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(...).

with(DifferentialGeometry): with(JetCalculus):

DGsetup([x], [u], E, 3):

omega1 := evalDG((u[1,1,1]*u[1] + x*u[1,1,1]*u[1,1] + 2*u[1,1]*u[1,1,1] + x*u[1]*u[1,1,1,1])*Dx);

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

EulerLagrange(omega1);

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

eta1 := HorizontalHomotopy(omega1);

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

omega1 &minus HorizontalExteriorDerivative(eta1);

$0\mathrm{Dx}$

omega2 := evalDG(Dx &w Cu[1] &w Cu[1,1,1,1] + Dx &w Cu[1,1] &w Cu[1,1,1]);

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

IntegrationByParts(omega2);

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

eta2 := HorizontalHomotopy(omega2);

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

omega2 &minus HorizontalExteriorDerivative(eta2);

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

HorizontalExteriorDerivative((u[1]*u[1,1,1])/u[1,1]^4);

omega3 := map(expand, evalDG((u[1,1]^2*u[1,1,1] - 4*u[1,1,1]^2*u[1] + u[1]*u[1,1,1,1]*u[1,1])/u[1,1]^5*Dx));

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

EulerLagrange(omega3);

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

eta3a := HorizontalHomotopy(omega3, integrationlimits = [infinity, 1]);

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

omega3 &minus HorizontalExteriorDerivative(eta3a);

$0\mathrm{Dx}$

opt := intmethod = "ExteriorDerivativeHomotopy", path = "zigzag", variableorder = [x, u[], u[1], u[1,1], u[1,1,1], u[1,1,1,1], u[1,1,1,1,1]], initialpoint = [u[1,1] = 1];

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

eta3b := HorizontalHomotopy(omega3, opt);

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

DGsetup([x, y], [u, v], E2, 2):

omega4 := evalDG((v[2, 2]*u[1, 1] + u[1]*v[1, 2, 2])*Dx + (v[2, 2]*u[1, 2] + u[1]*v[2, 2, 2])*Dy);

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

HorizontalExteriorDerivative(omega4);

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

eta4 := HorizontalHomotopy(omega4);

$\mathrm{\η4}≔v_{2,2}{u}_1$

omega4 &minus HorizontalExteriorDerivative(eta4);

$0\mathrm{Dx}$

omega5 := evalDG((v[2]*u[1, 1] + u[1]*v[1, 2] - v[1]*u[1, 2, 2] - u[1, 2]*v[1, 2])* Dx &w Dy);

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

EulerLagrange(omega5);

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

eta5a := HorizontalHomotopy(omega5);

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

omega5 &minus HorizontalExteriorDerivative(eta5a);

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

opt := intmethod = "ExteriorDerivativeHomotopy", path = "zigzag", variableorder = Tools:-DGinfo(E2, "FrameJetVariables");

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

eta5b := HorizontalHomotopy(omega5, opt);

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

omega5 &minus HorizontalExteriorDerivative(eta5b);

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