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);

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

HorizontalExteriorDerivative(omega1);

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

theta1 := ZigZag(omega1);

$\mathrm{\θ1}≔\mathrm{\_DG}\left(\left[\left[''form''\,E\,1\right]\,\left[\left[\left[6\right]\,u_{1,2}\right]\,\left[\left[7\right]\,u_{1,1}\right]\right]\right]\right)$

ExteriorDerivative(theta1);

$\mathrm{\_DG}\left(\left[\left[''form''\,E\,2\right]\,\left[\left[\left[1\,2\right]\,0\right]\right]\right]\right)$

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

$\mathrm{\_DG}\left(\left[\left[''biform''\,E\,\left[1\,0\right]\right]\,\left[\left[\left[1\right]\,u_{1,1}{u}_{1,1,2}+u_{1,2}{u}_{1,1,1}\right]\,\left[\left[2\right]\,u_{1,1}{u}_{1,2,2}+u_{1,2}{u}_{1,1,2}\right]\right]\right]\right)$

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);

$\mathrm{\ω2}≔\mathrm{\_DG}\left(\left[\left[''biform''\,E\,\left[2\,0\right]\right]\,\left[\left[\left[1\,2\right]\,-u_1{u}_{1,2,2}-u_2{u}_{1,2,2}-u_{1,2}^2-u_{2,2}{u}_{1,2}\right]\right]\right]\right)$

EulerLagrange(omega2);

$\mathrm{\_DG}\left(\left[\left[''biform''\,E\,\left[2\,1\right]\right]\,\left[\left[\left[1\,2\,3\right]\,0\right]\right]\right]\right)$

theta2 := ZigZag(omega2);

$\mathrm{\θ2}≔\mathrm{\_DG}\left(\left[\left[''form''\,E\,2\right]\,\left[\left[\left[1\,4\right]\,-\frac{u_{1,2}}{3}-\frac{u_{2,2}}{6}\right]\,\left[\left[1\,5\right]\,-\frac{2u_{1,2}}{3}\right]\,\left[\left[1\,7\right]\,-\frac{u_1}{3}-\frac{2u_2}{3}\right]\,\left[\left[1\,8\right]\,-\frac{u_1}{6}\right]\,\left[\left[2\,4\right]\,\frac{u_{2,2}}{3}\right]\,\left[\left[2\,5\right]\,\frac{u_{1,2}}{3}+\frac{u_{2,2}}{6}\right]\,\left[\left[2\,7\right]\,\frac{u_2}{3}\right]\,\left[\left[2\,8\right]\,\frac{u_2}{6}+\frac{u_1}{3}\right]\,\left[\left[3\,7\right]\,-\frac{1}{3}\right]\,\left[\left[3\,8\right]\,\frac{1}{6}\right]\,\left[\left[4\,5\right]\,\frac{1}{3}\right]\right]\right]\right)$

ExteriorDerivative(theta2);

$\mathrm{\_DG}\left(\left[\left[''form''\,E\,3\right]\,\left[\left[\left[1\,2\,3\right]\,0\right]\right]\right]\right)$

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

$\mathrm{\_DG}\left(\left[\left[''biform''\,E\,\left[2\,0\right]\right]\,\left[\left[\left[1\,2\right]\,-u_1{u}_{1,2,2}-u_2{u}_{1,2,2}-u_{1,2}^2-u_{2,2}{u}_{1,2}\right]\right]\right]\right)$

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

$\mathrm{\ω3}≔\mathrm{\_DG}\left(\left[\left[''biform''\,E\,\left[2\,1\right]\right]\,\left[\left[\left[1\,2\,3\right]\,-2u_1{u}_{2,2}-4u_2{u}_{1,2}\right]\right]\right]\right)$

IntegrationByParts(VerticalExteriorDerivative(omega3));

$\mathrm{\_DG}\left(\left[\left[''biform''\,E\,\left[2\,2\right]\right]\,\left[\left[\left[1\,2\,3\,4\right]\,0\right]\right]\right]\right)$

theta3 := ZigZag(omega3);

$\mathrm{\θ3}≔\mathrm{\_DG}\left(\left[\left[''form''\,E\,3\right]\,\left[\left[\left[1\,2\,4\right]\,-2u_2^2\right]\,\left[\left[1\,2\,5\right]\,-4u_1{u}_2\right]\,\left[\left[1\,3\,4\right]\,2u_2\right]\,\left[\left[1\,3\,5\right]\,2u_1\right]\,\left[\left[2\,3\,5\right]\,-2u_2\right]\right]\right]\right)$

ExteriorDerivative(theta3);

$\mathrm{\_DG}\left(\left[\left[''form''\,E\,4\right]\,\left[\left[\left[1\,2\,3\,4\right]\,0\right]\right]\right]\right)$

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

$\mathrm{\_DG}\left(\left[\left[''biform''\,E\,\left[2\,1\right]\right]\,\left[\left[\left[1\,2\,3\right]\,-2u_1{u}_{2,2}-4u_2{u}_{1,2}\right]\right]\right]\right)$