Let $\mathrm{\π}\:E \to M$be a fiber bundle and let ${\mathrm{\π}}^k\:J^k\left(E\right) \to M$ be the associated jet bundle. Let $\(x^i\, u^{\mathrm{\α}}\, u_{i_{}}^{\mathrm{\α}}\, u_{i_{}j}^{\mathrm{\α}}$, ...,$u_{\mathrm{ij} \cdot \cdot \cdot m}^{\mathrm{\α}}\)$ be the local coordinates on $J^k\left(E\right)$and let $X \= A^j\frac{\partial }{\partial x^i} \+B^{\mathrm{\β}}\frac{\partial }{\partial u^{\mathrm{\β}}}$ (*) be a generalized vector field on $E$. The coefficients $A^i$and $B^{\mathrm{\β} }$are functions on jet space. Then the evolutionary part of $X$is the generalized vertical vector field $X_{\mathrm{ev}} \= \left(B^{\mathrm{\β} }-A^{\mathrm{\ℓ}}{u}_{\mathrm{\ℓ}}^{\mathrm{\β}}\right)\frac{\partial }{\partial u^{\mathrm{\β}}}$.  Every vector field decomposes as a sum of its evolutionary and total parts $X \= X_{\mathrm{tot}} \+ X_{\mathrm{ev}}$.

The evolutionary part of a projectable vector field $X$has the following geometric interpretation (The vector (*) is projectable if $A^i\=A^i\left(x^j\right)$ and $B^{\mathrm{\beta } }$= $B^{\mathrm{\β}}\($$x^i\, u^{\mathrm{\alpha }}\)$). Let ${\mathrm{\φ}}_t\:E \to E$ be the flow of $X$. Then ${\mathrm{\phi }}_t$covers a map ${\mathrm{\ψ}}_t\:M\to M$. If $\mathrm{\σ}\:M\to E$ is a section of $E$, then the induced flow in the space of sections is defined by the section${\mathrm{\σ}}_t\left(x\right)\={\mathrm{\φ}}_t\left(\mathrm{\σ}\left({\mathrm{\ψ}}_{-t}\left(x\right)\right)\right)$. The derivative of ${\mathrm{\sigma }}_t$, evaluated at $t \= 0$, yields $X_{\mathrm{ev}}$ .

The command EvolutionaryVector is part of the DifferentialGeometry:-JetCalculus package.  It can be used in the form EvolutionaryVector(...) only after executing the commands with(DifferentialGeometry) and with(JetCalculus), but can always be used by executing DifferentialGeometry:-JetCalculus:-EvolutionaryVector(...).

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

$\mathrm{DGsetup}\left(\left[x\,y\right]\,\left[u\,v\right]\,\mathrm{J22}\,1\right)\:$

$\mathrm{X1}≔\mathrm{D\_x}$

$\mathrm{X1}≔\mathrm{D\_x}$

$\mathrm{totX1}≔\mathrm{TotalVector}\left(\mathrm{X1}\right)$

$\mathrm{totX1}≔\mathrm{D\_x}+u_1{\mathrm{D\_u}}_{\left[\right]}+v_1{\mathrm{D\_v}}_{\left[\right]}$

$\mathrm{evolX1}≔\mathrm{EvolutionaryVector}\left(\mathrm{X1}\right)$

$\mathrm{evolX1}≔-u_1{\mathrm{D\_u}}_{\left[\right]}-v_1{\mathrm{D\_v}}_{\left[\right]}$

$\mathrm{totX1}\&plus\mathrm{evolX1}$

$\mathrm{D\_x}$

$\mathrm{X2}≔\mathrm{D\_u}\left[\right]$

$\mathrm{X2}≔{\mathrm{D\_u}}_{\left[\right]}$

$\mathrm{totX2}≔\mathrm{TotalVector}\left(\mathrm{X2}\right)$

$\mathrm{totX2}≔0\mathrm{D\_x}$

$\mathrm{evolX2}≔\mathrm{EvolutionaryVector}\left(\mathrm{X2}\right)$

$\mathrm{evolX2}≔{\mathrm{D\_u}}_{\left[\right]}$

$\mathrm{totX2}\&plus\mathrm{evolX2}$

${\mathrm{D\_u}}_{\left[\right]}$

$\mathrm{X3}≔\mathrm{evalDG}\left(a\mathrm{D\_x}+b\mathrm{D\_y}+c\mathrm{D\_u}\left[\right]+d\mathrm{D\_v}\left[\right]\right)$

$\mathrm{X3}≔a\mathrm{D\_x}+b\mathrm{D\_y}+c{\mathrm{D\_u}}_{\left[\right]}+d{\mathrm{D\_v}}_{\left[\right]}$

$\mathrm{totX3}≔\mathrm{TotalVector}\left(\mathrm{X3}\right)$

$\mathrm{totX3}≔a\mathrm{D\_x}+b\mathrm{D\_y}+\left(u_{1}a+u_{2}b\right){\mathrm{D\_u}}_{\left[\right]}+\left(v_{1}a+v_{2}b\right){\mathrm{D\_v}}_{\left[\right]}$

$\mathrm{evolX3}≔\mathrm{EvolutionaryVector}\left(\mathrm{X3}\right)$

$\mathrm{evolX3}≔-\left(u_{1}a+u_{2}b-c\right){\mathrm{D\_u}}_{\left[\right]}-\left(v_{1}a+v_{2}b-d\right){\mathrm{D\_v}}_{\left[\right]}$

$\mathrm{totX3}\&plus\mathrm{evolX3}$

$a\mathrm{D\_x}+b\mathrm{D\_y}+c{\mathrm{D\_u}}_{\left[\right]}+d{\mathrm{D\_v}}_{\left[\right]}$

$\mathrm{DGsetup}\left(\left[x\,y\right]\,M\right)\:$$\mathrm{DGsetup}\left(\left[x\,y\right]\,\left[u\right]\,E\,2\right)\:$

$\mathrm{X4}≔\mathrm{evalDG}\left(-y\mathrm{D\_x}+x\mathrm{D\_y}+u\left[\right]\mathrm{D\_u}\left[\right]\right)$

$\mathrm{X4}≔-y\mathrm{D\_x}+x\mathrm{D\_y}+u_{\left[\right]}{\mathrm{D\_u}}_{\left[\right]}$

$\mathrm{evolX4}≔\mathrm{EvolutionaryVector}\left(\mathrm{X4}\right)$

$\mathrm{evolX4}≔-\left(u_{2}x-u_{1}y-u_{\left[\right]}\right){\mathrm{D\_u}}_{\left[\right]}$

$\mathrm{ChangeFrame}\left(M\right)$

$E$

$\mathrm{Y4}≔\mathrm{evalDG}\left(-y\mathrm{D\_x}+x\mathrm{D\_y}\right)$

$\mathrm{Y4}≔-y\mathrm{D\_x}+x\mathrm{D\_y}$

$\mathrm{\psi }≔\mathrm{eval}\left(\mathrm{Flow}\left(\mathrm{Y4}\,t\right)\,t=-t\right)$

$\mathrm{\psi }≔\left[x=y\sin \left(t\right)+x\cos \left(t\right)\,y=y\cos \left(t\right)-x\sin \left(t\right)\right]$

$\mathrm{\Phi }≔\mathrm{Flow}\left(\mathrm{X4}\,t\right)$

$\mathrm{\Phi }≔\left[x=-y\sin \left(t\right)+x\cos \left(t\right)\,y=y\cos \left(t\right)+x\sin \left(t\right)\,u_{\left[\right]}=u_{\left[\right]}{\ⅇ}^t\right]$

$\mathrm{\sigma }≔\mathrm{Transformation}\left(M\,E\,\left[x=x\,y=y\,u\left[\right]=U\left(x\,y\right)\right]\right)$

$\mathrm{\sigma }≔\left[x=x\,y=y\,u_{\left[\right]}=U\left(x\,y\right)\right]$

$\mathrm{sigma\_t}≔\mathrm{ComposeTransformations}\left(\mathrm{\Phi }\,\mathrm{\sigma }\,\mathrm{\psi }\right)$

$\mathrm{sigma\_t}≔\left[x=-\left(y\cos \left(t\right)-x\sin \left(t\right)\right)\sin \left(t\right)+\left(y\sin \left(t\right)+x\cos \left(t\right)\right)\cos \left(t\right)\,y=\left(y\cos \left(t\right)-x\sin \left(t\right)\right)\cos \left(t\right)+\left(y\sin \left(t\right)+x\cos \left(t\right)\right)\sin \left(t\right)\,u_{\left[\right]}=U\left(y\sin \left(t\right)+x\cos \left(t\right)\,y\cos \left(t\right)-x\sin \left(t\right)\right){\ⅇ}^t\right]$

$\mathrm{\Sigma }≔\mathrm{ApplyTransformation}\left(\mathrm{sigma\_t}\,\left[x\,y\right]\right)$

$\mathrm{\Sigma }≔\left[-\left(y\cos \left(t\right)-x\sin \left(t\right)\right)\sin \left(t\right)+\left(y\sin \left(t\right)+x\cos \left(t\right)\right)\cos \left(t\right)\,\left(y\cos \left(t\right)-x\sin \left(t\right)\right)\cos \left(t\right)+\left(y\sin \left(t\right)+x\cos \left(t\right)\right)\sin \left(t\right)\,U\left(y\sin \left(t\right)+x\cos \left(t\right)\,y\cos \left(t\right)-x\sin \left(t\right)\right){\ⅇ}^t\right]$

$\mathrm{eval}\left(\mathrm{diff}\left(\mathrm{\Sigma }\,t\right)\,t=0\right)$

$\left[0\,0\,{\mathrm{D}}_1\left(U\right)\left(x\,y\right)y-{\mathrm{D}}_2\left(U\right)\left(x\,y\right)x+U\left(x\,y\right)\right]$

$\mathrm{GetComponents}\left(\mathrm{evolX4}\,\left[\mathrm{D\_x}\,\mathrm{D\_y}\,\mathrm{D\_u}\left[\right]\right]\right)$

$\left[0\,0\,-x{u}_2+y{u}_1+u_{\left[\right]}\right]$