,}{2}{,}{2}}{}{b}\right){}{{v}}_{{1}{,}{1}{,}{1}}\right]{,}\left[\left[{5}\right]{,}\left({{u}}_{{0}{,}{1}{,}{2}}{}{a}{+}{{u}}_{{0}{,}{2}{,}{2}}{}{b}\right){}{{v}}_{{1}{,}{1}{,}{1}{,}{1}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{E2}}{,}\left[{"total"}{,}{3}\right]\right]{,}\left[\left[\left[{1}\right]{,}{{u}}_{{0}{,}{1}{,}{2}}{}{a}{+}{{u}}_{{0}{,}{2}{,}{2}}{}{b}\right]{,}\left[\left[{2}\right]{,}{{v}}_{{1}}{}\left({{u}}_{{0}{,}{1}{,}{2}}{}{a}{+}{{u}}_{{0}{,}{2}{,}{2}}{}{b}\right)\right]{,}\left[\left[{3}\right]{,}\left({{u}}_{{0}{,}{1}{,}{2}}{}{a}{+}{{u}}_{{0}{,}{2}{,}{2}}{}{b}\right){}{{v}}_{{1}{,}{1}}\right]{,}\left[\left[{4}\right]{,}\left({{u}}_{{0}{,}{1}{,}{2}}{}{a}{+}{{u}}_{{0}{,}{2}{,}{2}}{}{b}\right){}{{v}}_{{1}{,}{1}{,}{1}}\right]{,}\left[\left[{5}\right]{,}\left({{u}}_{{0}{,}{1}{,}{2}}{}{a}{+}{{u}}_{{0}{,}{2}{,}{2}}{}{b}\right){}{{v}}_{{1}{,}{1}{,}{1}{,}{1}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{\mathrm{E2}}{,}\left[{"total"}{,}{3}\right]\right]{,}\left[\left[\left[{1}\right]{,}{{u}}_{{0}{,}{1}{,}{2}}{}{a}{+}{{u}}_{{0}{,}{2}{,}{2}}{}{b}\right]{,}\left[\left[{2}\right]{,}{{v}}_{{1}}{}\left({{u}}_{{0}{,}{1}{,}{2}}{}{a}{+}{{u}}_{{0}{,}{2}{,}{2}}{}{b}\right)\right]{,}\left[\left[{3}\right]{,}\left({{u}}_{{0}{,}{1}{,}{2}}{}{a}{+}{{u}}_{{0}{,}{2}{,}{2}}{}{b}\right){}{{v}}_{{1}{,}{1}}\right]{,}\left[\left[{4}\right]{,}\left({{u}}_{{0}{,}{1}{,}{2}}{}{a}{+}{{u}}_{{0}{,}{2}{,}{2}}{}{b}\right){}{{v}}_{{1}{,}{1}{,}{1}}\right]{,}\left[\left[{5}\right]{,}\left({{u}}_{{0}{,}{1}{,}{2}}{}{a}{+}{{u}}_{{0}{,}{2}{,}{2}}{}{b}\right){}{{v}}_{{1}{,}{1}{,}{1}{,}{1}}\right]\right]\right]\right)$ (2.5)