Given a list of infinitesimals of a symmetry generator, or the corresponding infinitesimal generator differential operator, the Eta_k command returns a table-procedure $\mathrm{\eta }$ that can compute any particular prolongation of the symmetry. By table-procedure it is meant a Maple table that however acts as a procedure: it takes $n$ indices and works with them as a function of $n$ parameter does - see the examples and also indexing functions for table. Eta_k also works with anticommutative variables set using the Physics package.

The key idea in the implementation of Eta_k is that constructing a procedure optimized for a given symmetry is faster and more economical in computational resources than actually computing one or many prolongations from a single generic procedure. The table-procedure returned by Eta_k can in turn compute any prolongation, and will do so without repeating operations - that is the meaning of optimized in the previous sentence.

The table procedure $\mathrm{\eta }$ returned by Eta_k can be indexed using independent and dependent variables in jet notation, or their respective numeric positions in the lists of independent and dependent variables (see GetIndepVars and GetDepVars), and correspondingly, the output will be in jetvariables or jetnumbers notation. For example, if $\left[u\left(x\,t\right)\,v\left(x\,t\right)\right]$ are the dependent variables, you can index the table $\mathrm{\eta }$ returned by Eta_k as in ${\mathrm{\eta }}_{v,\left[x\,x\,t\right]}$, to receive the value of this prolongation in jetvariables notation. If you use jetnumbers, so the pair [1, 2] representing $\left[x\,t\right]$ as well as $\left[u\,v\right]$, as in ${\mathrm{\eta }}_{2,\left[1\,1\,2\right]}$, the output will be in jetnumbers notation, equivalent to apply ToJet with option jetnotation = jetnumbers over the output of ${\mathrm{\eta }}_{u,\left[x\,x\,t\right]}$. By using the option jetnotation - ..., where the right-hand-side could be jetvariables, jetvariableswithbrackets or jetODE, when indexing $\mathrm{\eta }$ with independent or dependent variables the output will appear in the jetnotation indicated.

Independent of this ability of returning a "prolongation procedure" without having in mind any particular prolongation, you can also specify to Eta_k the prolongation order using the optional argument $\mathrm{prolongation}=\left[m\,\left[i\,j\,\mathrm{...}\right]\right]$, where $m$ is a positive integer identifying a dependent variable and $\left[i\,j\,\mathrm{...}\right]$ is a list with the positions of independent variables, in which case that particular prolongation is returned, unexpanded; evaluating it further (for instance with eval) will automatically expand it. Or, if in addition to specifying the prolongation it is passed the optional argument expanded, then instead of the unexpanded form of the prolongation specified, the expanded form is returned.

To avoid having to remember the optional keywords, if you type the keyword misspelled, or just a portion of it, a matching against the correct keywords is performed, and when there is only one match, the input is automatically corrected.

$\mathrm{with}\left(\mathrm{PDEtools}\,\mathrm{declare}\,\mathrm{Eta\_k}\,\mathrm{InfinitesimalGenerator}\,\mathrm{ToJet}\right)\:$

$\mathrm{DepVars}≔\left[u\,v\right]\left(x\,t\right)$

$\mathrm{DepVars}≔\left[u\left(x\,t\right)\,v\left(x\,t\right)\right]$

$S≔\left[\mathrm{seq}\left(\mathrm{\xi }\left[j\right]\left(x\,t\,u\,v\right)\,j=\left[x\,t\right]\right)\,\mathrm{seq}\left(\mathrm{\eta }\left[j\right]\left(x\,t\,u\,v\right)\,j=\left[u\,v\right]\right)\right]$

$S≔\left[{\mathrm{\xi }}_x\left(x\,t\,u\,v\right)\,{\mathrm{\xi }}_t\left(x\,t\,u\,v\right)\,{\mathrm{\eta }}_u\left(x\,t\,u\,v\right)\,{\mathrm{\eta }}_v\left(x\,t\,u\,v\right)\right]$

$G≔\mathrm{InfinitesimalGenerator}\left(S\,\mathrm{DepVars}\right)$

$G≔f\→{\mathrm{\ξ}}_x\left(x\,t\,u\,v\right)\left(\frac{\partial }{\partial x}f\right)\+{\mathrm{\ξ}}_t\left(x\,t\,u\,v\right)\left(\frac{\partial }{\partial t}f\right)\+{\mathrm{\η}}_u\left(x\,t\,u\,v\right)\left(\frac{\partial }{\partial u}f\right)\+{\mathrm{\η}}_v\left(x\,t\,u\,v\right)\left(\frac{\partial }{\partial v}f\right)$

$\mathrm{{\rm H}}≔\mathrm{Eta\_k}\left(G\,\mathrm{DepVars}\right)$

$\mathrm{{\rm H}}≔\mathrm{\eta }$

$\mathrm{{\rm H}}\left[u\right]$

${\mathrm{\eta }}_u\left(x\,t\,u\,v\right)$

$\mathrm{{\rm H}}\left[1\right]$

${\mathrm{\eta }}_u\left(x\,t\,u\left[\right]\,v\left[\right]\right)$

$\mathrm{{\rm H}}\left[v,\left[x\right]\right]$

$-\left(\frac{\partial }{\partial u}{\mathrm{\xi }}_x\left(x\,t\,u\,v\right)\right)u_x{v}_x-\left(\frac{\partial }{\partial v}{\mathrm{\xi }}_x\left(x\,t\,u\,v\right)\right)v_x^2-\left(\frac{\partial }{\partial u}{\mathrm{\xi }}_t\left(x\,t\,u\,v\right)\right)u_x{v}_t-\left(\frac{\partial }{\partial v}{\mathrm{\xi }}_t\left(x\,t\,u\,v\right)\right)v_x{v}_t+u_x\left(\frac{\partial }{\partial u}{\mathrm{\eta }}_v\left(x\,t\,u\,v\right)\right)+v_x\left(\frac{\partial }{\partial v}{\mathrm{\eta }}_v\left(x\,t\,u\,v\right)\right)-\left(\frac{\partial }{\partial x}{\mathrm{\xi }}_x\left(x\,t\,u\,v\right)\right)v_x-\left(\frac{\partial }{\partial x}{\mathrm{\xi }}_t\left(x\,t\,u\,v\right)\right)v_t+\frac{\partial }{\partial x}{\mathrm{\eta }}_v\left(x\,t\,u\,v\right)$

$\mathrm{{\rm H}}\left[2,\left[1\right]\right]$

$-\left(\frac{\partial }{\partial u\left[\right]}{\mathrm{\xi }}_x\left(x\,t\,u\left[\right]\,v\left[\right]\right)\right)u_1{v}_1-\left(\frac{\partial }{\partial v\left[\right]}{\mathrm{\xi }}_x\left(x\,t\,u\left[\right]\,v\left[\right]\right)\right)v_1^2-\left(\frac{\partial }{\partial u\left[\right]}{\mathrm{\xi }}_t\left(x\,t\,u\left[\right]\,v\left[\right]\right)\right)u_1{v}_2-\left(\frac{\partial }{\partial v\left[\right]}{\mathrm{\xi }}_t\left(x\,t\,u\left[\right]\,v\left[\right]\right)\right)v_1{v}_2+u_1\left(\frac{\partial }{\partial u\left[\right]}{\mathrm{\eta }}_v\left(x\,t\,u\left[\right]\,v\left[\right]\right)\right)+v_1\left(\frac{\partial }{\partial v\left[\right]}{\mathrm{\eta }}_v\left(x\,t\,u\left[\right]\,v\left[\right]\right)\right)-\left(\frac{\partial }{\partial x}{\mathrm{\xi }}_x\left(x\,t\,u\left[\right]\,v\left[\right]\right)\right)v_1-\left(\frac{\partial }{\partial x}{\mathrm{\xi }}_t\left(x\,t\,u\left[\right]\,v\left[\right]\right)\right)v_2+\frac{\partial }{\partial x}{\mathrm{\eta }}_v\left(x\,t\,u\left[\right]\,v\left[\right]\right)$

$\mathrm{declare}\left(\left(\mathrm{\xi }\,\mathrm{\eta }\right)\left(x\,t\,u\,v\right)\right)$

$\mathrm{\xi }\left(x\,t\,u\,v\right)\mathrm{will\; now\; be\; displayed\; as}\mathrm{\xi }$

$\mathrm{\eta }\left(x\,t\,u\,v\right)\mathrm{will\; now\; be\; displayed\; as}\mathrm{\eta }$

$\mathrm{{\rm H}}\left[v,\left[x\right]\right]$

$-{\left({\mathrm{\xi }}_x\right)}_u{u}_x{v}_x-{\left({\mathrm{\xi }}_x\right)}_v{v}_x^2-{\left({\mathrm{\xi }}_t\right)}_u{u}_x{v}_t-{\left({\mathrm{\xi }}_t\right)}_v{v}_x{v}_t+u_x{\left({\mathrm{\eta }}_v\right)}_u+v_x{\left({\mathrm{\eta }}_v\right)}_v-{\left({\mathrm{\xi }}_x\right)}_x{v}_x-{\left({\mathrm{\xi }}_t\right)}_x{v}_t+{\left({\mathrm{\eta }}_v\right)}_x$

$\mathrm{{\rm H}}\left[u,\left[x\,x\right]\right]$

$-{\left({\mathrm{\xi }}_t\right)}_{v,v}{u}_t{v}_x^2-{\left({\mathrm{\xi }}_x\right)}_v{u}_x{v}_{x,x}-{\left({\mathrm{\xi }}_t\right)}_v{u}_t{v}_{x,x}-{\left({\mathrm{\xi }}_x\right)}_{v,v}{u}_x{v}_x^2-2{\left({\mathrm{\xi }}_t\right)}_v{u}_{x,t}{v}_x+2{\left({\mathrm{\eta }}_u\right)}_{u,v}{u}_x{v}_x-3{\left({\mathrm{\xi }}_x\right)}_u{u}_x{u}_{x,x}-2{\left({\mathrm{\xi }}_x\right)}_v{u}_{x,x}{v}_x-{\left({\mathrm{\xi }}_t\right)}_u{u}_t{u}_{x,x}-2{\left({\mathrm{\xi }}_t\right)}_u{u}_x{u}_{x,t}-{\left({\mathrm{\xi }}_t\right)}_{u,u}{u}_x^2{u}_t-2{\left({\mathrm{\xi }}_x\right)}_{u,v}{u}_x^2{v}_x+{\left({\mathrm{\eta }}_u\right)}_{x,x}-2{\left({\mathrm{\xi }}_x\right)}_{v,x}{u}_x{v}_x-2{\left({\mathrm{\xi }}_t\right)}_{u,v}{u}_x{u}_t{v}_x+2v_x{\left({\mathrm{\eta }}_u\right)}_{v,x}+2u_x{\left({\mathrm{\eta }}_u\right)}_{u,x}-2{\left({\mathrm{\xi }}_x\right)}_{u,x}{u}_x^2+{\left({\mathrm{\eta }}_u\right)}_{v,v}{v}_x^2+{\left({\mathrm{\eta }}_u\right)}_v{v}_{x,x}+{\left({\mathrm{\eta }}_u\right)}_{u,u}{u}_x^2+{\left({\mathrm{\eta }}_u\right)}_u{u}_{x,x}-2{\left({\mathrm{\xi }}_x\right)}_x{u}_{x,x}-2{\left({\mathrm{\xi }}_t\right)}_x{u}_{x,t}-{\left({\mathrm{\xi }}_x\right)}_{u,u}{u}_x^3-{\left({\mathrm{\xi }}_t\right)}_{x,x}{u}_t-{\left({\mathrm{\xi }}_x\right)}_{x,x}{u}_x-2{\left({\mathrm{\xi }}_t\right)}_{v,x}{u}_t{v}_x-2{\left({\mathrm{\xi }}_t\right)}_{u,x}{u}_x{u}_t$

$\mathrm{{\rm H}}\left[u,\left[t\,t\right]\right]$

$-2{\left({\mathrm{\xi }}_x\right)}_{t,v}{u}_x{v}_t-2{\left({\mathrm{\xi }}_x\right)}_{t,u}{u}_x{u}_t-{\left({\mathrm{\xi }}_x\right)}_{v,v}{u}_x{v}_t^2-{\left({\mathrm{\xi }}_t\right)}_v{u}_t{v}_{t,t}-{\left({\mathrm{\xi }}_x\right)}_v{u}_x{v}_{t,t}-{\left({\mathrm{\xi }}_t\right)}_{v,v}{u}_t{v}_t^2-2{\left({\mathrm{\xi }}_t\right)}_v{u}_{t,t}{v}_t-{\left({\mathrm{\xi }}_x\right)}_u{u}_x{u}_{t,t}-3{\left({\mathrm{\xi }}_t\right)}_u{u}_t{u}_{t,t}-2{\left({\mathrm{\xi }}_x\right)}_v{u}_{x,t}{v}_t+2{\left({\mathrm{\eta }}_u\right)}_{u,v}{u}_t{v}_t-2{\left({\mathrm{\xi }}_x\right)}_u{u}_t{u}_{x,t}-{\left({\mathrm{\xi }}_x\right)}_{u,u}{u}_x{u}_t^2-{\left({\mathrm{\xi }}_t\right)}_{t,t}{u}_t-{\left({\mathrm{\xi }}_x\right)}_{t,t}{u}_x+2v_t{\left({\mathrm{\eta }}_u\right)}_{t,v}+2u_t{\left({\mathrm{\eta }}_u\right)}_{t,u}-2{\left({\mathrm{\xi }}_t\right)}_{u,v}{u}_t^2{v}_t-2{\left({\mathrm{\xi }}_t\right)}_{t,u}{u}_t^2+{\left({\mathrm{\eta }}_u\right)}_{v,v}{v}_t^2+{\left({\mathrm{\eta }}_u\right)}_v{v}_{t,t}-2{\left({\mathrm{\xi }}_t\right)}_t{u}_{t,t}-2{\left({\mathrm{\xi }}_x\right)}_t{u}_{x,t}+{\left({\mathrm{\eta }}_u\right)}_u{u}_{t,t}+{\left({\mathrm{\eta }}_u\right)}_{u,u}{u}_t^2-{\left({\mathrm{\xi }}_t\right)}_{u,u}{u}_t^3+{\left({\mathrm{\eta }}_u\right)}_{t,t}-2{\left({\mathrm{\xi }}_x\right)}_{u,v}{u}_x{u}_t{v}_t-2{\left({\mathrm{\xi }}_t\right)}_{t,v}{u}_t{v}_t$

$\mathrm{{\rm H}}\left[u,\left[x\,t\right]\right]$

$-{\left({\mathrm{\xi }}_t\right)}_{t,x}{u}_t-{\left({\mathrm{\xi }}_x\right)}_{t,u}{u}_x^2+{\left({\mathrm{\eta }}_u\right)}_{t,u}{u}_x-{\left({\mathrm{\xi }}_x\right)}_t{u}_{x,x}+{\left({\mathrm{\eta }}_u\right)}_u{u}_{x,t}-{\left({\mathrm{\xi }}_t\right)}_t{u}_{x,t}-{\left({\mathrm{\xi }}_t\right)}_{u,x}{u}_t^2+u_t{\left({\mathrm{\eta }}_u\right)}_{u,x}+v_t{\left({\mathrm{\eta }}_u\right)}_{v,x}-{\left({\mathrm{\xi }}_t\right)}_{u,u}{u}_x{u}_t^2-{\left({\mathrm{\xi }}_t\right)}_{t,u}{u}_x{u}_t+{\left({\mathrm{\eta }}_u\right)}_{u,u}{u}_x{u}_t+{\left({\mathrm{\eta }}_u\right)}_{u,v}{u}_x{v}_t-{\left({\mathrm{\xi }}_x\right)}_u{u}_t{u}_{x,x}-{\left({\mathrm{\xi }}_x\right)}_v{u}_{x,x}{v}_t-{\left({\mathrm{\xi }}_x\right)}_{t,x}{u}_x-{\left({\mathrm{\xi }}_t\right)}_{u,v}{u}_x{u}_t{v}_t-{\left({\mathrm{\xi }}_x\right)}_{u,v}{u}_x{u}_t{v}_x-{\left({\mathrm{\xi }}_t\right)}_{v,v}{u}_t{v}_x{v}_t-{\left({\mathrm{\xi }}_x\right)}_{v,v}{u}_x{v}_x{v}_t-{\left({\mathrm{\xi }}_t\right)}_{v,x}{u}_t{v}_t-{\left({\mathrm{\xi }}_x\right)}_{u,x}{u}_x{u}_t-{\left({\mathrm{\xi }}_x\right)}_{v,x}{u}_x{v}_t-{\left({\mathrm{\xi }}_t\right)}_v{u}_{x,t}{v}_t-2{\left({\mathrm{\xi }}_x\right)}_u{u}_x{u}_{x,t}-2{\left({\mathrm{\xi }}_t\right)}_u{u}_t{u}_{x,t}-{\left({\mathrm{\xi }}_t\right)}_{u,v}{u}_t^2{v}_x-{\left({\mathrm{\xi }}_t\right)}_{t,v}{u}_t{v}_x-{\left({\mathrm{\xi }}_x\right)}_{t,v}{u}_x{v}_x+{\left({\mathrm{\eta }}_u\right)}_{u,v}{u}_t{v}_x+{\left({\mathrm{\eta }}_u\right)}_{t,x}+{\left({\mathrm{\eta }}_u\right)}_{t,v}{v}_x+{\left({\mathrm{\eta }}_u\right)}_v{v}_{x,t}-{\left({\mathrm{\xi }}_x\right)}_x{u}_{x,t}-{\left({\mathrm{\xi }}_t\right)}_x{u}_{t,t}+{\left({\mathrm{\eta }}_u\right)}_{v,v}{v}_x{v}_t-{\left({\mathrm{\xi }}_t\right)}_v{u}_t{v}_{x,t}-{\left({\mathrm{\xi }}_x\right)}_v{u}_x{v}_{x,t}-{\left({\mathrm{\xi }}_x\right)}_v{u}_{x,t}{v}_x-{\left({\mathrm{\xi }}_t\right)}_v{u}_{t,t}{v}_x-{\left({\mathrm{\xi }}_t\right)}_u{u}_x{u}_{t,t}-{\left({\mathrm{\xi }}_x\right)}_{u,u}{u}_x^2{u}_t-{\left({\mathrm{\xi }}_x\right)}_{u,v}{u}_x^2{v}_t$

$\mathrm{show}$

$\frac{{\partial }^2}{\partial t\partial x}{\mathrm{\eta }}_u\left(x\,t\,u\,v\right)-\left(\frac{{\partial }^2}{\partial u\partial v}{\mathrm{\xi }}_x\left(x\,t\,u\,v\right)\right)u_x^2{v}_t-\left(\frac{{\partial }^2}{\partial t\partial v}{\mathrm{\xi }}_x\left(x\,t\,u\,v\right)\right)u_x{v}_x+\left(\frac{{\partial }^2}{\partial u\partial v}{\mathrm{\eta }}_u\left(x\,t\,u\,v\right)\right)u_t{v}_x+\left(\frac{{\partial }^2}{\partial v^2}{\mathrm{\eta }}_u\left(x\,t\,u\,v\right)\right)v_x{v}_t-\left(\frac{{\partial }^2}{\partial u^2}{\mathrm{\xi }}_x\left(x\,t\,u\,v\right)\right)u_x^2{u}_t-\left(\frac{{\partial }^2}{\partial v\partial x}{\mathrm{\xi }}_x\left(x\,t\,u\,v\right)\right)u_x{v}_t-\left(\frac{{\partial }^2}{\partial u\partial v}{\mathrm{\xi }}_t\left(x\,t\,u\,v\right)\right)u_t^2{v}_x-\left(\frac{{\partial }^2}{\partial t\partial v}{\mathrm{\xi }}_t\left(x\,t\,u\,v\right)\right)u_t{v}_x-\left(\frac{{\partial }^2}{\partial u\partial x}{\mathrm{\xi }}_x\left(x\,t\,u\,v\right)\right)u_x{u}_t-\left(\frac{{\partial }^2}{\partial v\partial x}{\mathrm{\xi }}_t\left(x\,t\,u\,v\right)\right)u_t{v}_t+\left(\frac{{\partial }^2}{\partial u^2}{\mathrm{\eta }}_u\left(x\,t\,u\,v\right)\right)u_x{u}_t+\left(\frac{{\partial }^2}{\partial u\partial v}{\mathrm{\eta }}_u\left(x\,t\,u\,v\right)\right)u_x{v}_t-\left(\frac{{\partial }^2}{\partial u^2}{\mathrm{\xi }}_t\left(x\,t\,u\,v\right)\right)u_x{u}_t^2-\left(\frac{{\partial }^2}{\partial t\partial u}{\mathrm{\xi }}_t\left(x\,t\,u\,v\right)\right)u_x{u}_t-\left(\frac{{\partial }^2}{\partial u\partial x}{\mathrm{\xi }}_t\left(x\,t\,u\,v\right)\right)u_t^2+u_t\left(\frac{{\partial }^2}{\partial u\partial x}{\mathrm{\eta }}_u\left(x\,t\,u\,v\right)\right)+v_t\left(\frac{{\partial }^2}{\partial v\partial x}{\mathrm{\eta }}_u\left(x\,t\,u\,v\right)\right)-\left(\frac{{\partial }^2}{\partial t\partial x}{\mathrm{\xi }}_x\left(x\,t\,u\,v\right)\right)u_x+\left(\frac{\partial }{\partial v}{\mathrm{\eta }}_u\left(x\,t\,u\,v\right)\right)v_{x,t}-\left(\frac{\partial }{\partial x}{\mathrm{\xi }}_x\left(x\,t\,u\,v\right)\right)u_{x,t}-\left(\frac{\partial }{\partial x}{\mathrm{\xi }}_t\left(x\,t\,u\,v\right)\right)u_{t,t}-\left(\frac{\partial }{\partial t}{\mathrm{\xi }}_x\left(x\,t\,u\,v\right)\right)u_{x,x}+\left(\frac{\partial }{\partial u}{\mathrm{\eta }}_u\left(x\,t\,u\,v\right)\right)u_{x,t}-\left(\frac{\partial }{\partial t}{\mathrm{\xi }}_t\left(x\,t\,u\,v\right)\right)u_{x,t}-\left(\frac{\partial }{\partial u}{\mathrm{\xi }}_x\left(x\,t\,u\,v\right)\right)u_t{u}_{x,x}-\left(\frac{\partial }{\partial v}{\mathrm{\xi }}_x\left(x\,t\,u\,v\right)\right)u_{x,x}{v}_t-\left(\frac{\partial }{\partial v}{\mathrm{\xi }}_t\left(x\,t\,u\,v\right)\right)u_{x,t}{v}_t-2\left(\frac{\partial }{\partial u}{\mathrm{\xi }}_x\left(x\,t\,u\,v\right)\right)u_x{u}_{x,t}-2\left(\frac{\partial }{\partial u}{\mathrm{\xi }}_t\left(x\,t\,u\,v\right)\right)u_t{u}_{x,t}-\left(\frac{\partial }{\partial v}{\mathrm{\xi }}_t\left(x\,t\,u\,v\right)\right)u_t{v}_{x,t}-\left(\frac{\partial }{\partial v}{\mathrm{\xi }}_x\left(x\,t\,u\,v\right)\right)u_x{v}_{x,t}-\left(\frac{\partial }{\partial v}{\mathrm{\xi }}_x\left(x\,t\,u\,v\right)\right)u_{x,t}{v}_x-\left(\frac{\partial }{\partial v}{\mathrm{\xi }}_t\left(x\,t\,u\,v\right)\right)u_{t,t}{v}_x-\left(\frac{\partial }{\partial u}{\mathrm{\xi }}_t\left(x\,t\,u\,v\right)\right)u_x{u}_{t,t}-\left(\frac{{\partial }^2}{\partial t\partial x}{\mathrm{\xi }}_t\left(x\,t\,u\,v\right)\right)u_t-\left(\frac{{\partial }^2}{\partial t\partial u}{\mathrm{\xi }}_x\left(x\,t\,u\,v\right)\right)u_x^2+\left(\frac{{\partial }^2}{\partial t\partial u}{\mathrm{\eta }}_u\left(x\,t\,u\,v\right)\right)u_x+\left(\frac{{\partial }^2}{\partial t\partial v}{\mathrm{\eta }}_u\left(x\,t\,u\,v\right)\right)v_x-\left(\frac{{\partial }^2}{\partial u\partial v}{\mathrm{\xi }}_t\left(x\,t\,u\,v\right)\right)u_x{u}_t{v}_t-\left(\frac{{\partial }^2}{\partial u\partial v}{\mathrm{\xi }}_x\left(x\,t\,u\,v\right)\right)u_x{u}_t{v}_x-\left(\frac{{\partial }^2}{\partial v^2}{\mathrm{\xi }}_t\left(x\,t\,u\,v\right)\right)u_t{v}_x{v}_t-\left(\frac{{\partial }^2}{\partial v^2}{\mathrm{\xi }}_x\left(x\,t\,u\,v\right)\right)u_x{v}_x{v}_t$

$\mathrm{Eta\_k}\left(S\,\mathrm{DepVars}\,\mathrm{prolongation}=\left[u\,\left[x\,t\right]\right]\right)$

${\mathrm{\eta }}_{u,\left[x\,t\right]}$

$-{\left({\mathrm{\xi }}_t\right)}_{t,x}{u}_t-{\left({\mathrm{\xi }}_x\right)}_{t,u}{u}_x^2+{\left({\mathrm{\eta }}_u\right)}_{t,u}{u}_x-{\left({\mathrm{\xi }}_x\right)}_t{u}_{x,x}+{\left({\mathrm{\eta }}_u\right)}_u{u}_{x,t}-{\left({\mathrm{\xi }}_t\right)}_t{u}_{x,t}-{\left({\mathrm{\xi }}_t\right)}_{u,x}{u}_t^2+u_t{\left({\mathrm{\eta }}_u\right)}_{u,x}+v_t{\left({\mathrm{\eta }}_u\right)}_{v,x}-{\left({\mathrm{\xi }}_t\right)}_{u,u}{u}_x{u}_t^2-{\left({\mathrm{\xi }}_t\right)}_{t,u}{u}_x{u}_t+{\left({\mathrm{\eta }}_u\right)}_{u,u}{u}_x{u}_t+{\left({\mathrm{\eta }}_u\right)}_{u,v}{u}_x{v}_t-{\left({\mathrm{\xi }}_x\right)}_u{u}_t{u}_{x,x}-{\left({\mathrm{\xi }}_x\right)}_v{u}_{x,x}{v}_t-{\left({\mathrm{\xi }}_x\right)}_{t,x}{u}_x-{\left({\mathrm{\xi }}_t\right)}_{u,v}{u}_x{u}_t{v}_t-{\left({\mathrm{\xi }}_x\right)}_{u,v}{u}_x{u}_t{v}_x-{\left({\mathrm{\xi }}_t\right)}_{v,v}{u}_t{v}_x{v}_t-{\left({\mathrm{\xi }}_x\right)}_{v,v}{u}_x{v}_x{v}_t-{\left({\mathrm{\xi }}_t\right)}_{v,x}{u}_t{v}_t-{\left({\mathrm{\xi }}_x\right)}_{u,x}{u}_x{u}_t-{\left({\mathrm{\xi }}_x\right)}_{v,x}{u}_x{v}_t-{\left({\mathrm{\xi }}_t\right)}_v{u}_{x,t}{v}_t-2{\left({\mathrm{\xi }}_x\right)}_u{u}_x{u}_{x,t}-2{\left({\mathrm{\xi }}_t\right)}_u{u}_t{u}_{x,t}-{\left({\mathrm{\xi }}_t\right)}_{u,v}{u}_t^2{v}_x-{\left({\mathrm{\xi }}_t\right)}_{t,v}{u}_t{v}_x-{\left({\mathrm{\xi }}_x\right)}_{t,v}{u}_x{v}_x+{\left({\mathrm{\eta }}_u\right)}_{u,v}{u}_t{v}_x+{\left({\mathrm{\eta }}_u\right)}_{t,x}+{\left({\mathrm{\eta }}_u\right)}_{t,v}{v}_x+{\left({\mathrm{\eta }}_u\right)}_v{v}_{x,t}-{\left({\mathrm{\xi }}_x\right)}_x{u}_{x,t}-{\left({\mathrm{\xi }}_t\right)}_x{u}_{t,t}+{\left({\mathrm{\eta }}_u\right)}_{v,v}{v}_x{v}_t-{\left({\mathrm{\xi }}_t\right)}_v{u}_t{v}_{x,t}-{\left({\mathrm{\xi }}_x\right)}_v{u}_x{v}_{x,t}-{\left({\mathrm{\xi }}_x\right)}_v{u}_{x,t}{v}_x-{\left({\mathrm{\xi }}_t\right)}_v{u}_{t,t}{v}_x-{\left({\mathrm{\xi }}_t\right)}_u{u}_x{u}_{t,t}-{\left({\mathrm{\xi }}_x\right)}_{u,u}{u}_x^2{u}_t-{\left({\mathrm{\xi }}_x\right)}_{u,v}{u}_x^2{v}_t$

$\mathrm{Eta\_k}\left(S\,\mathrm{DepVars}\,\mathrm{prolongation}=\left[u\,\left[x\,t\right]\right]\,\mathrm{expanded}\right)$

$-{\left({\mathrm{\xi }}_t\right)}_{t,x}{u}_t-{\left({\mathrm{\xi }}_x\right)}_{t,u}{u}_x^2+{\left({\mathrm{\eta }}_u\right)}_{t,u}{u}_x-{\left({\mathrm{\xi }}_x\right)}_t{u}_{x,x}+{\left({\mathrm{\eta }}_u\right)}_u{u}_{x,t}-{\left({\mathrm{\xi }}_t\right)}_t{u}_{x,t}-{\left({\mathrm{\xi }}_t\right)}_{u,x}{u}_t^2+u_t{\left({\mathrm{\eta }}_u\right)}_{u,x}+v_t{\left({\mathrm{\eta }}_u\right)}_{v,x}-{\left({\mathrm{\xi }}_t\right)}_{u,u}{u}_x{u}_t^2-{\left({\mathrm{\xi }}_t\right)}_{t,u}{u}_x{u}_t+{\left({\mathrm{\eta }}_u\right)}_{u,u}{u}_x{u}_t+{\left({\mathrm{\eta }}_u\right)}_{u,v}{u}_x{v}_t-{\left({\mathrm{\xi }}_x\right)}_u{u}_t{u}_{x,x}-{\left({\mathrm{\xi }}_x\right)}_v{u}_{x,x}{v}_t-{\left({\mathrm{\xi }}_x\right)}_{t,x}{u}_x-{\left({\mathrm{\xi }}_t\right)}_{u,v}{u}_x{u}_t{v}_t-{\left({\mathrm{\xi }}_x\right)}_{u,v}{u}_x{u}_t{v}_x-{\left({\mathrm{\xi }}_t\right)}_{v,v}{u}_t{v}_x{v}_t-{\left({\mathrm{\xi }}_x\right)}_{v,v}{u}_x{v}_x{v}_t-{\left({\mathrm{\xi }}_t\right)}_{v,x}{u}_t{v}_t-{\left({\mathrm{\xi }}_x\right)}_{u,x}{u}_x{u}_t-{\left({\mathrm{\xi }}_x\right)}_{v,x}{u}_x{v}_t-{\left({\mathrm{\xi }}_t\right)}_v{u}_{x,t}{v}_t-2{\left({\mathrm{\xi }}_x\right)}_u{u}_x{u}_{x,t}-2{\left({\mathrm{\xi }}_t\right)}_u{u}_t{u}_{x,t}-{\left({\mathrm{\xi }}_t\right)}_{u,v}{u}_t^2{v}_x-{\left({\mathrm{\xi }}_t\right)}_{t,v}{u}_t{v}_x-{\left({\mathrm{\xi }}_x\right)}_{t,v}{u}_x{v}_x+{\left({\mathrm{\eta }}_u\right)}_{u,v}{u}_t{v}_x+{\left({\mathrm{\eta }}_u\right)}_{t,x}+{\left({\mathrm{\eta }}_u\right)}_{t,v}{v}_x+{\left({\mathrm{\eta }}_u\right)}_v{v}_{x,t}-{\left({\mathrm{\xi }}_x\right)}_x{u}_{x,t}-{\left({\mathrm{\xi }}_t\right)}_x{u}_{t,t}+{\left({\mathrm{\eta }}_u\right)}_{v,v}{v}_x{v}_t-{\left({\mathrm{\xi }}_t\right)}_v{u}_t{v}_{x,t}-{\left({\mathrm{\xi }}_x\right)}_v{u}_x{v}_{x,t}-{\left({\mathrm{\xi }}_x\right)}_v{u}_{x,t}{v}_x-{\left({\mathrm{\xi }}_t\right)}_v{u}_{t,t}{v}_x-{\left({\mathrm{\xi }}_t\right)}_u{u}_x{u}_{t,t}-{\left({\mathrm{\xi }}_x\right)}_{u,u}{u}_x^2{u}_t-{\left({\mathrm{\xi }}_x\right)}_{u,v}{u}_x^2{v}_t$

$-$

$0$

$\mathrm{Eta\_k}\left(S\,\mathrm{DepVars}\,\mathrm{prolongation}=\left[1\,\left[1\,2\right]\right]\,\mathrm{expanded}\right)$

$-{\left({\mathrm{\xi }}_x\right)}_{t,x}{u}_1-{\left({\mathrm{\xi }}_x\right)}_{x,v\left[\right]}{u}_1{v}_2-{\left({\mathrm{\xi }}_x\right)}_{x,u\left[\right]}{u}_1{u}_2-{\left({\mathrm{\xi }}_t\right)}_{x,v\left[\right]}{u}_2{v}_2-{\left({\mathrm{\xi }}_t\right)}_{u\left[\right],v\left[\right]}{u}_1{u}_2{v}_2-{\left({\mathrm{\xi }}_x\right)}_{u\left[\right],v\left[\right]}{u}_1{u}_2{v}_1+{\left({\mathrm{\eta }}_u\right)}_{u\left[\right],u\left[\right]}{u}_1{u}_2+{\left({\mathrm{\eta }}_u\right)}_{u\left[\right],v\left[\right]}{u}_1{v}_2-{\left({\mathrm{\xi }}_x\right)}_{u\left[\right]}{u}_2{u}_{1,1}-{\left({\mathrm{\xi }}_x\right)}_{v\left[\right]}{u}_{1,1}{v}_2-{\left({\mathrm{\xi }}_t\right)}_{t,u\left[\right]}{u}_1{u}_2+{\left({\mathrm{\eta }}_u\right)}_{t,x}+u_2{\left({\mathrm{\eta }}_u\right)}_{x,u\left[\right]}+v_2{\left({\mathrm{\eta }}_u\right)}_{x,v\left[\right]}-{\left({\mathrm{\xi }}_t\right)}_{t,x}{u}_2+{\left({\mathrm{\eta }}_u\right)}_{t,v\left[\right]}{v}_1+{\left({\mathrm{\eta }}_u\right)}_{v\left[\right]}{v}_{1,2}-{\left({\mathrm{\xi }}_x\right)}_x{u}_{1,2}-{\left({\mathrm{\xi }}_t\right)}_x{u}_{2,2}-{\left({\mathrm{\xi }}_t\right)}_{v\left[\right],v\left[\right]}{u}_2{v}_1{v}_2-{\left({\mathrm{\xi }}_x\right)}_{v\left[\right],v\left[\right]}{u}_1{v}_1{v}_2-{\left({\mathrm{\xi }}_t\right)}_{u\left[\right],u\left[\right]}{u}_1{u}_2^2-{\left({\mathrm{\xi }}_x\right)}_{u\left[\right],v\left[\right]}{u}_1^2{v}_2-{\left({\mathrm{\xi }}_x\right)}_{u\left[\right],u\left[\right]}{u}_1^2{u}_2-{\left({\mathrm{\xi }}_x\right)}_{v\left[\right]}{u}_1{v}_{1,2}-{\left({\mathrm{\xi }}_x\right)}_{v\left[\right]}{u}_{1,2}{v}_1-{\left({\mathrm{\xi }}_t\right)}_{v\left[\right]}{u}_{2,2}{v}_1-{\left({\mathrm{\xi }}_t\right)}_{u\left[\right]}{u}_1{u}_{2,2}+{\left({\mathrm{\eta }}_u\right)}_{v\left[\right],v\left[\right]}{v}_1{v}_2-{\left({\mathrm{\xi }}_t\right)}_{v\left[\right]}{u}_2{v}_{1,2}+{\left({\mathrm{\eta }}_u\right)}_{u\left[\right],v\left[\right]}{u}_2{v}_1-{\left({\mathrm{\xi }}_x\right)}_{t,u\left[\right]}{u}_1^2+{\left({\mathrm{\eta }}_u\right)}_{t,u\left[\right]}{u}_1-{\left({\mathrm{\xi }}_x\right)}_t{u}_{1,1}+{\left({\mathrm{\eta }}_u\right)}_{u\left[\right]}{u}_{1,2}-{\left({\mathrm{\xi }}_t\right)}_t{u}_{1,2}-{\left({\mathrm{\xi }}_x\right)}_{t,v\left[\right]}{u}_1{v}_1-{\left({\mathrm{\xi }}_t\right)}_{x,u\left[\right]}{u}_2^2-{\left({\mathrm{\xi }}_t\right)}_{t,v\left[\right]}{u}_2{v}_1-{\left({\mathrm{\xi }}_t\right)}_{u\left[\right],v\left[\right]}{u}_2^2{v}_1-2{\left({\mathrm{\xi }}_t\right)}_{u\left[\right]}{u}_2{u}_{1,2}-2{\left({\mathrm{\xi }}_x\right)}_{u\left[\right]}{u}_1{u}_{1,2}-{\left({\mathrm{\xi }}_t\right)}_{v\left[\right]}{u}_{1,2}{v}_2$