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 space of $p$-forms ${\mathrm{\Ω}}^p\left(J^{\mathrm{\∞}}\left(E\right)\right)$ decomposes into a direct sum ${\mathrm{\Ω}}^p\left(J^{\infty } \right) \= \underset{r\+s \=p}{\overset{}{⨁}} {\mathrm{\Omega }}^{\left(r\,s\right)}\left(J^{\infty }\left(E\right)\right)$, where  ${\mathrm{\Omega }}^{\left(r\,s\right)} \left(J^{\infty }\left(E\right)\right)$is the space of bi-forms of horizontal degree $r$ and vertical degree $s\.$The horizontal exterior derivative  is a mapping $d_{H }\:{\mathrm{\Omega }}^{\left(r\,s\right)}\left(J^{\infty }\left(E\right)\right)\to {\mathrm{\Omega }}^{\left(r\+1\,s\right)}\left(J^{\infty }\left(E\right)\right)$ with the following properties. A form $\mathrm{\ω} \in {\mathrm{\Omega }}^{\left(r\,s\right)} \left(J^{\infty }\left(E\right)\right)$is called $d_H$closed if $d_{H }\mathrm{\ω} \= 0$and $d_H$exact if there is a bi-form $\mathrm{\η}\in {\mathrm{\Ω}}^{\left(r-1\,s\right)} \left(J^{\infty }\left(E\right)\right)$such that $\mathrm{\ω} \= d_H \mathrm{\η}$. Since $d_H\circ d_H \=0\,$every $d_{H }$exact bi-form is $d_H$closed.

If $\mathrm{\ω}$ is a bi-form of degree $\left(r\,s\right)$with $r\>0$ then HorizontalHomotopy($\mathrm{\ω}$) returns a bi-form of degree ($r-1\, s\)$.

For $s \>0$ the operators $h_{{}^{}H}^{r\,s}$ are total differential operators and therefore, unlike the usual homotopy operators for the de Rham complex or the vertical homotopy operators for bi-forms on jet spaces, do not involve any quadratures. For $s\= 0$the horizontal homotopy does involve quadratures and the optional arguments used in the commands DeRhamHomotopy or VerticalHomotopy can be invoked.

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

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

$\mathrm{DGsetup}\left(\left[x\right]\,\left[u\right]\,E\,3\right)\:$

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

$\mathrm{\ω1}≔\left(x{u}_1{u}_{1,1,1,1}+x{u}_{1,1,1}{u}_{1,1}+u_{1,1,1}{u}_1+2u_{1,1}{u}_{1,1,1}\right)\mathrm{Dx}$

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

$0\mathrm{Dx}\bigwedge {\mathrm{Cu}}_{\left[\right]}$

$\mathrm{\η1}≔\mathrm{HorizontalHomotopy}\left(\mathrm{\ω1}\right)$

$\mathrm{\η1}≔x{u}_{1,1,1}{u}_1+u_{1,1}^2$

$\mathrm{\ω1}\&minus\mathrm{HorizontalExteriorDerivative}\left(\mathrm{\η1}\right)$

$0\mathrm{Dx}$

$\mathrm{\ω2}≔\mathrm{evalDG}\left(\left(\mathrm{Dx}\&w\mathrm{Cu}\left[1\right]\right)\&w\mathrm{Cu}\left[1,1,1,1\right]+\left(\mathrm{Dx}\&w\mathrm{Cu}\left[1,1\right]\right)\&w\mathrm{Cu}\left[1,1,1\right]\right)$

$\mathrm{\ω2}≔\mathrm{Dx}\bigwedge {\mathrm{Cu}}_1\bigwedge {\mathrm{Cu}}_{1,1,1,1}+\mathrm{Dx}\bigwedge {\mathrm{Cu}}_{1,1}\bigwedge {\mathrm{Cu}}_{1,1,1}$

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

$0\mathrm{Dx}\bigwedge {\mathrm{Cu}}_{\left[\right]}\bigwedge {\mathrm{Cu}}_1$

$\mathrm{\η2}≔\mathrm{HorizontalHomotopy}\left(\mathrm{\ω2}\right)$

$\mathrm{\η2}≔{\mathrm{Cu}}_1\bigwedge {\mathrm{Cu}}_{1,1,1}$

$\mathrm{\ω2}\&minus\mathrm{HorizontalExteriorDerivative}\left(\mathrm{\η2}\right)$

$0\mathrm{Dx}\bigwedge {\mathrm{Cu}}_{\left[\right]}\bigwedge {\mathrm{Cu}}_1$

$\mathrm{HorizontalExteriorDerivative}\left(\frac{u\left[1\right]u\left[1,1,1\right]}{{u\left[1,1\right]}^4}\right)$

$\mathrm{\ω3}≔\mathrm{map}\left(\mathrm{expand}\,\mathrm{evalDG}\left(\frac{{u\left[1,1\right]}^{2}u\left[1,1,1\right]-4{u\left[1,1,1\right]}^{2}u\left[1\right]+u\left[1\right]u\left[1,1,1,1\right]u\left[1,1\right]}{{u\left[1,1\right]}^5}\mathrm{Dx}\right)\right)$

$\mathrm{\ω3}≔\left(\frac{u_{1,1,1}}{u_{1,1}^3}-\frac{4u_1{u}_{1,1,1}^2}{u_{1,1}^5}+\frac{u_1{u}_{1,1,1,1}}{u_{1,1}^4}\right)\mathrm{Dx}$

$\mathrm{EulerLagrange}\left(\mathrm{\ω3}\right)$

$0\mathrm{Dx}\bigwedge {\mathrm{Cu}}_{\left[\right]}$

$\mathrm{eta3a}≔\mathrm{HorizontalHomotopy}\left(\mathrm{\ω3}\,\mathrm{integrationlimits}=\left[\mathrm{\infty }\,1\right]\right)$

$\mathrm{eta3a}≔\frac{u_{1,1,1}{u}_1}{u_{1,1}^4}$

$\mathrm{\ω3}\&minus\mathrm{HorizontalExteriorDerivative}\left(\mathrm{eta3a}\right)$

$0\mathrm{Dx}$

$\mathrm{opt}≔\mathrm{intmethod}=''ExteriorDerivativeHomotopy'',\mathrm{path}=''zigzag'',\mathrm{variableorder}=\left[x\,u\left[\right]\,u\left[1\right]\,u\left[1,1\right]\,u\left[1,1,1\right]\,u\left[1,1,1,1\right]\,u\left[1,1,1,1,1\right]\right],\mathrm{initialpoint}=\left[u\left[1,1\right]=1\right]$

$\mathrm{opt}≔\mathrm{intmethod}=''ExteriorDerivativeHomotopy'',\mathrm{path}=''zigzag'',\mathrm{variableorder}=\left[x\,u_{\left[\right]}\,u_1\,u_{1,1}\,u_{1,1,1}\,u_{1,1,1,1}\,u_{1,1,1,1,1}\right],\mathrm{initialpoint}=\left[u_{1,1}=1\right]$

$\mathrm{eta3b}≔\mathrm{HorizontalHomotopy}\left(\mathrm{\ω3}\,\mathrm{opt}\right)$

$\mathrm{eta3b}≔\frac{u_{1,1,1}{u}_1}{u_{1,1}^4}$

$\mathrm{DGsetup}\left(\left[x\,y\right]\,\left[u\,v\right]\,\mathrm{E2}\,2\right)\:$

$\mathrm{\ω4}≔\mathrm{evalDG}\left(\left(v\left[2,2\right]u\left[1,1\right]+u\left[1\right]v\left[1,2,2\right]\right)\mathrm{Dx}+\left(v\left[2,2\right]u\left[1,2\right]+u\left[1\right]v\left[2,2,2\right]\right)\mathrm{Dy}\right)$

$\mathrm{\ω4}≔\left(u_1{v}_{1,2,2}+v_{2,2}{u}_{1,1}\right)\mathrm{Dx}+\left(u_1{v}_{2,2,2}+v_{2,2}{u}_{1,2}\right)\mathrm{Dy}$

$\mathrm{HorizontalExteriorDerivative}\left(\mathrm{\ω4}\right)$

$0\mathrm{Dx}\bigwedge \mathrm{Dy}$

$\mathrm{\η4}≔\mathrm{HorizontalHomotopy}\left(\mathrm{\ω4}\right)$

$\mathrm{\η4}≔v_{2,2}{u}_1$

$\mathrm{\ω4}\&minus\mathrm{HorizontalExteriorDerivative}\left(\mathrm{\η4}\right)$

$0\mathrm{Dx}$

$\mathrm{\ω5}≔\mathrm{evalDG}\left(\left(v\left[2\right]u\left[1,1\right]+u\left[1\right]v\left[1,2\right]-v\left[1\right]u\left[1,2,2\right]-u\left[1,2\right]v\left[1,2\right]\right)\mathrm{Dx}\&w\mathrm{Dy}\right)$

$\mathrm{\ω5}≔\left(u_1{v}_{1,2}+v_2{u}_{1,1}-u_{1,2}{v}_{1,2}-v_1{u}_{1,2,2}\right)\mathrm{Dx}\bigwedge \mathrm{Dy}$

$\mathrm{EulerLagrange}\left(\mathrm{\ω5}\right)$

$0\mathrm{Dx}\bigwedge \mathrm{Dy}\bigwedge {\mathrm{Cu}}_{\left[\right]}$

$\mathrm{eta5a}≔\mathrm{HorizontalHomotopy}\left(\mathrm{\ω5}\right)$

$\mathrm{eta5a}≔\left(\frac{7}{12}{v}_1{u}_{1,2}-\frac{1}{4}{v}_1{u}_1-\frac{1}{6}{v}_{1,1}{u}_2+\frac{1}{12}{u}_1{v}_{1,2}-\frac{1}{4}{v}_{\left[\right]}{u}_{1,1}-\frac{1}{4}{v}_{\left[\right]}{u}_{1,1,2}+\frac{1}{12}{v}_{1,1,2}{u}_{\left[\right]}\right)\mathrm{Dx}+\left(-\frac{1}{6}{v}_1{u}_{2,2}+\frac{3}{4}{v}_2{u}_1-\frac{1}{4}{v}_2{u}_{1,2}-\frac{1}{12}{v}_{1,2}{u}_2-\frac{1}{4}{v}_{\left[\right]}{u}_{1,2}-\frac{1}{4}{v}_{\left[\right]}{u}_{1,2,2}+\frac{1}{12}{v}_{1,2,2}{u}_{\left[\right]}\right)\mathrm{Dy}$

$\mathrm{\ω5}\&minus\mathrm{HorizontalExteriorDerivative}\left(\mathrm{eta5a}\right)$

$0\mathrm{Dx}\bigwedge \mathrm{Dy}$

$\mathrm{opt}≔\mathrm{intmethod}=''ExteriorDerivativeHomotopy'',\mathrm{path}=''zigzag'',\mathrm{variableorder}=\mathrm{Tools}:-\mathrm{DGinfo}\left(\mathrm{E2}\,''FrameJetVariables''\right)$

$\mathrm{opt}≔\mathrm{intmethod}=''ExteriorDerivativeHomotopy'',\mathrm{path}=''zigzag'',\mathrm{variableorder}=\left[x\,y\,u_{\left[\right]}\,v_{\left[\right]}\,u_1\,u_2\,v_1\,v_2\,u_{1,1}\,u_{1,2}\,u_{2,2}\,v_{1,1}\,v_{1,2}\,v_{2,2}\,u_{1,1,1}\,u_{1,1,2}\,u_{1,2,2}\,u_{2,2,2}\,v_{1,1,1}\,v_{1,1,2}\,v_{1,2,2}\,v_{2,2,2}\right]$

$\mathrm{eta5b}≔\mathrm{HorizontalHomotopy}\left(\mathrm{\ω5}\,\mathrm{opt}\right)$

$\mathrm{eta5b}≔v_1{u}_{1,2}\mathrm{Dx}+v_2{u}_1\mathrm{Dy}$

$\mathrm{\ω5}\&minus\mathrm{HorizontalExteriorDerivative}\left(\mathrm{eta5b}\right)$

$0\mathrm{Dx}\bigwedge \mathrm{Dy}$