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 $x^i$. The push forward of the total vector field $X \= X^i {\mathrm{D}}_i$on $J^k\left(E\right)$is the total vector $X \= Y^a {\mathrm{D}}_a$, where $Y^a \= \left({\mathrm{D}}_i{\mathrm{\φ}}^a\right)X^i$.

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(...).

$\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{\φ}≔\mathrm{Transformation}\left(\mathrm{E1}\,\mathrm{E2}\,\left[t\=u_{2\,0}\,v_{\[\]}\=xy\right]\right)$

$\mathrm{\phi }≔\mathrm{\_DG}\left(\left[\left[''transformation''\,\left[\left[\mathrm{E1}\,2\right]\,\left[\mathrm{E2}\,0\right]\right]\,\left[\right]\,\left[\left[\begin{array}{cccccccc}0& 0& 0& 0& 0& 0& 0& 0\\ y& x& 0& 0& 0& 0& 0& 0\end{array}\right]\right]\right]\,\left[\left[u_{2,0}\,t\right]\,\left[xy\,v_{\left[\right]}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''transformation''\,\left[\left[\mathrm{E1}\,2\right]\,\left[\mathrm{E2}\,0\right]\right]\,\left[\right]\,\left[\left[\begin{array}{cccccccc}0& 0& 0& 0& 0& 0& 0& 0\\ y& x& 0& 0& 0& 0& 0& 0\end{array}\right]\right]\right]\,\left[\left[u_{2,0}\,t\right]\,\left[xy\,v_{\left[\right]}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''transformation''\,\left[\left[\mathrm{E1}\,2\right]\,\left[\mathrm{E2}\,0\right]\right]\,\left[\right]\,\left[\left[\begin{array}{cccccccc}0& 0& 0& 0& 0& 0& 0& 0\\ y& x& 0& 0& 0& 0& 0& 0\end{array}\right]\right]\right]\,\left[\left[u_{2,0}\,t\right]\,\left[xy\,v_{\left[\right]}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''transformation''\,\left[\left[\mathrm{E1}\,2\right]\,\left[\mathrm{E2}\,0\right]\right]\,\left[\right]\,\left[\left[\begin{array}{cccccccc}0& 0& 0& 0& 0& 0& 0& 0\\ y& x& 0& 0& 0& 0& 0& 0\end{array}\right]\right]\right]\,\left[\left[u_{2,0}\,t\right]\,\left[xy\,v_{\left[\right]}\right]\right]\right]\right)$

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

$\mathrm{J1}≔\left[\begin{array}{cc}{u}_{0,1,2}& u_{0,2,2}\end{array}\right]$

$X≔a\mathrm{D\_x}\+b\mathrm{D\_y}$

$X≔\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{E1}\,\left[\right]\right]\,\left[\left[\left[1\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{E1}\,\left[\right]\right]\,\left[\left[\left[1\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{E1}\,\left[\right]\right]\,\left[\left[\left[1\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{E1}\,\left[\right]\right]\,\left[\left[\left[1\right]\,1\right]\right]\right]\right)a+\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{E1}\,\left[\right]\right]\,\left[\left[\left[2\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{E1}\,\left[\right]\right]\,\left[\left[\left[2\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{E1}\,\left[\right]\right]\,\left[\left[\left[2\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{E1}\,\left[\right]\right]\,\left[\left[\left[2\right]\,1\right]\right]\right]\right)b$

$\mathrm{totX}≔\mathrm{Prolong}\left(\mathrm{TotalVector}\left(X\right)\,3\right)$

$\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)$

$\mathrm{PushforwardTotalVector}\left(\mathrm{\φ}\,\mathrm{totX}\right)$

$\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{E2}\,\left[''total''\,3\right]\right]\,\left[\left[\left[1\right]\,u_{0,1,2}a+u_{0,2,2}b\right]\,\left[\left[2\right]\,v_1\left(u_{0,1,2}a+u_{0,2,2}b\right)\right]\,\left[\left[3\right]\,\left(u_{0,1,2}a+u_{0,2,2}b\right)v_{1,1}\right]\,\left[\left[4\right]\,\left(u_{0,1,2}a+u_{0,2,2}b\right)v_{1,1,1}\right]\,\left[\left[5\right]\,\left(u_{0,1,2}a+u_{0,2,2}b\right)v_{1,1,1,1}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{E2}\,\left[''total''\,3\right]\right]\,\left[\left[\left[1\right]\,u_{0,1,2}a+u_{0,2,2}b\right]\,\left[\left[2\right]\,v_1\left(u_{0,1,2}a+u_{0,2,2}b\right)\right]\,\left[\left[3\right]\,\left(u_{0,1,2}a+u_{0,2,2}b\right)v_{1,1}\right]\,\left[\left[4\right]\,\left(u_{0,1,2}a+u_{0,2,2}b\right)v_{1,1,1}\right]\,\left[\left[5\right]\,\left(u_{0,1,2}a+u_{0,2,2}b\right)v_{1,1,1,1}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{E2}\,\left[''total''\,3\right]\right]\,\left[\left[\left[1\right]\,u_{0,1,2}a+u_{0,2,2}b\right]\,\left[\left[2\right]\,v_1\left(u_{0,1,2}a+u_{0,2,2}b\right)\right]\,\left[\left[3\right]\,\left(u_{0,1,2}a+u_{0,2,2}b\right)v_{1,1}\right]\,\left[\left[4\right]\,\left(u_{0,1,2}a+u_{0,2,2}b\right)v_{1,1,1}\right]\,\left[\left[5\right]\,\left(u_{0,1,2}a+u_{0,2,2}b\right)v_{1,1,1,1}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{E2}\,\left[''total''\,3\right]\right]\,\left[\left[\left[1\right]\,u_{0,1,2}a+u_{0,2,2}b\right]\,\left[\left[2\right]\,v_1\left(u_{0,1,2}a+u_{0,2,2}b\right)\right]\,\left[\left[3\right]\,\left(u_{0,1,2}a+u_{0,2,2}b\right)v_{1,1}\right]\,\left[\left[4\right]\,\left(u_{0,1,2}a+u_{0,2,2}b\right)v_{1,1,1}\right]\,\left[\left[5\right]\,\left(u_{0,1,2}a+u_{0,2,2}b\right)v_{1,1,1,1}\right]\right]\right]\right)$