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 on $J^{\mathrm{\∞}}\left(E\right)$decomposes as a direct sum of bi-forms

If $\mathrm{\ω}$ is a bi-form of degree $\left(r\,s\right)\,$then ZigZag($\mathbf{\omega }$) returns the differential form $\mathrm{\θ}$of degree $r\+s$.

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

with(DifferentialGeometry): with(JetCalculus):

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

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

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

HorizontalExteriorDerivative(omega1);

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

theta1 := ZigZag(omega1);

$\mathrm{\θ1}≔u_{1,2}{\mathrm{du}}_{1,1}+u_{1,1}{\mathrm{du}}_{1,2}$

ExteriorDerivative(theta1);

$0\mathrm{dx}\bigwedge \mathrm{dy}$

convert(theta1, DGbiform, [1, 0]);

$\left(u_{1,1}{u}_{1,1,2}+u_{1,2}{u}_{1,1,1}\right)\mathrm{Dx}+\left(u_{1,1}{u}_{1,2,2}+u_{1,2}{u}_{1,1,2}\right)\mathrm{Dy}$

omega2 := evalDG((- u[2, 2]*u[1, 2] - u[2]*u[1, 2, 2] - u[1]*u[1, 2, 2] - u[1, 2]^2)*Dx &w Dy);

$\mathrm{\ω2}≔-\left(u_1{u}_{1,2,2}+u_2{u}_{1,2,2}+u_{1,2}^2+u_{2,2}{u}_{1,2}\right)\mathrm{Dx}\bigwedge \mathrm{Dy}$

EulerLagrange(omega2);

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

theta2 := ZigZag(omega2);

$\mathrm{\θ2}≔-\left(\frac{u_{1,2}}{3}+\frac{u_{2,2}}{6}\right)\mathrm{dx}\bigwedge {\mathrm{du}}_1-\frac{2u_{1,2}}{3}\mathrm{dx}\bigwedge {\mathrm{du}}_2-\left(\frac{u_1}{3}+\frac{2u_2}{3}\right)\mathrm{dx}\bigwedge {\mathrm{du}}_{1,2}-\frac{u_1}{6}\mathrm{dx}\bigwedge {\mathrm{du}}_{2,2}+\frac{u_{2,2}}{3}\mathrm{dy}\bigwedge {\mathrm{du}}_1+\left(\frac{u_{1,2}}{3}+\frac{u_{2,2}}{6}\right)\mathrm{dy}\bigwedge {\mathrm{du}}_2+\frac{u_2}{3}\mathrm{dy}\bigwedge {\mathrm{du}}_{1,2}+\left(\frac{u_2}{6}+\frac{u_1}{3}\right)\mathrm{dy}\bigwedge {\mathrm{du}}_{2,2}-\frac{1}{3}{\mathrm{du}}_{\left[\right]}\bigwedge {\mathrm{du}}_{1,2}+\frac{1}{6}{\mathrm{du}}_{\left[\right]}\bigwedge {\mathrm{du}}_{2,2}+\frac{1}{3}{\mathrm{du}}_1\bigwedge {\mathrm{du}}_2$

ExteriorDerivative(theta2);

$0\mathrm{dx}\bigwedge \mathrm{dy}\bigwedge {\mathrm{du}}_{\left[\right]}$

convert(theta2, DGbiform, [2, 0]);

$-\left(u_1{u}_{1,2,2}+u_2{u}_{1,2,2}+u_{1,2}^2+u_{2,2}{u}_{1,2}\right)\mathrm{Dx}\bigwedge \mathrm{Dy}$

omega3 := EulerLagrange(u[1]*u[2]^2*Dx &w Dy);

$\mathrm{\ω3}≔-\left(2u_1{u}_{2,2}+4u_2{u}_{1,2}\right)\mathrm{Dx}\bigwedge \mathrm{Dy}\bigwedge {\mathrm{Cu}}_{\left[\right]}$

IntegrationByParts(VerticalExteriorDerivative(omega3));

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

theta3 := ZigZag(omega3);

$\mathrm{\θ3}≔-2u_2^2\mathrm{dx}\bigwedge \mathrm{dy}\bigwedge {\mathrm{du}}_1-4u_1{u}_2\mathrm{dx}\bigwedge \mathrm{dy}\bigwedge {\mathrm{du}}_2+2u_2\mathrm{dx}\bigwedge {\mathrm{du}}_{\left[\right]}\bigwedge {\mathrm{du}}_1+2u_1\mathrm{dx}\bigwedge {\mathrm{du}}_{\left[\right]}\bigwedge {\mathrm{du}}_2-2u_2\mathrm{dy}\bigwedge {\mathrm{du}}_{\left[\right]}\bigwedge {\mathrm{du}}_2$

ExteriorDerivative(theta3);

$0\mathrm{dx}\bigwedge \mathrm{dy}\bigwedge {\mathrm{du}}_{\left[\right]}\bigwedge {\mathrm{du}}_1$

convert(theta3, DGbiform, [2, 1]);

$-\left(2u_1{u}_{2,2}+4u_2{u}_{1,2}\right)\mathrm{Dx}\bigwedge \mathrm{Dy}\bigwedge {\mathrm{Cu}}_{\left[\right]}$