TotalJacobian - Maple Help

JetCalculus[TotalJacobian] - find the Jacobian of a transformation using total derivatives

Calling Sequences

TotalJacobian(${\mathbf{φ}}$)

Parameters

- 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 . Then the total Jacobian of is the matrix $\left[{\mathrm{D}}_{i}{\mathrm{φ}}^{a}\right]$, where ${\mathrm{D}}_{i}$ denotes the total derivative with respect to ${x}^{i}$.
 • TotalJacobian returns the matrix $\left[{\mathrm{D}}_{i}{\mathrm{φ}}^{a}\right]$.
 • The command TotalJacobian is part of the DifferentialGeometry:-JetCalculus package. It can be used in the form TotalJacobian(...) only after executing the commands with(DifferentialGeometry) and with(JetCalculus), but can always be used by executing DifferentialGeometry:-JetCalculus:-TotalJacobian(...).

Examples

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

Example 1.

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

 > $\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 > $\mathrm{φ1}≔\mathrm{Transformation}\left(\mathrm{E1},\mathrm{E2},\left[t={u}_{1,1},{v}_{[]}=xy\right]\right)$
 ${\mathrm{φ1}}{≔}{\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}& {1}& {0}& {0}\\ {y}& {x}& {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]\right]{,}\left[\left[{{u}}_{{1}{,}{1}}{,}{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}& {1}& {0}& {0}\\ {y}& {x}& {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]\right]{,}\left[\left[{{u}}_{{1}{,}{1}}{,}{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}& {1}& {0}& {0}\\ {y}& {x}& {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]\right]{,}\left[\left[{{u}}_{{1}{,}{1}}{,}{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}& {1}& {0}& {0}\\ {y}& {x}& {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]\right]{,}\left[\left[{{u}}_{{1}{,}{1}}{,}{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}& {1}& {0}& {0}\\ {y}& {x}& {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]\right]{,}\left[\left[{{u}}_{{1}{,}{1}}{,}{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}& {1}& {0}& {0}\\ {y}& {x}& {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]\right]{,}\left[\left[{{u}}_{{1}{,}{1}}{,}{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}& {1}& {0}& {0}\\ {y}& {x}& {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]\right]{,}\left[\left[{{u}}_{{1}{,}{1}}{,}{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}& {1}& {0}& {0}\\ {y}& {x}& {0}& {0}& {0}& {0}& {0}& {0}\end{array}\right]\right]\right]{,}\left[\left[{{u}}_{{1}{,}{1}}{,}{t}\right]{,}\left[{x}{}{y}{,}{{v}}_{\left[\right]}\right]\right]\right]\right)$ (2.1)
 E1 > $\mathrm{J1}≔\mathrm{TotalJacobian}\left(\mathrm{φ1}\right)$
 ${\mathrm{J1}}{≔}\left[\begin{array}{cc}{{u}}_{{1}{,}{1}{,}{1}}& {{u}}_{{1}{,}{1}{,}{2}}\end{array}\right]$ (2.2)

Define a transformation  and compute its total Jacobian (a 3$×$2 matrix).

 E1 > $\mathrm{φ2}≔\mathrm{Transformation}\left(\mathrm{E1},\mathrm{E3},\left[p=x{u}_{1},q=y{u}_{[]},r={u}_{2,2},{w}_{[]}={u}_{1}\right]\right)$
 ${\mathrm{φ2}}{≔}{\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{\mathrm{E1}}{,}{2}\right]{,}\left[{\mathrm{E3}}{,}{0}\right]\right]{,}\left[\right]{,}\left[\left[\begin{array}{cccccccc}{{u}}_{{1}}& {0}& {0}& {x}& {0}& {0}& {0}& {0}\\ {0}& {{u}}_{\left[\right]}& {y}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {1}& {0}& {0}& {0}& {0}\end{array}\right]\right]\right]{,}\left[\left[{x}{}{{u}}_{{1}}{,}{p}\right]{,}\left[{y}{}{{u}}_{\left[\right]}{,}{q}\right]{,}\left[{{u}}_{{2}{,}{2}}{,}{r}\right]{,}\left[{{u}}_{{1}}{,}{{w}}_{\left[\right]}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{\mathrm{E1}}{,}{2}\right]{,}\left[{\mathrm{E3}}{,}{0}\right]\right]{,}\left[\right]{,}\left[\left[\begin{array}{cccccccc}{{u}}_{{1}}& {0}& {0}& {x}& {0}& {0}& {0}& {0}\\ {0}& {{u}}_{\left[\right]}& {y}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {1}& {0}& {0}& {0}& {0}\end{array}\right]\right]\right]{,}\left[\left[{x}{}{{u}}_{{1}}{,}{p}\right]{,}\left[{y}{}{{u}}_{\left[\right]}{,}{q}\right]{,}\left[{{u}}_{{2}{,}{2}}{,}{r}\right]{,}\left[{{u}}_{{1}}{,}{{w}}_{\left[\right]}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{\mathrm{E1}}{,}{2}\right]{,}\left[{\mathrm{E3}}{,}{0}\right]\right]{,}\left[\right]{,}\left[\left[\begin{array}{cccccccc}{{u}}_{{1}}& {0}& {0}& {x}& {0}& {0}& {0}& {0}\\ {0}& {{u}}_{\left[\right]}& {y}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {1}& {0}& {0}& {0}& {0}\end{array}\right]\right]\right]{,}\left[\left[{x}{}{{u}}_{{1}}{,}{p}\right]{,}\left[{y}{}{{u}}_{\left[\right]}{,}{q}\right]{,}\left[{{u}}_{{2}{,}{2}}{,}{r}\right]{,}\left[{{u}}_{{1}}{,}{{w}}_{\left[\right]}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{\mathrm{E1}}{,}{2}\right]{,}\left[{\mathrm{E3}}{,}{0}\right]\right]{,}\left[\right]{,}\left[\left[\begin{array}{cccccccc}{{u}}_{{1}}& {0}& {0}& {x}& {0}& {0}& {0}& {0}\\ {0}& {{u}}_{\left[\right]}& {y}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {1}& {0}& {0}& {0}& {0}\end{array}\right]\right]\right]{,}\left[\left[{x}{}{{u}}_{{1}}{,}{p}\right]{,}\left[{y}{}{{u}}_{\left[\right]}{,}{q}\right]{,}\left[{{u}}_{{2}{,}{2}}{,}{r}\right]{,}\left[{{u}}_{{1}}{,}{{w}}_{\left[\right]}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{\mathrm{E1}}{,}{2}\right]{,}\left[{\mathrm{E3}}{,}{0}\right]\right]{,}\left[\right]{,}\left[\left[\begin{array}{cccccccc}{{u}}_{{1}}& {0}& {0}& {x}& {0}& {0}& {0}& {0}\\ {0}& {{u}}_{\left[\right]}& {y}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {1}& {0}& {0}& {0}& {0}\end{array}\right]\right]\right]{,}\left[\left[{x}{}{{u}}_{{1}}{,}{p}\right]{,}\left[{y}{}{{u}}_{\left[\right]}{,}{q}\right]{,}\left[{{u}}_{{2}{,}{2}}{,}{r}\right]{,}\left[{{u}}_{{1}}{,}{{w}}_{\left[\right]}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{\mathrm{E1}}{,}{2}\right]{,}\left[{\mathrm{E3}}{,}{0}\right]\right]{,}\left[\right]{,}\left[\left[\begin{array}{cccccccc}{{u}}_{{1}}& {0}& {0}& {x}& {0}& {0}& {0}& {0}\\ {0}& {{u}}_{\left[\right]}& {y}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {1}& {0}& {0}& {0}& {0}\end{array}\right]\right]\right]{,}\left[\left[{x}{}{{u}}_{{1}}{,}{p}\right]{,}\left[{y}{}{{u}}_{\left[\right]}{,}{q}\right]{,}\left[{{u}}_{{2}{,}{2}}{,}{r}\right]{,}\left[{{u}}_{{1}}{,}{{w}}_{\left[\right]}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{\mathrm{E1}}{,}{2}\right]{,}\left[{\mathrm{E3}}{,}{0}\right]\right]{,}\left[\right]{,}\left[\left[\begin{array}{cccccccc}{{u}}_{{1}}& {0}& {0}& {x}& {0}& {0}& {0}& {0}\\ {0}& {{u}}_{\left[\right]}& {y}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {1}& {0}& {0}& {0}& {0}\end{array}\right]\right]\right]{,}\left[\left[{x}{}{{u}}_{{1}}{,}{p}\right]{,}\left[{y}{}{{u}}_{\left[\right]}{,}{q}\right]{,}\left[{{u}}_{{2}{,}{2}}{,}{r}\right]{,}\left[{{u}}_{{1}}{,}{{w}}_{\left[\right]}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{\mathrm{E1}}{,}{2}\right]{,}\left[{\mathrm{E3}}{,}{0}\right]\right]{,}\left[\right]{,}\left[\left[\begin{array}{cccccccc}{{u}}_{{1}}& {0}& {0}& {x}& {0}& {0}& {0}& {0}\\ {0}& {{u}}_{\left[\right]}& {y}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {1}& {0}& {0}& {0}& {0}\end{array}\right]\right]\right]{,}\left[\left[{x}{}{{u}}_{{1}}{,}{p}\right]{,}\left[{y}{}{{u}}_{\left[\right]}{,}{q}\right]{,}\left[{{u}}_{{2}{,}{2}}{,}{r}\right]{,}\left[{{u}}_{{1}}{,}{{w}}_{\left[\right]}\right]\right]\right]\right)$ (2.3)
 E1 > $\mathrm{J2}≔\mathrm{TotalJacobian}\left(\mathrm{φ2}\right)$
 ${\mathrm{J2}}{≔}\left[\begin{array}{cc}{x}{}{{u}}_{{1}{,}{1}}{+}{{u}}_{{1}}& {x}{}{{u}}_{{1}{,}{2}}\\ {y}{}{{u}}_{{1}}& {y}{}{{u}}_{{2}}{+}{{u}}_{\left[\right]}\\ {{u}}_{{1}{,}{2}{,}{2}}& {{u}}_{{2}{,}{2}{,}{2}}\end{array}\right]$ (2.4)

Define a transformation  and compute its total Jacobian (a 2$×$2 matrix).

 E1 > $\mathrm{φ3}≔\mathrm{Transformation}\left(\mathrm{E1},\mathrm{E1},\left[x=xy,y={u}_{[]}{u}_{2},{u}_{[]}=y\right]\right)$
 ${\mathrm{φ3}}{≔}{\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{\mathrm{E1}}{,}{1}\right]{,}\left[{\mathrm{E1}}{,}{0}\right]\right]{,}\left[\right]{,}\left[\left[\begin{array}{ccccc}{y}& {x}& {0}& {0}& {0}\\ {0}& {0}& {{u}}_{{2}}& {0}& {{u}}_{\left[\right]}\\ {0}& {1}& {0}& {0}& {0}\end{array}\right]\right]\right]{,}\left[\left[{x}{}{y}{,}{x}\right]{,}\left[{{u}}_{\left[\right]}{}{{u}}_{{2}}{,}{y}\right]{,}\left[{y}{,}{{u}}_{\left[\right]}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{\mathrm{E1}}{,}{1}\right]{,}\left[{\mathrm{E1}}{,}{0}\right]\right]{,}\left[\right]{,}\left[\left[\begin{array}{ccccc}{y}& {x}& {0}& {0}& {0}\\ {0}& {0}& {{u}}_{{2}}& {0}& {{u}}_{\left[\right]}\\ {0}& {1}& {0}& {0}& {0}\end{array}\right]\right]\right]{,}\left[\left[{x}{}{y}{,}{x}\right]{,}\left[{{u}}_{\left[\right]}{}{{u}}_{{2}}{,}{y}\right]{,}\left[{y}{,}{{u}}_{\left[\right]}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{\mathrm{E1}}{,}{1}\right]{,}\left[{\mathrm{E1}}{,}{0}\right]\right]{,}\left[\right]{,}\left[\left[\begin{array}{ccccc}{y}& {x}& {0}& {0}& {0}\\ {0}& {0}& {{u}}_{{2}}& {0}& {{u}}_{\left[\right]}\\ {0}& {1}& {0}& {0}& {0}\end{array}\right]\right]\right]{,}\left[\left[{x}{}{y}{,}{x}\right]{,}\left[{{u}}_{\left[\right]}{}{{u}}_{{2}}{,}{y}\right]{,}\left[{y}{,}{{u}}_{\left[\right]}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{\mathrm{E1}}{,}{1}\right]{,}\left[{\mathrm{E1}}{,}{0}\right]\right]{,}\left[\right]{,}\left[\left[\begin{array}{ccccc}{y}& {x}& {0}& {0}& {0}\\ {0}& {0}& {{u}}_{{2}}& {0}& {{u}}_{\left[\right]}\\ {0}& {1}& {0}& {0}& {0}\end{array}\right]\right]\right]{,}\left[\left[{x}{}{y}{,}{x}\right]{,}\left[{{u}}_{\left[\right]}{}{{u}}_{{2}}{,}{y}\right]{,}\left[{y}{,}{{u}}_{\left[\right]}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{\mathrm{E1}}{,}{1}\right]{,}\left[{\mathrm{E1}}{,}{0}\right]\right]{,}\left[\right]{,}\left[\left[\begin{array}{ccccc}{y}& {x}& {0}& {0}& {0}\\ {0}& {0}& {{u}}_{{2}}& {0}& {{u}}_{\left[\right]}\\ {0}& {1}& {0}& {0}& {0}\end{array}\right]\right]\right]{,}\left[\left[{x}{}{y}{,}{x}\right]{,}\left[{{u}}_{\left[\right]}{}{{u}}_{{2}}{,}{y}\right]{,}\left[{y}{,}{{u}}_{\left[\right]}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{\mathrm{E1}}{,}{1}\right]{,}\left[{\mathrm{E1}}{,}{0}\right]\right]{,}\left[\right]{,}\left[\left[\begin{array}{ccccc}{y}& {x}& {0}& {0}& {0}\\ {0}& {0}& {{u}}_{{2}}& {0}& {{u}}_{\left[\right]}\\ {0}& {1}& {0}& {0}& {0}\end{array}\right]\right]\right]{,}\left[\left[{x}{}{y}{,}{x}\right]{,}\left[{{u}}_{\left[\right]}{}{{u}}_{{2}}{,}{y}\right]{,}\left[{y}{,}{{u}}_{\left[\right]}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{\mathrm{E1}}{,}{1}\right]{,}\left[{\mathrm{E1}}{,}{0}\right]\right]{,}\left[\right]{,}\left[\left[\begin{array}{ccccc}{y}& {x}& {0}& {0}& {0}\\ {0}& {0}& {{u}}_{{2}}& {0}& {{u}}_{\left[\right]}\\ {0}& {1}& {0}& {0}& {0}\end{array}\right]\right]\right]{,}\left[\left[{x}{}{y}{,}{x}\right]{,}\left[{{u}}_{\left[\right]}{}{{u}}_{{2}}{,}{y}\right]{,}\left[{y}{,}{{u}}_{\left[\right]}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"transformation"}{,}\left[\left[{\mathrm{E1}}{,}{1}\right]{,}\left[{\mathrm{E1}}{,}{0}\right]\right]{,}\left[\right]{,}\left[\left[\begin{array}{ccccc}{y}& {x}& {0}& {0}& {0}\\ {0}& {0}& {{u}}_{{2}}& {0}& {{u}}_{\left[\right]}\\ {0}& {1}& {0}& {0}& {0}\end{array}\right]\right]\right]{,}\left[\left[{x}{}{y}{,}{x}\right]{,}\left[{{u}}_{\left[\right]}{}{{u}}_{{2}}{,}{y}\right]{,}\left[{y}{,}{{u}}_{\left[\right]}\right]\right]\right]\right)$ (2.5)
 E1 > $\mathrm{J3}≔\mathrm{TotalJacobian}\left(\mathrm{φ3}\right)$
 ${\mathrm{J3}}{≔}\left[\begin{array}{cc}{y}& {x}\\ {{u}}_{\left[\right]}{}{{u}}_{{1}{,}{2}}{+}{{u}}_{{2}}{}{{u}}_{{1}}& {{u}}_{\left[\right]}{}{{u}}_{{2}{,}{2}}{+}{{u}}_{{2}}^{{2}}\end{array}\right]$ (2.6)