Let $E\to M$and $F\to N$ be two fiber bundles with associated jet spaces $J^k\left(E\right) \to M$and $J^{\mathrm{\ℓ}}\left(F\right) \to N$ and with jet coordinates $\(x^i\, u^{\mathrm{\α}}\, u_{i_{}}^{\mathrm{\α}}\, u_{i_{}j}^{\mathrm{\α}}$, ..., $u_{\mathrm{ij} \cdot \cdot \cdot k}^{\mathrm{\α}}\)$and $\(y^a\, v^{\mathrm{\ρ}}\, v_{i_{}}^{\mathrm{\ρ}}\, v_{i_{}j}^{ \mathrm{\ρ}}$, ..., $v_{\mathrm{ij} \cdot \cdot \cdot \mathrm{\ℓ}}^{\mathrm{\ρ}}\)$respectively. Let $\mathrm{\φ}\:J^k\left(E\right) \to J^{\ell }\left(F\right)$be a transformation and let ${\mathrm{\φ}}^a\= {\mathrm{\φ}}^a$$\(x^i\, u^{\mathrm{\α}}\, u_{i_{}}^{\mathrm{\α}}\, u_{i_{}j}^{\mathrm{\α}}$, ..., $u_{\mathrm{ij} \cdot \cdot \cdot k}^{\mathrm{\α}}\)$be the $y^a$components of $\mathrm{\φ}$. Then the total Jacobian of $\mathrm{\φ}$is the $m \times n$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 $m \times n$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(...).

$\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\=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[xy\,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[xy\,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[xy\,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[xy\,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[xy\,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[xy\,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[xy\,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[xy\,v_{\left[\right]}\right]\right]\right]\right)$

$\mathrm{J1}≔\mathrm{TotalJacobian}\left(\mathrm{\φ1}\right)$

$\mathrm{J1}≔\left[\begin{array}{cc}{u}_{1,1,1}& u_{1,1,2}\end{array}\right]$

$\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)$

$\mathrm{J2}≔\mathrm{TotalJacobian}\left(\mathrm{\φ2}\right)$

$\mathrm{J2}≔\left[\begin{array}{cc}x{u}_{1,1}+u_1& x{u}_{1,2}\\ y{u}_1& y{u}_2+u_{\left[\right]}\\ u_{1,2,2}& u_{2,2,2}\end{array}\right]$

$\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[xy\,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[xy\,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[xy\,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[xy\,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[xy\,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[xy\,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[xy\,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[xy\,x\right]\,\left[u_{\left[\right]}{u}_2\,y\right]\,\left[y\,u_{\left[\right]}\right]\right]\right]\right)$

$\mathrm{J3}≔\mathrm{TotalJacobian}\left(\mathrm{\φ3}\right)$

$\mathrm{J3}≔\left[\begin{array}{cc}y& x\\ u_{\left[\right]}{u}_{1,2}+u_2{u}_1& u_{\left[\right]}{u}_{2,2}+u_2^2\end{array}\right]$