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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 > $\mathrm{φ1}≔\mathrm{Transformation}\left(\mathrm{E1},\mathrm{E2},\left[t=u\left[1,1\right],v\left[\right]=xy\right]\right)$
 ${\mathrm{φ1}}{≔}\left[{t}{=}{{u}}_{{1}{,}{1}}{,}{{v}}_{\left[\right]}{=}{x}{}{y}\right]$ (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 > $\mathrm{φ2}≔\mathrm{Transformation}\left(\mathrm{E1},\mathrm{E3},\left[p=xu\left[1\right],q=yu\left[\right],r=u\left[2,2\right],w\left[\right]=u\left[1\right]\right]\right)$
 ${\mathrm{φ2}}{≔}\left[{p}{=}{x}{}{{u}}_{{1}}{,}{q}{=}{y}{}{{u}}_{\left[\right]}{,}{r}{=}{{u}}_{{2}{,}{2}}{,}{{w}}_{\left[\right]}{=}{{u}}_{{1}}\right]$ (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 > $\mathrm{φ3}≔\mathrm{Transformation}\left(\mathrm{E1},\mathrm{E1},\left[x=xy,y=u\left[\right]u\left[2\right],u\left[\right]=y\right]\right)$
 ${\mathrm{φ3}}{≔}\left[{x}{=}{x}{}{y}{,}{y}{=}{{u}}_{\left[\right]}{}{{u}}_{{2}}{,}{{u}}_{\left[\right]}{=}{y}\right]$ (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)