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JetCalculus[Noether] - find the conservation law for the Euler-Lagrange equations from a given symmetry of the Lagrangian

Calling Sequences

Noether(X, l)

Parameters

X         -  a vector field representing a symmetry of the Lagrangian $\mathrm{λ}$

λ         - a Lagrangian for a variational principle, defined as a top degree horizontal form of the jet space of a bundle $\mathrm{π}:E\to M$

Description

 • The celebrated theorem of E. Noether provides a formula for the calculation of a first integral or conservation law for any symmetry of the Lagrangian. This formula, which is very complicated for high order Lagrangians is easily implemented using the horizontal homotopy for the variational bicomplex.
 • Within the framework of the JetCalculus package conservation laws are represented by differential forms of degree $n-1$, where $n$ is the dimension of $M$, whose horizontal or total exterior derivative vanishes by virtue of the Euler Lagrange equations.
 • The vector field $X$ is a symmetry of the Lagrangian $\mathrm{λ}$ if the Lie derivative of l with respect to the prolongation of $X$ (to the order of λ) vanishes.
 • The command Noether is part of the DifferentialGeometry:-JetCalculus package. It can be used in the form Noether(...) only after executing the commands with(DifferentialGeometry) and with(JetCalculus), but can always be used by executing DifferentialGeometry:-JetCalculus:-Noether(...).

Examples

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

Example 1.

We set a simple single integral problem with 2 dependent variables and compute the Euler-Lagrange equations

 > $\mathrm{DGsetup}\left(\left[t\right],\left[x,y\right],E,2\right):$
 E > $L≔\frac{1\left({x}_{1}^{2}+{y}_{1}^{2}-\frac{1}{\sqrt{{x}_{[]}^{2}+{y}_{[]}^{2}}}\right)}{2}$
 ${L}{≔}\frac{{{x}}_{{1}}^{{2}}}{{2}}{+}\frac{{{y}}_{{1}}^{{2}}}{{2}}{-}\frac{{1}}{{2}{}\sqrt{{{x}}_{\left[\right]}^{{2}}{+}{{y}}_{\left[\right]}^{{2}}}}$ (2.1)
 E > $\mathrm{EL}≔\mathrm{EulerLagrange}\left(L\right)$
 ${\mathrm{EL}}{≔}\left[\frac{{{x}}_{\left[\right]}}{{2}{}{\left({{x}}_{\left[\right]}^{{2}}{+}{{y}}_{\left[\right]}^{{2}}\right)}^{{3}}{{2}}}}{-}{{x}}_{{1}{,}{1}}{,}\frac{{{y}}_{\left[\right]}}{{2}{}{\left({{x}}_{\left[\right]}^{{2}}{+}{{y}}_{\left[\right]}^{{2}}\right)}^{{3}}{{2}}}}{-}{{y}}_{{1}{,}{1}}\right]$ (2.2)
 E > $\mathrm{Eq}≔\mathrm{solve}\left(\mathrm{EL},\left\{{x}_{1,1},{y}_{1,1}\right\}\right)$
 ${\mathrm{Eq}}{≔}\left\{{{x}}_{{1}{,}{1}}{=}\frac{\sqrt{{{x}}_{\left[\right]}^{{2}}{+}{{y}}_{\left[\right]}^{{2}}}{}{{x}}_{\left[\right]}}{{2}{}\left({{x}}_{\left[\right]}^{{4}}{+}{2}{}{{x}}_{\left[\right]}^{{2}}{}{{y}}_{\left[\right]}^{{2}}{+}{{y}}_{\left[\right]}^{{4}}\right)}{,}{{y}}_{{1}{,}{1}}{=}\frac{\sqrt{{{x}}_{\left[\right]}^{{2}}{+}{{y}}_{\left[\right]}^{{2}}}{}{{y}}_{\left[\right]}}{{2}{}\left({{x}}_{\left[\right]}^{{4}}{+}{2}{}{{x}}_{\left[\right]}^{{2}}{}{{y}}_{\left[\right]}^{{2}}{+}{{y}}_{\left[\right]}^{{4}}\right)}\right\}$ (2.3)

The Lagrangian $L$ is invariant under rotations in the $x-y$ plane. Let us check this. To be technically correct we should work with the differential 1-form $\mathrm{λ}$ defined by $L$.

 E > $\mathrm{λ}≔L&mult\mathrm{Dt}$
 ${\mathrm{\lambda }}{≔}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{1}{,}{0}\right]\right]{,}\left[\left[\left[{1}\right]{,}\frac{{{x}}_{{1}}^{{2}}{}\sqrt{{{x}}_{\left[\right]}^{{2}}{+}{{y}}_{\left[\right]}^{{2}}}{+}{{y}}_{{1}}^{{2}}{}\sqrt{{{x}}_{\left[\right]}^{{2}}{+}{{y}}_{\left[\right]}^{{2}}}{-}{1}}{{2}{}\sqrt{{{x}}_{\left[\right]}^{{2}}{+}{{y}}_{\left[\right]}^{{2}}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{1}{,}{0}\right]\right]{,}\left[\left[\left[{1}\right]{,}\frac{{{x}}_{{1}}^{{2}}{}\sqrt{{{x}}_{\left[\right]}^{{2}}{+}{{y}}_{\left[\right]}^{{2}}}{+}{{y}}_{{1}}^{{2}}{}\sqrt{{{x}}_{\left[\right]}^{{2}}{+}{{y}}_{\left[\right]}^{{2}}}{-}{1}}{{2}{}\sqrt{{{x}}_{\left[\right]}^{{2}}{+}{{y}}_{\left[\right]}^{{2}}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{1}{,}{0}\right]\right]{,}\left[\left[\left[{1}\right]{,}\frac{{{x}}_{{1}}^{{2}}{}\sqrt{{{x}}_{\left[\right]}^{{2}}{+}{{y}}_{\left[\right]}^{{2}}}{+}{{y}}_{{1}}^{{2}}{}\sqrt{{{x}}_{\left[\right]}^{{2}}{+}{{y}}_{\left[\right]}^{{2}}}{-}{1}}{{2}{}\sqrt{{{x}}_{\left[\right]}^{{2}}{+}{{y}}_{\left[\right]}^{{2}}}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{1}{,}{0}\right]\right]{,}\left[\left[\left[{1}\right]{,}\frac{{{x}}_{{1}}^{{2}}{}\sqrt{{{x}}_{\left[\right]}^{{2}}{+}{{y}}_{\left[\right]}^{{2}}}{+}{{y}}_{{1}}^{{2}}{}\sqrt{{{x}}_{\left[\right]}^{{2}}{+}{{y}}_{\left[\right]}^{{2}}}{-}{1}}{{2}{}\sqrt{{{x}}_{\left[\right]}^{{2}}{+}{{y}}_{\left[\right]}^{{2}}}}\right]\right]\right]\right)$ (2.4)
 E > $X≔{x}_{[]}{\mathrm{D_y}}_{[]}-{y}_{[]}{\mathrm{D_x}}_{[]}$
 ${X}{≔}{-}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{E}{,}\left[\right]\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{E}{,}\left[\right]\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{E}{,}\left[\right]\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{E}{,}\left[\right]\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){}{{y}}_{\left[\right]}{+}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{E}{,}\left[\right]\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{E}{,}\left[\right]\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{E}{,}\left[\right]\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{E}{,}\left[\right]\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right){}{{x}}_{\left[\right]}$ (2.5)
 E > $\mathrm{X1}≔\mathrm{Prolong}\left(X,1\right)$
 ${\mathrm{X1}}{≔}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{E}{,}\left[{"projectable"}{,}{1}\right]\right]{,}\left[\left[\left[{2}\right]{,}{-}{{y}}_{\left[\right]}\right]{,}\left[\left[{3}\right]{,}{{x}}_{\left[\right]}\right]{,}\left[\left[{4}\right]{,}{-}{{y}}_{{1}}\right]{,}\left[\left[{5}\right]{,}{{x}}_{{1}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{E}{,}\left[{"projectable"}{,}{1}\right]\right]{,}\left[\left[\left[{2}\right]{,}{-}{{y}}_{\left[\right]}\right]{,}\left[\left[{3}\right]{,}{{x}}_{\left[\right]}\right]{,}\left[\left[{4}\right]{,}{-}{{y}}_{{1}}\right]{,}\left[\left[{5}\right]{,}{{x}}_{{1}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{E}{,}\left[{"projectable"}{,}{1}\right]\right]{,}\left[\left[\left[{2}\right]{,}{-}{{y}}_{\left[\right]}\right]{,}\left[\left[{3}\right]{,}{{x}}_{\left[\right]}\right]{,}\left[\left[{4}\right]{,}{-}{{y}}_{{1}}\right]{,}\left[\left[{5}\right]{,}{{x}}_{{1}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{E}{,}\left[{"projectable"}{,}{1}\right]\right]{,}\left[\left[\left[{2}\right]{,}{-}{{y}}_{\left[\right]}\right]{,}\left[\left[{3}\right]{,}{{x}}_{\left[\right]}\right]{,}\left[\left[{4}\right]{,}{-}{{y}}_{{1}}\right]{,}\left[\left[{5}\right]{,}{{x}}_{{1}}\right]\right]\right]\right)$ (2.6)
 E > $\mathrm{LieDerivative}\left(X,\mathrm{λ}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{1}{,}{0}\right]\right]{,}\left[\left[\left[{1}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{1}{,}{0}\right]\right]{,}\left[\left[\left[{1}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{1}{,}{0}\right]\right]{,}\left[\left[\left[{1}\right]{,}{0}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{E}{,}\left[{1}{,}{0}\right]\right]{,}\left[\left[\left[{1}\right]{,}{0}\right]\right]\right]\right)$ (2.7)

Now we find the first integral associated to the symmetry $X$:

 E > $F≔\mathrm{Noether}\left(X,\mathrm{λ}\right)$
 ${F}{≔}{-}{{y}}_{{1}}{}{{x}}_{\left[\right]}{+}{{x}}_{{1}}{}{{y}}_{\left[\right]}$ (2.8)

To check that this is indeed a first integral, take the total derivative of $F$ with respect to $t$ and substitute from the Euler-Lagrange equations.

 E > $\mathrm{tdF}≔\mathrm{TotalDiff}\left(F,t\right)$
 ${\mathrm{tdF}}{≔}{-}{{x}}_{\left[\right]}{}{{y}}_{{1}{,}{1}}{+}{{y}}_{\left[\right]}{}{{x}}_{{1}{,}{1}}$ (2.9)
 E > $\mathrm{eval}\left(\mathrm{tdF},\mathrm{Eq}\right)$
 ${0}$ (2.10)

Example 2.

We use the command InfinitesimalSymmetriesOfGeometricObjectFields to find the symmetries of the Lagrangian for the wave equation in (2+1) dimensions.

We then use the command Noether to calculate the associated conservation laws.

 E > $\mathrm{with}\left(\mathrm{GroupActions}\right):$
 E > $\mathrm{DGsetup}\left(\left[x,y,t\right],\left[u\right],J,1\right)$
 ${\mathrm{frame name: J}}$ (2.11)
 J > $\mathrm{λ}≔\mathrm{evalDG}\left(\left({u}_{1}^{2}+{u}_{2}^{2}-{u}_{3}^{2}\right)\left(\mathrm{Dx}&w\mathrm{Dy}\right)&w\mathrm{Dt}\right)$
 ${\mathrm{\lambda }}{≔}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{J}{,}\left[{3}{,}{0}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{{u}}_{{1}}^{{2}}{+}{{u}}_{{2}}^{{2}}{-}{{u}}_{{3}}^{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{J}{,}\left[{3}{,}{0}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{{u}}_{{1}}^{{2}}{+}{{u}}_{{2}}^{{2}}{-}{{u}}_{{3}}^{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{J}{,}\left[{3}{,}{0}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{{u}}_{{1}}^{{2}}{+}{{u}}_{{2}}^{{2}}{-}{{u}}_{{3}}^{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{J}{,}\left[{3}{,}{0}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{{u}}_{{1}}^{{2}}{+}{{u}}_{{2}}^{{2}}{-}{{u}}_{{3}}^{{2}}\right]\right]\right]\right)$ (2.12)
 J > $\mathrm{Gamma}≔\mathrm{InfinitesimalSymmetriesOfGeometricObjectFields}\left(\left[\mathrm{λ}\right],\mathrm{output}="list"\right)$
 ${\mathrm{Γ}}{≔}\left[{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{-}{2}{}{x}\right]{,}\left[\left[{2}\right]{,}{-}{2}{}{y}\right]{,}\left[\left[{3}\right]{,}{-}{2}{}{t}\right]{,}\left[\left[{4}\right]{,}{{u}}_{\left[\right]}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{-}{2}{}{x}\right]{,}\left[\left[{2}\right]{,}{-}{2}{}{y}\right]{,}\left[\left[{3}\right]{,}{-}{2}{}{t}\right]{,}\left[\left[{4}\right]{,}{{u}}_{\left[\right]}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{-}{2}{}{x}\right]{,}\left[\left[{2}\right]{,}{-}{2}{}{y}\right]{,}\left[\left[{3}\right]{,}{-}{2}{}{t}\right]{,}\left[\left[{4}\right]{,}{{u}}_{\left[\right]}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{-}{2}{}{x}\right]{,}\left[\left[{2}\right]{,}{-}{2}{}{y}\right]{,}\left[\left[{3}\right]{,}{-}{2}{}{t}\right]{,}\left[\left[{4}\right]{,}{{u}}_{\left[\right]}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{t}\right]{,}\left[\left[{3}\right]{,}{x}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{t}\right]{,}\left[\left[{3}\right]{,}{x}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{t}\right]{,}\left[\left[{3}\right]{,}{x}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{t}\right]{,}\left[\left[{3}\right]{,}{x}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{2}\right]{,}{t}\right]{,}\left[\left[{3}\right]{,}{y}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{2}\right]{,}{t}\right]{,}\left[\left[{3}\right]{,}{y}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{2}\right]{,}{t}\right]{,}\left[\left[{3}\right]{,}{y}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{2}\right]{,}{t}\right]{,}\left[\left[{3}\right]{,}{y}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{3}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{-}{y}\right]{,}\left[\left[{2}\right]{,}{x}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{-}{y}\right]{,}\left[\left[{2}\right]{,}{x}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{-}{y}\right]{,}\left[\left[{2}\right]{,}{x}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{-}{y}\right]{,}\left[\left[{2}\right]{,}{x}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{2}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{1}\right]\right]\right]\right)\right]$ (2.13)

Let us find the conservation law $\mathrm{ω1}$ associated to the symmetry of infinitesimal translations in the dependent variable $u$. We check the horizontal exterior derivative of $\mathrm{ω1}$ vanishes on solutions to the 2+1 wave equation.

 J > $\mathrm{X1}≔{\mathrm{D_u}}_{[]}$
 ${\mathrm{X1}}{≔}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{4}\right]{,}{1}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{4}\right]{,}{1}\right]\right]\right]\right)$ (2.14)
 J > $\mathrm{ω1}≔\mathrm{Noether}\left(\mathrm{X1},\mathrm{λ}\right)$
 ${\mathrm{ω1}}{≔}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{J}{,}\left[{2}{,}{0}\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{2}{}{{u}}_{{3}}\right]{,}\left[\left[{1}{,}{3}\right]{,}{2}{}{{u}}_{{2}}\right]{,}\left[\left[{2}{,}{3}\right]{,}{-}{2}{}{{u}}_{{1}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{J}{,}\left[{2}{,}{0}\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{2}{}{{u}}_{{3}}\right]{,}\left[\left[{1}{,}{3}\right]{,}{2}{}{{u}}_{{2}}\right]{,}\left[\left[{2}{,}{3}\right]{,}{-}{2}{}{{u}}_{{1}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{J}{,}\left[{2}{,}{0}\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{2}{}{{u}}_{{3}}\right]{,}\left[\left[{1}{,}{3}\right]{,}{2}{}{{u}}_{{2}}\right]{,}\left[\left[{2}{,}{3}\right]{,}{-}{2}{}{{u}}_{{1}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{J}{,}\left[{2}{,}{0}\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{2}{}{{u}}_{{3}}\right]{,}\left[\left[{1}{,}{3}\right]{,}{2}{}{{u}}_{{2}}\right]{,}\left[\left[{2}{,}{3}\right]{,}{-}{2}{}{{u}}_{{1}}\right]\right]\right]\right)$ (2.15)
 J > $\mathrm{HorizontalExteriorDerivative}\left(\mathrm{ω1}\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{J}{,}\left[{3}{,}{0}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{-}{2}{}{{u}}_{{1}{,}{1}}{-}{2}{}{{u}}_{{2}{,}{2}}{+}{2}{}{{u}}_{{3}{,}{3}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{J}{,}\left[{3}{,}{0}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{-}{2}{}{{u}}_{{1}{,}{1}}{-}{2}{}{{u}}_{{2}{,}{2}}{+}{2}{}{{u}}_{{3}{,}{3}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{J}{,}\left[{3}{,}{0}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{-}{2}{}{{u}}_{{1}{,}{1}}{-}{2}{}{{u}}_{{2}{,}{2}}{+}{2}{}{{u}}_{{3}{,}{3}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{J}{,}\left[{3}{,}{0}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{-}{2}{}{{u}}_{{1}{,}{1}}{-}{2}{}{{u}}_{{2}{,}{2}}{+}{2}{}{{u}}_{{3}{,}{3}}\right]\right]\right]\right)$ (2.16)

Let us find the conservation law $\mathrm{ω2}$ associated to the symmetry of infinitesimal scaling of the independent and dependent variables. We check the horizontal exterior derivative of $\mathrm{ω2}$ vanishes on solutions to the 2+1 wave equation.

 J > $\mathrm{X2}≔\mathrm{evalDG}\left(x\mathrm{D_x}+y\mathrm{D_y}+t\mathrm{D_t}-\frac{1{u}_{[]}{\mathrm{D_u}}_{[]}}{2}\right)$
 ${\mathrm{X2}}{≔}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{x}\right]{,}\left[\left[{2}\right]{,}{y}\right]{,}\left[\left[{3}\right]{,}{t}\right]{,}\left[\left[{4}\right]{,}{-}\frac{{{u}}_{\left[\right]}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{x}\right]{,}\left[\left[{2}\right]{,}{y}\right]{,}\left[\left[{3}\right]{,}{t}\right]{,}\left[\left[{4}\right]{,}{-}\frac{{{u}}_{\left[\right]}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{x}\right]{,}\left[\left[{2}\right]{,}{y}\right]{,}\left[\left[{3}\right]{,}{t}\right]{,}\left[\left[{4}\right]{,}{-}\frac{{{u}}_{\left[\right]}}{{2}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{x}\right]{,}\left[\left[{2}\right]{,}{y}\right]{,}\left[\left[{3}\right]{,}{t}\right]{,}\left[\left[{4}\right]{,}{-}\frac{{{u}}_{\left[\right]}}{{2}}\right]\right]\right]\right)$ (2.17)
 J > $\mathrm{ω2}≔\mathrm{Noether}\left(\mathrm{X2},\mathrm{λ}\right)$
 J > $\mathrm{HorizontalExteriorDerivative}\left(\mathrm{ω1}\right)$
 ${\mathrm{ω2}}{≔}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{J}{,}\left[{2}{,}{0}\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{-}{t}{}{{u}}_{{1}}^{{2}}{-}{t}{}{{u}}_{{2}}^{{2}}{-}{t}{}{{u}}_{{3}}^{{2}}{-}{2}{}{x}{}{{u}}_{{1}}{}{{u}}_{{3}}{-}{2}{}{y}{}{{u}}_{{2}}{}{{u}}_{{3}}{-}{{u}}_{\left[\right]}{}{{u}}_{{3}}\right]{,}\left[\left[{1}{,}{3}\right]{,}{-}{2}{}{t}{}{{u}}_{{2}}{}{{u}}_{{3}}{-}{2}{}{x}{}{{u}}_{{1}}{}{{u}}_{{2}}{+}{y}{}{{u}}_{{1}}^{{2}}{-}{y}{}{{u}}_{{2}}^{{2}}{-}{y}{}{{u}}_{{3}}^{{2}}{-}{{u}}_{\left[\right]}{}{{u}}_{{2}}\right]{,}\left[\left[{2}{,}{3}\right]{,}{2}{}{t}{}{{u}}_{{1}}{}{{u}}_{{3}}{+}{x}{}{{u}}_{{1}}^{{2}}{-}{x}{}{{u}}_{{2}}^{{2}}{+}{x}{}{{u}}_{{3}}^{{2}}{+}{2}{}{y}{}{{u}}_{{1}}{}{{u}}_{{2}}{+}{{u}}_{\left[\right]}{}{{u}}_{{1}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{J}{,}\left[{2}{,}{0}\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{-}{t}{}{{u}}_{{1}}^{{2}}{-}{t}{}{{u}}_{{2}}^{{2}}{-}{t}{}{{u}}_{{3}}^{{2}}{-}{2}{}{x}{}{{u}}_{{1}}{}{{u}}_{{3}}{-}{2}{}{y}{}{{u}}_{{2}}{}{{u}}_{{3}}{-}{{u}}_{\left[\right]}{}{{u}}_{{3}}\right]{,}\left[\left[{1}{,}{3}\right]{,}{-}{2}{}{t}{}{{u}}_{{2}}{}{{u}}_{{3}}{-}{2}{}{x}{}{{u}}_{{1}}{}{{u}}_{{2}}{+}{y}{}{{u}}_{{1}}^{{2}}{-}{y}{}{{u}}_{{2}}^{{2}}{-}{y}{}{{u}}_{{3}}^{{2}}{-}{{u}}_{\left[\right]}{}{{u}}_{{2}}\right]{,}\left[\left[{2}{,}{3}\right]{,}{2}{}{t}{}{{u}}_{{1}}{}{{u}}_{{3}}{+}{x}{}{{u}}_{{1}}^{{2}}{-}{x}{}{{u}}_{{2}}^{{2}}{+}{x}{}{{u}}_{{3}}^{{2}}{+}{2}{}{y}{}{{u}}_{{1}}{}{{u}}_{{2}}{+}{{u}}_{\left[\right]}{}{{u}}_{{1}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{J}{,}\left[{2}{,}{0}\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{-}{t}{}{{u}}_{{1}}^{{2}}{-}{t}{}{{u}}_{{2}}^{{2}}{-}{t}{}{{u}}_{{3}}^{{2}}{-}{2}{}{x}{}{{u}}_{{1}}{}{{u}}_{{3}}{-}{2}{}{y}{}{{u}}_{{2}}{}{{u}}_{{3}}{-}{{u}}_{\left[\right]}{}{{u}}_{{3}}\right]{,}\left[\left[{1}{,}{3}\right]{,}{-}{2}{}{t}{}{{u}}_{{2}}{}{{u}}_{{3}}{-}{2}{}{x}{}{{u}}_{{1}}{}{{u}}_{{2}}{+}{y}{}{{u}}_{{1}}^{{2}}{-}{y}{}{{u}}_{{2}}^{{2}}{-}{y}{}{{u}}_{{3}}^{{2}}{-}{{u}}_{\left[\right]}{}{{u}}_{{2}}\right]{,}\left[\left[{2}{,}{3}\right]{,}{2}{}{t}{}{{u}}_{{1}}{}{{u}}_{{3}}{+}{x}{}{{u}}_{{1}}^{{2}}{-}{x}{}{{u}}_{{2}}^{{2}}{+}{x}{}{{u}}_{{3}}^{{2}}{+}{2}{}{y}{}{{u}}_{{1}}{}{{u}}_{{2}}{+}{{u}}_{\left[\right]}{}{{u}}_{{1}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{J}{,}\left[{2}{,}{0}\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{-}{t}{}{{u}}_{{1}}^{{2}}{-}{t}{}{{u}}_{{2}}^{{2}}{-}{t}{}{{u}}_{{3}}^{{2}}{-}{2}{}{x}{}{{u}}_{{1}}{}{{u}}_{{3}}{-}{2}{}{y}{}{{u}}_{{2}}{}{{u}}_{{3}}{-}{{u}}_{\left[\right]}{}{{u}}_{{3}}\right]{,}\left[\left[{1}{,}{3}\right]{,}{-}{2}{}{t}{}{{u}}_{{2}}{}{{u}}_{{3}}{-}{2}{}{x}{}{{u}}_{{1}}{}{{u}}_{{2}}{+}{y}{}{{u}}_{{1}}^{{2}}{-}{y}{}{{u}}_{{2}}^{{2}}{-}{y}{}{{u}}_{{3}}^{{2}}{-}{{u}}_{\left[\right]}{}{{u}}_{{2}}\right]{,}\left[\left[{2}{,}{3}\right]{,}{2}{}{t}{}{{u}}_{{1}}{}{{u}}_{{3}}{+}{x}{}{{u}}_{{1}}^{{2}}{-}{x}{}{{u}}_{{2}}^{{2}}{+}{x}{}{{u}}_{{3}}^{{2}}{+}{2}{}{y}{}{{u}}_{{1}}{}{{u}}_{{2}}{+}{{u}}_{\left[\right]}{}{{u}}_{{1}}\right]\right]\right]\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{J}{,}\left[{3}{,}{0}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{-}{2}{}{{u}}_{{1}{,}{1}}{-}{2}{}{{u}}_{{2}{,}{2}}{+}{2}{}{{u}}_{{3}{,}{3}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{J}{,}\left[{3}{,}{0}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{-}{2}{}{{u}}_{{1}{,}{1}}{-}{2}{}{{u}}_{{2}{,}{2}}{+}{2}{}{{u}}_{{3}{,}{3}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{J}{,}\left[{3}{,}{0}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{-}{2}{}{{u}}_{{1}{,}{1}}{-}{2}{}{{u}}_{{2}{,}{2}}{+}{2}{}{{u}}_{{3}{,}{3}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{J}{,}\left[{3}{,}{0}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{-}{2}{}{{u}}_{{1}{,}{1}}{-}{2}{}{{u}}_{{2}{,}{2}}{+}{2}{}{{u}}_{{3}{,}{3}}\right]\right]\right]\right)$ (2.18)
 J > $\mathrm{factor}\left(\mathrm{HorizontalExteriorDerivative}\left(\mathrm{ω2}\right)\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{J}{,}\left[{3}{,}{0}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}\left({{u}}_{{1}{,}{1}}{+}{{u}}_{{2}{,}{2}}{-}{{u}}_{{3}{,}{3}}\right){}\left({2}{}{{u}}_{{3}}{}{t}{+}{2}{}{{u}}_{{1}}{}{x}{+}{2}{}{{u}}_{{2}}{}{y}{+}{{u}}_{\left[\right]}\right)\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{J}{,}\left[{3}{,}{0}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}\left({{u}}_{{1}{,}{1}}{+}{{u}}_{{2}{,}{2}}{-}{{u}}_{{3}{,}{3}}\right){}\left({2}{}{{u}}_{{3}}{}{t}{+}{2}{}{{u}}_{{1}}{}{x}{+}{2}{}{{u}}_{{2}}{}{y}{+}{{u}}_{\left[\right]}\right)\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{J}{,}\left[{3}{,}{0}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}\left({{u}}_{{1}{,}{1}}{+}{{u}}_{{2}{,}{2}}{-}{{u}}_{{3}{,}{3}}\right){}\left({2}{}{{u}}_{{3}}{}{t}{+}{2}{}{{u}}_{{1}}{}{x}{+}{2}{}{{u}}_{{2}}{}{y}{+}{{u}}_{\left[\right]}\right)\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{J}{,}\left[{3}{,}{0}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}\left({{u}}_{{1}{,}{1}}{+}{{u}}_{{2}{,}{2}}{-}{{u}}_{{3}{,}{3}}\right){}\left({2}{}{{u}}_{{3}}{}{t}{+}{2}{}{{u}}_{{1}}{}{x}{+}{2}{}{{u}}_{{2}}{}{y}{+}{{u}}_{\left[\right]}\right)\right]\right]\right]\right)$ (2.19)

Finally, let us find the conservation law $\mathrm{ω3}$ associated to the symmetry of infinitesimal boosts in the independent variables $x$ and $t$. We check that the horizontal exterior derivative of $\mathrm{ω3}$ vanishes on solutions to the 2+1 wave equation.

 J > $\mathrm{X3}≔\mathrm{evalDG}\left(x\mathrm{D_t}+t\mathrm{D_x}\right)$
 ${\mathrm{X3}}{≔}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{t}\right]{,}\left[\left[{3}\right]{,}{x}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{t}\right]{,}\left[\left[{3}\right]{,}{x}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{t}\right]{,}\left[\left[{3}\right]{,}{x}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"vector"}{,}{J}{,}\left[\right]\right]{,}\left[\left[\left[{1}\right]{,}{t}\right]{,}\left[\left[{3}\right]{,}{x}\right]\right]\right]\right)$ (2.20)
 J > $\mathrm{ω3}≔\mathrm{Noether}\left(\mathrm{X3},\mathrm{λ}\right)$
 ${\mathrm{ω3}}{≔}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{J}{,}\left[{2}{,}{0}\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{-}{2}{}{t}{}{{u}}_{{1}}{}{{u}}_{{3}}{-}{x}{}{{u}}_{{1}}^{{2}}{-}{x}{}{{u}}_{{2}}^{{2}}{-}{x}{}{{u}}_{{3}}^{{2}}\right]{,}\left[\left[{1}{,}{3}\right]{,}{-}{2}{}{{u}}_{{2}}{}\left({t}{}{{u}}_{{1}}{+}{x}{}{{u}}_{{3}}\right)\right]{,}\left[\left[{2}{,}{3}\right]{,}{t}{}{{u}}_{{1}}^{{2}}{-}{t}{}{{u}}_{{2}}^{{2}}{+}{t}{}{{u}}_{{3}}^{{2}}{+}{2}{}{x}{}{{u}}_{{1}}{}{{u}}_{{3}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{J}{,}\left[{2}{,}{0}\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{-}{2}{}{t}{}{{u}}_{{1}}{}{{u}}_{{3}}{-}{x}{}{{u}}_{{1}}^{{2}}{-}{x}{}{{u}}_{{2}}^{{2}}{-}{x}{}{{u}}_{{3}}^{{2}}\right]{,}\left[\left[{1}{,}{3}\right]{,}{-}{2}{}{{u}}_{{2}}{}\left({t}{}{{u}}_{{1}}{+}{x}{}{{u}}_{{3}}\right)\right]{,}\left[\left[{2}{,}{3}\right]{,}{t}{}{{u}}_{{1}}^{{2}}{-}{t}{}{{u}}_{{2}}^{{2}}{+}{t}{}{{u}}_{{3}}^{{2}}{+}{2}{}{x}{}{{u}}_{{1}}{}{{u}}_{{3}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{J}{,}\left[{2}{,}{0}\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{-}{2}{}{t}{}{{u}}_{{1}}{}{{u}}_{{3}}{-}{x}{}{{u}}_{{1}}^{{2}}{-}{x}{}{{u}}_{{2}}^{{2}}{-}{x}{}{{u}}_{{3}}^{{2}}\right]{,}\left[\left[{1}{,}{3}\right]{,}{-}{2}{}{{u}}_{{2}}{}\left({t}{}{{u}}_{{1}}{+}{x}{}{{u}}_{{3}}\right)\right]{,}\left[\left[{2}{,}{3}\right]{,}{t}{}{{u}}_{{1}}^{{2}}{-}{t}{}{{u}}_{{2}}^{{2}}{+}{t}{}{{u}}_{{3}}^{{2}}{+}{2}{}{x}{}{{u}}_{{1}}{}{{u}}_{{3}}\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{J}{,}\left[{2}{,}{0}\right]\right]{,}\left[\left[\left[{1}{,}{2}\right]{,}{-}{2}{}{t}{}{{u}}_{{1}}{}{{u}}_{{3}}{-}{x}{}{{u}}_{{1}}^{{2}}{-}{x}{}{{u}}_{{2}}^{{2}}{-}{x}{}{{u}}_{{3}}^{{2}}\right]{,}\left[\left[{1}{,}{3}\right]{,}{-}{2}{}{{u}}_{{2}}{}\left({t}{}{{u}}_{{1}}{+}{x}{}{{u}}_{{3}}\right)\right]{,}\left[\left[{2}{,}{3}\right]{,}{t}{}{{u}}_{{1}}^{{2}}{-}{t}{}{{u}}_{{2}}^{{2}}{+}{t}{}{{u}}_{{3}}^{{2}}{+}{2}{}{x}{}{{u}}_{{1}}{}{{u}}_{{3}}\right]\right]\right]\right)$ (2.21)
 J > $\mathrm{factor}\left(\mathrm{HorizontalExteriorDerivative}\left(\mathrm{ω3}\right)\right)$
 ${\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{J}{,}\left[{3}{,}{0}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{2}{}\left({{u}}_{{1}{,}{1}}{+}{{u}}_{{2}{,}{2}}{-}{{u}}_{{3}{,}{3}}\right){}\left({t}{}{{u}}_{{1}}{+}{x}{}{{u}}_{{3}}\right)\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{J}{,}\left[{3}{,}{0}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{2}{}\left({{u}}_{{1}{,}{1}}{+}{{u}}_{{2}{,}{2}}{-}{{u}}_{{3}{,}{3}}\right){}\left({t}{}{{u}}_{{1}}{+}{x}{}{{u}}_{{3}}\right)\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{J}{,}\left[{3}{,}{0}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{2}{}\left({{u}}_{{1}{,}{1}}{+}{{u}}_{{2}{,}{2}}{-}{{u}}_{{3}{,}{3}}\right){}\left({t}{}{{u}}_{{1}}{+}{x}{}{{u}}_{{3}}\right)\right]\right]\right]\right){,}{\mathrm{_DG}}{}\left(\left[\left[{"biform"}{,}{J}{,}\left[{3}{,}{0}\right]\right]{,}\left[\left[\left[{1}{,}{2}{,}{3}\right]{,}{2}{}\left({{u}}_{{1}{,}{1}}{+}{{u}}_{{2}{,}{2}}{-}{{u}}_{{3}{,}{3}}\right){}\left({t}{}{{u}}_{{1}}{+}{x}{}{{u}}_{{3}}\right)\right]\right]\right]\right)$ (2.22)