Let $\mathrm{\π}\:E\to M$be a fiber bundle, with base dimension $n$ and fiber dimension $m$ and let ${\mathrm{\π}}^{\mathrm{\∞}} \:J^{\mathrm{\∞}}\left(E\right) \to M$ be the infinite jet bundle of $E$. The  $p$-forms ${\mathrm{\Ω}}^p\left(J^{\mathrm{\∞}}\left(E\right)\right)$ can be graded by horizontal and vertical (or contact) degree and, with respect to this grading, the exterior derivative operator can be decomposed as $d \= d_{H } \+ d_V\.$The horizontal exterior derivative $d_H$raises the horizontal degree by 1 and the vertical exterior derivative $d_V$raises the vertical degree by 1. For details, see HorizontalExteriorDerivative.

The command VerticalExteriorDerivative($\mathrm{\ω}$) returns the vertical exterior derivative $d_V\left(\mathrm{\ω}\right)$.

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

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

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

$F≔f\left(x\,y\,u\left[\right]\,v\left[\right]\,u\left[1\right]\,u\left[2\right]\,v\left[1\right]\,v\left[2\right]\right)\:$

$\mathrm{PDEtools}\left[\mathrm{declare}\right]\left(F\,\mathrm{quiet}\right)\:$

$\mathrm{VerticalExteriorDerivative}\left(F\right)$

$\mathrm{\_DG}\left(\left[\left[''biform''\,E\,\left[0\,1\right]\right]\,\left[\left[\left[3\right]\,f_{u_{\left[\right]}}\right]\,\left[\left[4\right]\,f_{v_{\left[\right]}}\right]\,\left[\left[5\right]\,f_{u_1}\right]\,\left[\left[6\right]\,f_{u_2}\right]\,\left[\left[7\right]\,f_{v_1}\right]\,\left[\left[8\right]\,f_{v_2}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''biform''\,E\,\left[0\,1\right]\right]\,\left[\left[\left[3\right]\,f_{u_{\left[\right]}}\right]\,\left[\left[4\right]\,f_{v_{\left[\right]}}\right]\,\left[\left[5\right]\,f_{u_1}\right]\,\left[\left[6\right]\,f_{u_2}\right]\,\left[\left[7\right]\,f_{v_1}\right]\,\left[\left[8\right]\,f_{v_2}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''biform''\,E\,\left[0\,1\right]\right]\,\left[\left[\left[3\right]\,f_{u_{\left[\right]}}\right]\,\left[\left[4\right]\,f_{v_{\left[\right]}}\right]\,\left[\left[5\right]\,f_{u_1}\right]\,\left[\left[6\right]\,f_{u_2}\right]\,\left[\left[7\right]\,f_{v_1}\right]\,\left[\left[8\right]\,f_{v_2}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''biform''\,E\,\left[0\,1\right]\right]\,\left[\left[\left[3\right]\,f_{u_{\left[\right]}}\right]\,\left[\left[4\right]\,f_{v_{\left[\right]}}\right]\,\left[\left[5\right]\,f_{u_1}\right]\,\left[\left[6\right]\,f_{u_2}\right]\,\left[\left[7\right]\,f_{v_1}\right]\,\left[\left[8\right]\,f_{v_2}\right]\right]\right]\right)$

$\mathrm{\ω1}≔\mathrm{evalDG}\left(u\left[2,2\right]\mathrm{Dx}+v\left[1,1,1\right]\mathrm{Dy}\right)$

$\mathrm{\ω1}≔\mathrm{\_DG}\left(\left[\left[''biform''\,E\,\left[1\,0\right]\right]\,\left[\left[\left[1\right]\,u_{2,2}\right]\,\left[\left[2\right]\,v_{1,1,1}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''biform''\,E\,\left[1\,0\right]\right]\,\left[\left[\left[1\right]\,u_{2,2}\right]\,\left[\left[2\right]\,v_{1,1,1}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''biform''\,E\,\left[1\,0\right]\right]\,\left[\left[\left[1\right]\,u_{2,2}\right]\,\left[\left[2\right]\,v_{1,1,1}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''biform''\,E\,\left[1\,0\right]\right]\,\left[\left[\left[1\right]\,u_{2,2}\right]\,\left[\left[2\right]\,v_{1,1,1}\right]\right]\right]\right)$

$\mathrm{VerticalExteriorDerivative}\left(\mathrm{\ω1}\right)$

$\mathrm{\_DG}\left(\left[\left[''biform''\,E\,\left[1\,1\right]\right]\,\left[\left[\left[1\,11\right]\,\mathrm{-1}\right]\,\left[\left[2\,19\right]\,\mathrm{-1}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''biform''\,E\,\left[1\,1\right]\right]\,\left[\left[\left[1\,11\right]\,\mathrm{-1}\right]\,\left[\left[2\,19\right]\,\mathrm{-1}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''biform''\,E\,\left[1\,1\right]\right]\,\left[\left[\left[1\,11\right]\,\mathrm{-1}\right]\,\left[\left[2\,19\right]\,\mathrm{-1}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''biform''\,E\,\left[1\,1\right]\right]\,\left[\left[\left[1\,11\right]\,\mathrm{-1}\right]\,\left[\left[2\,19\right]\,\mathrm{-1}\right]\right]\right]\right)$

$\mathrm{\ω2}≔\mathrm{evalDG}\left(v\left[1,1\right]u\left[2,2,2\right]\mathrm{Cu}\left[2\right]\&w\mathrm{Cv}\left[2\right]\right)$

$\mathrm{\ω2}≔\mathrm{\_DG}\left(\left[\left[''biform''\,E\,\left[0\,2\right]\right]\,\left[\left[\left[6\,8\right]\,v_{1,1}{u}_{2,2,2}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''biform''\,E\,\left[0\,2\right]\right]\,\left[\left[\left[6\,8\right]\,v_{1,1}{u}_{2,2,2}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''biform''\,E\,\left[0\,2\right]\right]\,\left[\left[\left[6\,8\right]\,v_{1,1}{u}_{2,2,2}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''biform''\,E\,\left[0\,2\right]\right]\,\left[\left[\left[6\,8\right]\,v_{1,1}{u}_{2,2,2}\right]\right]\right]\right)$

$\mathrm{VerticalExteriorDerivative}\left(\mathrm{\ω2}\right)$

$\mathrm{\_DG}\left(\left[\left[''biform''\,E\,\left[0\,3\right]\right]\,\left[\left[\left[6\,8\,12\right]\,u_{2,2,2}\right]\,\left[\left[6\,8\,18\right]\,v_{1,1}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''biform''\,E\,\left[0\,3\right]\right]\,\left[\left[\left[6\,8\,12\right]\,u_{2,2,2}\right]\,\left[\left[6\,8\,18\right]\,v_{1,1}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''biform''\,E\,\left[0\,3\right]\right]\,\left[\left[\left[6\,8\,12\right]\,u_{2,2,2}\right]\,\left[\left[6\,8\,18\right]\,v_{1,1}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''biform''\,E\,\left[0\,3\right]\right]\,\left[\left[\left[6\,8\,12\right]\,u_{2,2,2}\right]\,\left[\left[6\,8\,18\right]\,v_{1,1}\right]\right]\right]\right)$