Given a list of infinitesimals of a symmetry generator, the InfinitesimalGenerator command returns a differential operator representing the infinitesimal generator. Note the distinction here made between infinitesimals and infinitesimal generator: the latter is an operator constructed as a linear combination of differential operators, while the infinitesimals are represented by the list you can construct with the coefficients of each differential operator entering the infinitesimal generator. InfinitesimalGenerator also works with anticommutative variables set using the Physics package.

You can also pass the symmetry S directly as an infinitesimal generator operator, for instance with the purpose of having it rewritten in different jetnotation or prolonged - see related options below.

By default the infinitesimal generator returned is not prolonged more than S - to request any other prolongation pass the optional argument $\mathrm{prolongation}=n$ where n is a nonnegative integer. As a handy shortcut notation, instead of the optional argument $\mathrm{prolongation}=n$ you can also just specify $n$, a nonnegative integer, as the third argument passed to InfinitesimalGenerator.

By default, the infinitesimal generator is also returned not expanded. To request it otherwise, pass the optional argument expanded or expanded = true. To understand this option, recall that given a prolongation order - say k - there exist as many prolonged infinitesimals as partial derivatives of order $0<k$ exist in the jet space where the symmetry generator acts, that is, $m\left(\genfrac{}{}{0ex}{}{n+k}{k}\right)-1$, where $n$ and $m$ are respectively the numbers of independent and dependent variables. In addition, the infinitesimal generator to be returned includes a sum over the independent and another one over the dependent variables, where each of the summands has for coefficient the corresponding prolongation of the infinitesimal $\mathrm{\eta }$ (see Eta_k) associated to each dependent variable. All these sums return expanded when the optional argument expanded is given. As an indicator of what this means, in a problem with 2 independent and 2 dependent variables, the length of the infinitesimal generator prolonged to order 2, when the option expanded is given, is more than 40 times larger than when it is not given.

By default, the infinitesimal generator is returned as an operator, that is a mapping to be applied to a function (see the Examples section). Equivalent representations would be: an expression, i.e., the result of applying the operator to some generic function, or a list with the infinitesimal components. To compute these representations use the option output = ... where the right-hand side is respectively expression or list. When output = expression, differentiation is represented by the inert Diff command and the function to which the mapping is applied is represented generically by a label, typically $\mathrm{\_\&Phi;}$ (provided that $\mathrm{\_\&Phi;}$ is not assigned already, or found in the infinitesimals S).

The jetnotation used in the output is the one of S unless indicated otherwise using the option jetnotation = ... where the right-hand side is any of jetvariables (default), jetvariableswithtbrackets, jetnumbers' or jetODE; for details about the available jet notations see ToJet.

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.

$\mathrm{with}\left(\mathrm{PDEtools}\&comma;\mathrm{InfinitesimalGenerator}\&comma;\mathrm{declare}\right)$

$\left[\mathrm{InfinitesimalGenerator}\&comma;\mathrm{declare}\right]$

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

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

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

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

$G\left[1\right]≔\mathrm{InfinitesimalGenerator}\left(S\&comma;\mathrm{DepVars}\&comma;\mathrm{prolongation}=1\right)$

$G_1≔f\&rarr;\mathrm{add}\left({\mathrm{\&xi;}}_{x_j}\left(\frac{\partial }{\partial x_j}f\right)\&comma;j\&equals;1..2\right)\&plus;\mathrm{add}\left({\mathrm{\&eta;}}_{u_m}\left(\frac{\partial }{\partial u_m}f\right)\&plus;{\mathrm{\&eta;}}_{u_m\&comma;\left[x\right]}\left(\frac{\partial }{\partial {u_m}_x}f\right)\&plus;{\mathrm{\&eta;}}_{u_m\&comma;\left[t\right]}\left(\frac{\partial }{\partial {u_m}_t}f\right)\&comma;m\&equals;1..2\right)$

$G\left[2\right]≔\mathrm{InfinitesimalGenerator}\left(S\&comma;\mathrm{DepVars}\&comma;\mathrm{prolongation}=1\&comma;\mathrm{expanded}\right)$

$G_2≔f\&rarr;{\mathrm{\&xi;}}_x\left(x\&comma;t\&comma;u\&comma;v\right)\left(\frac{\partial }{\partial x}f\right)\&plus;{\mathrm{\&xi;}}_t\left(x\&comma;t\&comma;u\&comma;v\right)\left(\frac{\partial }{\partial t}f\right)\&plus;{\mathrm{\&eta;}}_u\left(x\&comma;t\&comma;u\&comma;v\right)\left(\frac{\partial }{\partial u}f\right)\&plus;{\mathrm{\&eta;}}_v\left(x\&comma;t\&comma;u\&comma;v\right)\left(\frac{\partial }{\partial v}f\right)\&plus;\left(-\left(\frac{\partial }{\partial u}{\mathrm{\&xi;}}_x\left(x\&comma;t\&comma;u\&comma;v\right)\right)u_x^2-\left(\frac{\partial }{\partial v}{\mathrm{\&xi;}}_x\left(x\&comma;t\&comma;u\&comma;v\right)\right)u_x{v}_x-\left(\frac{\partial }{\partial u}{\mathrm{\&xi;}}_t\left(x\&comma;t\&comma;u\&comma;v\right)\right)u_x{u}_t-\left(\frac{\partial }{\partial v}{\mathrm{\&xi;}}_t\left(x\&comma;t\&comma;u\&comma;v\right)\right)u_t{v}_x\&plus;u_x\left(\frac{\partial }{\partial u}{\mathrm{\&eta;}}_u\left(x\&comma;t\&comma;u\&comma;v\right)\right)\&plus;v_x\left(\frac{\partial }{\partial v}{\mathrm{\&eta;}}_u\left(x\&comma;t\&comma;u\&comma;v\right)\right)-\left(\frac{\partial }{\partial x}{\mathrm{\&xi;}}_x\left(x\&comma;t\&comma;u\&comma;v\right)\right)u_x-\left(\frac{\partial }{\partial x}{\mathrm{\&xi;}}_t\left(x\&comma;t\&comma;u\&comma;v\right)\right)u_t\&plus;\frac{\partial }{\partial x}{\mathrm{\&eta;}}_u\left(x\&comma;t\&comma;u\&comma;v\right)\right)\left(\frac{\partial }{\partial u_x}f\right)\&plus;\left(-\left(\frac{\partial }{\partial u}{\mathrm{\&xi;}}_x\left(x\&comma;t\&comma;u\&comma;v\right)\right)u_x{v}_x-\left(\frac{\partial }{\partial v}{\mathrm{\&xi;}}_x\left(x\&comma;t\&comma;u\&comma;v\right)\right)v_x^2-\left(\frac{\partial }{\partial u}{\mathrm{\&xi;}}_t\left(x\&comma;t\&comma;u\&comma;v\right)\right)u_x{v}_t-\left(\frac{\partial }{\partial v}{\mathrm{\&xi;}}_t\left(x\&comma;t\&comma;u\&comma;v\right)\right)v_x{v}_t\&plus;u_x\left(\frac{\partial }{\partial u}{\mathrm{\&eta;}}_v\left(x\&comma;t\&comma;u\&comma;v\right)\right)\&plus;v_x\left(\frac{\partial }{\partial v}{\mathrm{\&eta;}}_v\left(x\&comma;t\&comma;u\&comma;v\right)\right)-\left(\frac{\partial }{\partial x}{\mathrm{\&xi;}}_x\left(x\&comma;t\&comma;u\&comma;v\right)\right)v_x-\left(\frac{\partial }{\partial x}{\mathrm{\&xi;}}_t\left(x\&comma;t\&comma;u\&comma;v\right)\right)v_t\&plus;\frac{\partial }{\partial x}{\mathrm{\&eta;}}_v\left(x\&comma;t\&comma;u\&comma;v\right)\right)\left(\frac{\partial }{\partial v_x}f\right)\&plus;\left(-\left(\frac{\partial }{\partial u}{\mathrm{\&xi;}}_x\left(x\&comma;t\&comma;u\&comma;v\right)\right)u_x{u}_t-\left(\frac{\partial }{\partial v}{\mathrm{\&xi;}}_x\left(x\&comma;t\&comma;u\&comma;v\right)\right)u_x{v}_t-\left(\frac{\partial }{\partial u}{\mathrm{\&xi;}}_t\left(x\&comma;t\&comma;u\&comma;v\right)\right)u_t^2-\left(\frac{\partial }{\partial v}{\mathrm{\&xi;}}_t\left(x\&comma;t\&comma;u\&comma;v\right)\right)u_t{v}_t-\left(\frac{\partial }{\partial t}{\mathrm{\&xi;}}_x\left(x\&comma;t\&comma;u\&comma;v\right)\right)u_x-\left(\frac{\partial }{\partial t}{\mathrm{\&xi;}}_t\left(x\&comma;t\&comma;u\&comma;v\right)\right)u_t\&plus;u_t\left(\frac{\partial }{\partial u}{\mathrm{\&eta;}}_u\left(x\&comma;t\&comma;u\&comma;v\right)\right)\&plus;v_t\left(\frac{\partial }{\partial v}{\mathrm{\&eta;}}_u\left(x\&comma;t\&comma;u\&comma;v\right)\right)\&plus;\frac{\partial }{\partial t}{\mathrm{\&eta;}}_u\left(x\&comma;t\&comma;u\&comma;v\right)\right)\left(\frac{\partial }{\partial u_t}f\right)\&plus;\left(-\left(\frac{\partial }{\partial u}{\mathrm{\&xi;}}_x\left(x\&comma;t\&comma;u\&comma;v\right)\right)u_t{v}_x-\left(\frac{\partial }{\partial v}{\mathrm{\&xi;}}_x\left(x\&comma;t\&comma;u\&comma;v\right)\right)v_x{v}_t-\left(\frac{\partial }{\partial u}{\mathrm{\&xi;}}_t\left(x\&comma;t\&comma;u\&comma;v\right)\right)u_t{v}_t-\left(\frac{\partial }{\partial v}{\mathrm{\&xi;}}_t\left(x\&comma;t\&comma;u\&comma;v\right)\right)v_t^2-\left(\frac{\partial }{\partial t}{\mathrm{\&xi;}}_x\left(x\&comma;t\&comma;u\&comma;v\right)\right)v_x-\left(\frac{\partial }{\partial t}{\mathrm{\&xi;}}_t\left(x\&comma;t\&comma;u\&comma;v\right)\right)v_t\&plus;u_t\left(\frac{\partial }{\partial u}{\mathrm{\&eta;}}_v\left(x\&comma;t\&comma;u\&comma;v\right)\right)\&plus;v_t\left(\frac{\partial }{\partial v}{\mathrm{\&eta;}}_v\left(x\&comma;t\&comma;u\&comma;v\right)\right)\&plus;\frac{\partial }{\partial t}{\mathrm{\&eta;}}_v\left(x\&comma;t\&comma;u\&comma;v\right)\right)\left(\frac{\partial }{\partial v_t}f\right)$

$L≔\mathrm{\Lambda }\left(x\&comma;t\&comma;u\&comma;v\&comma;u\left[x\right]\&comma;u\left[t\right]\&comma;v\left[x\right]\&comma;v\left[t\right]\&comma;u\left[x,x\right]\&comma;u\left[x,t\right]\&comma;u\left[t,t\right]\&comma;v\left[x,x\right]\&comma;v\left[x,t\right]\&comma;v\left[t,t\right]\right)\&colon;$

$\mathrm{declare}\left(\left(\mathrm{\xi }\&comma;\mathrm{\eta }\right)\left(x\&comma;t\&comma;u\&comma;v\right)\&comma;L\right)$

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

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

$\mathrm{\Lambda }\left(x\&comma;t\&comma;u\&comma;v\&comma;u_x\&comma;u_t\&comma;v_x\&comma;v_t\&comma;u_{x,x}\&comma;u_{x,t}\&comma;u_{t,t}\&comma;v_{x,x}\&comma;v_{x,t}\&comma;v_{t,t}\right)\mathrm{will\; now\; be\; displayed\; as}\mathrm{\Lambda }$

$G\left[1\right]\left(L\right)$

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

$G\left[1\right]\left(L\right)-G\left[2\right]\left(L\right)$

$0$

$\mathrm{InfinitesimalGenerator}\left(S\&comma;\mathrm{DepVars}\&comma;\mathrm{prolongation}=1\&comma;\mathrm{expanded}\&comma;\mathrm{output}=\mathrm{expression}\right)$

${\mathrm{\xi }}_x\left(\frac{\&DifferentialD;\mathrm{\_\&Phi;}}{\&DifferentialD;x}\right)+{\mathrm{\xi }}_t\left(\frac{\&DifferentialD;\mathrm{\_\&Phi;}}{\&DifferentialD;t}\right)+{\mathrm{\eta }}_u\left(\frac{\&DifferentialD;\mathrm{\_\&Phi;}}{\&DifferentialD;u}\right)+{\mathrm{\eta }}_v\left(\frac{\&DifferentialD;\mathrm{\_\&Phi;}}{\&DifferentialD;v}\right)+\left(-{\left({\mathrm{\xi }}_x\right)}_u{u}_x^2-{\left({\mathrm{\xi }}_x\right)}_v{u}_x{v}_x-{\left({\mathrm{\xi }}_t\right)}_u{u}_x{u}_t-{\left({\mathrm{\xi }}_t\right)}_v{u}_t{v}_x+u_x{\left({\mathrm{\eta }}_u\right)}_u+v_x{\left({\mathrm{\eta }}_u\right)}_v-{\left({\mathrm{\xi }}_x\right)}_x{u}_x-{\left({\mathrm{\xi }}_t\right)}_x{u}_t+{\left({\mathrm{\eta }}_u\right)}_x\right)\left(\frac{\&DifferentialD;\mathrm{\_\&Phi;}}{\&DifferentialD;u_x}\right)+\left(-{\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\right)\left(\frac{\&DifferentialD;\mathrm{\_\&Phi;}}{\&DifferentialD;v_x}\right)+\left(-{\left({\mathrm{\xi }}_x\right)}_u{u}_x{u}_t-{\left({\mathrm{\xi }}_x\right)}_v{u}_x{v}_t-{\left({\mathrm{\xi }}_t\right)}_u{u}_t^2-{\left({\mathrm{\xi }}_t\right)}_v{u}_t{v}_t-{\left({\mathrm{\xi }}_x\right)}_t{u}_x-{\left({\mathrm{\xi }}_t\right)}_t{u}_t+u_t{\left({\mathrm{\eta }}_u\right)}_u+v_t{\left({\mathrm{\eta }}_u\right)}_v+{\left({\mathrm{\eta }}_u\right)}_t\right)\left(\frac{\&DifferentialD;\mathrm{\_\&Phi;}}{\&DifferentialD;u_t}\right)+\left(-{\left({\mathrm{\xi }}_x\right)}_u{u}_t{v}_x-{\left({\mathrm{\xi }}_x\right)}_v{v}_x{v}_t-{\left({\mathrm{\xi }}_t\right)}_u{u}_t{v}_t-{\left({\mathrm{\xi }}_t\right)}_v{v}_t^2-{\left({\mathrm{\xi }}_x\right)}_t{v}_x-{\left({\mathrm{\xi }}_t\right)}_t{v}_t+u_t{\left({\mathrm{\eta }}_v\right)}_u+v_t{\left({\mathrm{\eta }}_v\right)}_v+{\left({\mathrm{\eta }}_v\right)}_t\right)\left(\frac{\&DifferentialD;\mathrm{\_\&Phi;}}{\&DifferentialD;v_t}\right)$

$\mathrm{value}\left(\mathrm{subs}\left(\mathrm{\_\&Phi;}=f\left(x\&comma;t\&comma;u\&comma;v\right)\&comma;\right)\right)$

${\mathrm{\xi }}_x{f}_x+{\mathrm{\xi }}_t{f}_t+{\mathrm{\eta }}_u{f}_u+{\mathrm{\eta }}_v{f}_v$

$\mathrm{InfinitesimalGenerator}\left(S\&comma;\mathrm{DepVars}\&comma;\mathrm{prolongation}=1\&comma;\mathrm{expanded}\&comma;\mathrm{output}=\mathrm{list}\right)$

$\left[{\mathrm{\xi }}_x\&comma;{\mathrm{\xi }}_t\&comma;{\mathrm{\eta }}_u\&comma;{\mathrm{\eta }}_v\&comma;-{\left({\mathrm{\xi }}_x\right)}_u{u}_x^2-{\left({\mathrm{\xi }}_x\right)}_v{u}_x{v}_x-{\left({\mathrm{\xi }}_t\right)}_u{u}_x{u}_t-{\left({\mathrm{\xi }}_t\right)}_v{u}_t{v}_x+u_x{\left({\mathrm{\eta }}_u\right)}_u+v_x{\left({\mathrm{\eta }}_u\right)}_v-{\left({\mathrm{\xi }}_x\right)}_x{u}_x-{\left({\mathrm{\xi }}_t\right)}_x{u}_t+{\left({\mathrm{\eta }}_u\right)}_x\&comma;-{\left({\mathrm{\xi }}_x\right)}_u{u}_x{v}_x-{\left({\mathrm{\xi }}_x\right)}_v{v}_x^2-{\left({\mathrm{\xi }}_t\right)}_u{u}_x{v}_t-{\left({\mathrm{\xi }}_t\right)}_v{v}_x{v}_t+u_x{\left({\mathrm{\eta }}_v\right)}_u+v_x{\left({\mathrm{\eta }}_v\right)}_v-{\left({\mathrm{\xi }}_x\right)}_x{v}_x-{\left({\mathrm{\xi }}_t\right)}_x{v}_t+{\left({\mathrm{\eta }}_v\right)}_x\&comma;-{\left({\mathrm{\xi }}_x\right)}_u{u}_x{u}_t-{\left({\mathrm{\xi }}_x\right)}_v{u}_x{v}_t-{\left({\mathrm{\xi }}_t\right)}_u{u}_t^2-{\left({\mathrm{\xi }}_t\right)}_v{u}_t{v}_t-{\left({\mathrm{\xi }}_x\right)}_t{u}_x-{\left({\mathrm{\xi }}_t\right)}_t{u}_t+u_t{\left({\mathrm{\eta }}_u\right)}_u+v_t{\left({\mathrm{\eta }}_u\right)}_v+{\left({\mathrm{\eta }}_u\right)}_t\&comma;-{\left({\mathrm{\xi }}_x\right)}_u{u}_t{v}_x-{\left({\mathrm{\xi }}_x\right)}_v{v}_x{v}_t-{\left({\mathrm{\xi }}_t\right)}_u{u}_t{v}_t-{\left({\mathrm{\xi }}_t\right)}_v{v}_t^2-{\left({\mathrm{\xi }}_x\right)}_t{v}_x-{\left({\mathrm{\xi }}_t\right)}_t{v}_t+u_t{\left({\mathrm{\eta }}_v\right)}_u+v_t{\left({\mathrm{\eta }}_v\right)}_v+{\left({\mathrm{\eta }}_v\right)}_t\right]$

$\mathrm{show}$

$\left[{\mathrm{\xi }}_x\left(x\&comma;t\&comma;u\&comma;v\right)\&comma;{\mathrm{\xi }}_t\left(x\&comma;t\&comma;u\&comma;v\right)\&comma;{\mathrm{\eta }}_u\left(x\&comma;t\&comma;u\&comma;v\right)\&comma;{\mathrm{\eta }}_v\left(x\&comma;t\&comma;u\&comma;v\right)\&comma;-\left(\frac{\partial }{\partial u}{\mathrm{\xi }}_x\left(x\&comma;t\&comma;u\&comma;v\right)\right)u_x^2-\left(\frac{\partial }{\partial v}{\mathrm{\xi }}_x\left(x\&comma;t\&comma;u\&comma;v\right)\right)u_x{v}_x-\left(\frac{\partial }{\partial u}{\mathrm{\xi }}_t\left(x\&comma;t\&comma;u\&comma;v\right)\right)u_x{u}_t-\left(\frac{\partial }{\partial v}{\mathrm{\xi }}_t\left(x\&comma;t\&comma;u\&comma;v\right)\right)u_t{v}_x+u_x\left(\frac{\partial }{\partial u}{\mathrm{\eta }}_u\left(x\&comma;t\&comma;u\&comma;v\right)\right)+v_x\left(\frac{\partial }{\partial v}{\mathrm{\eta }}_u\left(x\&comma;t\&comma;u\&comma;v\right)\right)-\left(\frac{\partial }{\partial x}{\mathrm{\xi }}_x\left(x\&comma;t\&comma;u\&comma;v\right)\right)u_x-\left(\frac{\partial }{\partial x}{\mathrm{\xi }}_t\left(x\&comma;t\&comma;u\&comma;v\right)\right)u_t+\frac{\partial }{\partial x}{\mathrm{\eta }}_u\left(x\&comma;t\&comma;u\&comma;v\right)\&comma;-\left(\frac{\partial }{\partial u}{\mathrm{\xi }}_x\left(x\&comma;t\&comma;u\&comma;v\right)\right)u_x{v}_x-\left(\frac{\partial }{\partial v}{\mathrm{\xi }}_x\left(x\&comma;t\&comma;u\&comma;v\right)\right)v_x^2-\left(\frac{\partial }{\partial u}{\mathrm{\xi }}_t\left(x\&comma;t\&comma;u\&comma;v\right)\right)u_x{v}_t-\left(\frac{\partial }{\partial v}{\mathrm{\xi }}_t\left(x\&comma;t\&comma;u\&comma;v\right)\right)v_x{v}_t+u_x\left(\frac{\partial }{\partial u}{\mathrm{\eta }}_v\left(x\&comma;t\&comma;u\&comma;v\right)\right)+v_x\left(\frac{\partial }{\partial v}{\mathrm{\eta }}_v\left(x\&comma;t\&comma;u\&comma;v\right)\right)-\left(\frac{\partial }{\partial x}{\mathrm{\xi }}_x\left(x\&comma;t\&comma;u\&comma;v\right)\right)v_x-\left(\frac{\partial }{\partial x}{\mathrm{\xi }}_t\left(x\&comma;t\&comma;u\&comma;v\right)\right)v_t+\frac{\partial }{\partial x}{\mathrm{\eta }}_v\left(x\&comma;t\&comma;u\&comma;v\right)\&comma;-\left(\frac{\partial }{\partial u}{\mathrm{\xi }}_x\left(x\&comma;t\&comma;u\&comma;v\right)\right)u_x{u}_t-\left(\frac{\partial }{\partial v}{\mathrm{\xi }}_x\left(x\&comma;t\&comma;u\&comma;v\right)\right)u_x{v}_t-\left(\frac{\partial }{\partial u}{\mathrm{\xi }}_t\left(x\&comma;t\&comma;u\&comma;v\right)\right)u_t^2-\left(\frac{\partial }{\partial v}{\mathrm{\xi }}_t\left(x\&comma;t\&comma;u\&comma;v\right)\right)u_t{v}_t-\left(\frac{\partial }{\partial t}{\mathrm{\xi }}_x\left(x\&comma;t\&comma;u\&comma;v\right)\right)u_x-\left(\frac{\partial }{\partial t}{\mathrm{\xi }}_t\left(x\&comma;t\&comma;u\&comma;v\right)\right)u_t+u_t\left(\frac{\partial }{\partial u}{\mathrm{\eta }}_u\left(x\&comma;t\&comma;u\&comma;v\right)\right)+v_t\left(\frac{\partial }{\partial v}{\mathrm{\eta }}_u\left(x\&comma;t\&comma;u\&comma;v\right)\right)+\frac{\partial }{\partial t}{\mathrm{\eta }}_u\left(x\&comma;t\&comma;u\&comma;v\right)\&comma;-\left(\frac{\partial }{\partial u}{\mathrm{\xi }}_x\left(x\&comma;t\&comma;u\&comma;v\right)\right)u_t{v}_x-\left(\frac{\partial }{\partial v}{\mathrm{\xi }}_x\left(x\&comma;t\&comma;u\&comma;v\right)\right)v_x{v}_t-\left(\frac{\partial }{\partial u}{\mathrm{\xi }}_t\left(x\&comma;t\&comma;u\&comma;v\right)\right)u_t{v}_t-\left(\frac{\partial }{\partial v}{\mathrm{\xi }}_t\left(x\&comma;t\&comma;u\&comma;v\right)\right)v_t^2-\left(\frac{\partial }{\partial t}{\mathrm{\xi }}_x\left(x\&comma;t\&comma;u\&comma;v\right)\right)v_x-\left(\frac{\partial }{\partial t}{\mathrm{\xi }}_t\left(x\&comma;t\&comma;u\&comma;v\right)\right)v_t+u_t\left(\frac{\partial }{\partial u}{\mathrm{\eta }}_v\left(x\&comma;t\&comma;u\&comma;v\right)\right)+v_t\left(\frac{\partial }{\partial v}{\mathrm{\eta }}_v\left(x\&comma;t\&comma;u\&comma;v\right)\right)+\frac{\partial }{\partial t}{\mathrm{\eta }}_v\left(x\&comma;t\&comma;u\&comma;v\right)\right]$

$\mathrm{InfinitesimalGenerator}\left(G\left[2\right]\&comma;\mathrm{DepVars}\&comma;\mathrm{expanded}\&comma;\mathrm{notation}=\mathrm{jetnumbers}\right)$

$f\&rarr;{\mathrm{\&xi;}}_x\left(x\&comma;t\&comma;u_{\&lsqb;\&rsqb;}\&comma;v_{\&lsqb;\&rsqb;}\right)\left(\frac{\partial }{\partial x}f\right)\&plus;{\mathrm{\&xi;}}_t\left(x\&comma;t\&comma;u_{\&lsqb;\&rsqb;}\&comma;v_{\&lsqb;\&rsqb;}\right)\left(\frac{\partial }{\partial t}f\right)\&plus;{\mathrm{\&eta;}}_u\left(x\&comma;t\&comma;u_{\&lsqb;\&rsqb;}\&comma;v_{\&lsqb;\&rsqb;}\right)\left(\frac{\partial }{\partial u_{\&lsqb;\&rsqb;}}f\right)\&plus;{\mathrm{\&eta;}}_v\left(x\&comma;t\&comma;u_{\&lsqb;\&rsqb;}\&comma;v_{\&lsqb;\&rsqb;}\right)\left(\frac{\partial }{\partial v_{\&lsqb;\&rsqb;}}f\right)\&plus;\left(-\left(\frac{\partial }{\partial u_{\&lsqb;\&rsqb;}}{\mathrm{\&xi;}}_x\left(x\&comma;t\&comma;u_{\&lsqb;\&rsqb;}\&comma;v_{\&lsqb;\&rsqb;}\right)\right)u_1^2-\left(\frac{\partial }{\partial v_{\&lsqb;\&rsqb;}}{\mathrm{\&xi;}}_x\left(x\&comma;t\&comma;u_{\&lsqb;\&rsqb;}\&comma;v_{\&lsqb;\&rsqb;}\right)\right)u_1{v}_1-\left(\frac{\partial }{\partial u_{\&lsqb;\&rsqb;}}{\mathrm{\&xi;}}_t\left(x\&comma;t\&comma;u_{\&lsqb;\&rsqb;}\&comma;v_{\&lsqb;\&rsqb;}\right)\right)u_1{u}_2-\left(\frac{\partial }{\partial v_{\&lsqb;\&rsqb;}}{\mathrm{\&xi;}}_t\left(x\&comma;t\&comma;u_{\&lsqb;\&rsqb;}\&comma;v_{\&lsqb;\&rsqb;}\right)\right)u_2{v}_1\&plus;u_1\left(\frac{\partial }{\partial u_{\&lsqb;\&rsqb;}}{\mathrm{\&eta;}}_u\left(x\&comma;t\&comma;u_{\&lsqb;\&rsqb;}\&comma;v_{\&lsqb;\&rsqb;}\right)\right)\&plus;v_1\left(\frac{\partial }{\partial v_{\&lsqb;\&rsqb;}}{\mathrm{\&eta;}}_u\left(x\&comma;t\&comma;u_{\&lsqb;\&rsqb;}\&comma;v_{\&lsqb;\&rsqb;}\right)\right)-\left(\frac{\partial }{\partial x}{\mathrm{\&xi;}}_x\left(x\&comma;t\&comma;u_{\&lsqb;\&rsqb;}\&comma;v_{\&lsqb;\&rsqb;}\right)\right)u_1-\left(\frac{\partial }{\partial x}{\mathrm{\&xi;}}_t\left(x\&comma;t\&comma;u_{\&lsqb;\&rsqb;}\&comma;v_{\&lsqb;\&rsqb;}\right)\right)u_2\&plus;\frac{\partial }{\partial x}{\mathrm{\&eta;}}_u\left(x\&comma;t\&comma;u_{\&lsqb;\&rsqb;}\&comma;v_{\&lsqb;\&rsqb;}\right)\right)\left(\frac{\partial }{\partial u_1}f\right)\&plus;\left(-\left(\frac{\partial }{\partial u_{\&lsqb;\&rsqb;}}{\mathrm{\&xi;}}_x\left(x\&comma;t\&comma;u_{\&lsqb;\&rsqb;}\&comma;v_{\&lsqb;\&rsqb;}\right)\right)u_1{v}_1-\left(\frac{\partial }{\partial v_{\&lsqb;\&rsqb;}}{\mathrm{\&xi;}}_x\left(x\&comma;t\&comma;u_{\&lsqb;\&rsqb;}\&comma;v_{\&lsqb;\&rsqb;}\right)\right)v_1^2-\left(\frac{\partial }{\partial u_{\&lsqb;\&rsqb;}}{\mathrm{\&xi;}}_t\left(x\&comma;t\&comma;u_{\&lsqb;\&rsqb;}\&comma;v_{\&lsqb;\&rsqb;}\right)\right)u_1{v}_2-\left(\frac{\partial }{\partial v_{\&lsqb;\&rsqb;}}{\mathrm{\&xi;}}_t\left(x\&comma;t\&comma;u_{\&lsqb;\&rsqb;}\&comma;v_{\&lsqb;\&rsqb;}\right)\right)v_1{v}_2\&plus;u_1\left(\frac{\partial }{\partial u_{\&lsqb;\&rsqb;}}{\mathrm{\&eta;}}_v\left(x\&comma;t\&comma;u_{\&lsqb;\&rsqb;}\&comma;v_{\&lsqb;\&rsqb;}\right)\right)\&plus;v_1\left(\frac{\partial }{\partial v_{\&lsqb;\&rsqb;}}{\mathrm{\&eta;}}_v\left(x\&comma;t\&comma;u_{\&lsqb;\&rsqb;}\&comma;v_{\&lsqb;\&rsqb;}\right)\right)-\left(\frac{\partial }{\partial x}{\mathrm{\&xi;}}_x\left(x\&comma;t\&comma;u_{\&lsqb;\&rsqb;}\&comma;v_{\&lsqb;\&rsqb;}\right)\right)v_1-\left(\frac{\partial }{\partial x}{\mathrm{\&xi;}}_t\left(x\&comma;t\&comma;u_{\&lsqb;\&rsqb;}\&comma;v_{\&lsqb;\&rsqb;}\right)\right)v_2\&plus;\frac{\partial }{\partial x}{\mathrm{\&eta;}}_v\left(x\&comma;t\&comma;u_{\&lsqb;\&rsqb;}\&comma;v_{\&lsqb;\&rsqb;}\right)\right)\left(\frac{\partial }{\partial v_1}f\right)\&plus;\left(-\left(\frac{\partial }{\partial u_{\&lsqb;\&rsqb;}}{\mathrm{\&xi;}}_x\left(x\&comma;t\&comma;u_{\&lsqb;\&rsqb;}\&comma;v_{\&lsqb;\&rsqb;}\right)\right)u_1{u}_2-\left(\frac{\partial }{\partial v_{\&lsqb;\&rsqb;}}{\mathrm{\&xi;}}_x\left(x\&comma;t\&comma;u_{\&lsqb;\&rsqb;}\&comma;v_{\&lsqb;\&rsqb;}\right)\right)u_1{v}_2-\left(\frac{\partial }{\partial u_{\&lsqb;\&rsqb;}}{\mathrm{\&xi;}}_t\left(x\&comma;t\&comma;u_{\&lsqb;\&rsqb;}\&comma;v_{\&lsqb;\&rsqb;}\right)\right)u_2^2-\left(\frac{\partial }{\partial v_{\&lsqb;\&rsqb;}}{\mathrm{\&xi;}}_t\left(x\&comma;t\&comma;u_{\&lsqb;\&rsqb;}\&comma;v_{\&lsqb;\&rsqb;}\right)\right)u_2{v}_2-\left(\frac{\partial }{\partial t}{\mathrm{\&xi;}}_x\left(x\&comma;t\&comma;u_{\&lsqb;\&rsqb;}\&comma;v_{\&lsqb;\&rsqb;}\right)\right)u_1-\left(\frac{\partial }{\partial t}{\mathrm{\&xi;}}_t\left(x\&comma;t\&comma;u_{\&lsqb;\&rsqb;}\&comma;v_{\&lsqb;\&rsqb;}\right)\right)u_2\&plus;u_2\left(\frac{\partial }{\partial u_{\&lsqb;\&rsqb;}}{\mathrm{\&eta;}}_u\left(x\&comma;t\&comma;u_{\&lsqb;\&rsqb;}\&comma;v_{\&lsqb;\&rsqb;}\right)\right)\&plus;v_2\left(\frac{\partial }{\partial v_{\&lsqb;\&rsqb;}}{\mathrm{\&eta;}}_u\left(x\&comma;t\&comma;u_{\&lsqb;\&rsqb;}\&comma;v_{\&lsqb;\&rsqb;}\right)\right)\&plus;\frac{\partial }{\partial t}{\mathrm{\&eta;}}_u\left(x\&comma;t\&comma;u_{\&lsqb;\&rsqb;}\&comma;v_{\&lsqb;\&rsqb;}\right)\right)\left(\frac{\partial }{\partial u_2}f\right)\&plus;\left(-\left(\frac{\partial }{\partial u_{\&lsqb;\&rsqb;}}{\mathrm{\&xi;}}_x\left(x\&comma;t\&comma;u_{\&lsqb;\&rsqb;}\&comma;v_{\&lsqb;\&rsqb;}\right)\right)u_2{v}_1-\left(\frac{\partial }{\partial v_{\&lsqb;\&rsqb;}}{\mathrm{\&xi;}}_x\left(x\&comma;t\&comma;u_{\&lsqb;\&rsqb;}\&comma;v_{\&lsqb;\&rsqb;}\right)\right)v_1{v}_2-\left(\frac{\partial }{\partial u_{\&lsqb;\&rsqb;}}{\mathrm{\&xi;}}_t\left(x\&comma;t\&comma;u_{\&lsqb;\&rsqb;}\&comma;v_{\&lsqb;\&rsqb;}\right)\right)u_2{v}_2-\left(\frac{\partial }{\partial v_{\&lsqb;\&rsqb;}}{\mathrm{\&xi;}}_t\left(x\&comma;t\&comma;u_{\&lsqb;\&rsqb;}\&comma;v_{\&lsqb;\&rsqb;}\right)\right)v_2^2-\left(\frac{\partial }{\partial t}{\mathrm{\&xi;}}_x\left(x\&comma;t\&comma;u_{\&lsqb;\&rsqb;}\&comma;v_{\&lsqb;\&rsqb;}\right)\right)v_1-\left(\frac{\partial }{\partial t}{\mathrm{\&xi;}}_t\left(x\&comma;t\&comma;u_{\&lsqb;\&rsqb;}\&comma;v_{\&lsqb;\&rsqb;}\right)\right)v_2\&plus;u_2\left(\frac{\partial }{\partial u_{\&lsqb;\&rsqb;}}{\mathrm{\&eta;}}_v\left(x\&comma;t\&comma;u_{\&lsqb;\&rsqb;}\&comma;v_{\&lsqb;\&rsqb;}\right)\right)\&plus;v_2\left(\frac{\partial }{\partial v_{\&lsqb;\&rsqb;}}{\mathrm{\&eta;}}_v\left(x\&comma;t\&comma;u_{\&lsqb;\&rsqb;}\&comma;v_{\&lsqb;\&rsqb;}\right)\right)\&plus;\frac{\partial }{\partial t}{\mathrm{\&eta;}}_v\left(x\&comma;t\&comma;u_{\&lsqb;\&rsqb;}\&comma;v_{\&lsqb;\&rsqb;}\right)\right)\left(\frac{\partial }{\partial v_2}f\right)$

$\mathrm{with}\left(\mathrm{Physics}\right)$

$\left[\mathrm{`*`}\&comma;\mathrm{`.`}\&comma;\mathrm{Annihilation}\&comma;\mathrm{AntiCommutator}\&comma;\mathrm{Antisymmetrize}\&comma;\mathrm{Assume}\&comma;\mathrm{Bra}\&comma;\mathrm{Bracket}\&comma;\mathrm{Check}\&comma;\mathrm{Christoffel}\&comma;\mathrm{Coefficients}\&comma;\mathrm{Commutator}\&comma;\mathrm{CompactDisplay}\&comma;\mathrm{Coordinates}\&comma;\mathrm{Creation}\&comma;\mathrm{D\_}\&comma;\mathrm{Dagger}\&comma;\mathrm{Decompose}\&comma;\mathrm{Define}\&comma;\mathrm{D\&gamma;}\&comma;\mathrm{DiracConjugate}\&comma;\mathrm{Einstein}\&comma;\mathrm{EnergyMomentum}\&comma;\mathrm{Expand}\&comma;\mathrm{ExteriorDerivative}\&comma;\mathrm{Factor}\&comma;\mathrm{FeynmanDiagrams}\&comma;\mathrm{FeynmanIntegral}\&comma;\mathrm{Fundiff}\&comma;\mathrm{Geodesics}\&comma;\mathrm{GrassmannParity}\&comma;\mathrm{Gtaylor}\&comma;\mathrm{Intc}\&comma;\mathrm{Inverse}\&comma;\mathrm{Ket}\&comma;\mathrm{KillingVectors}\&comma;\mathrm{KroneckerDelta}\&comma;\mathrm{LagrangeEquations}\&comma;\mathrm{LeviCivita}\&comma;\mathrm{Library}\&comma;\mathrm{LieBracket}\&comma;\mathrm{LieDerivative}\&comma;\mathrm{Normal}\&comma;\mathrm{NumericalRelativity}\&comma;\mathrm{Parameters}\&comma;\mathrm{PerformOnAnticommutativeSystem}\&comma;\mathrm{Projector}\&comma;\mathrm{Psigma}\&comma;\mathrm{Redefine}\&comma;\mathrm{Ricci}\&comma;\mathrm{Riemann}\&comma;\mathrm{Setup}\&comma;\mathrm{Simplify}\&comma;\mathrm{SortProducts}\&comma;\mathrm{SpaceTimeVector}\&comma;\mathrm{StandardModel}\&comma;\mathrm{Substitute}\&comma;\mathrm{SubstituteTensor}\&comma;\mathrm{SubstituteTensorIndices}\&comma;\mathrm{SumOverRepeatedIndices}\&comma;\mathrm{Symmetrize}\&comma;\mathrm{TensorArray}\&comma;\mathrm{Tetrads}\&comma;\mathrm{ThreePlusOne}\&comma;\mathrm{ToContravariant}\&comma;\mathrm{ToCovariant}\&comma;\mathrm{ToFieldComponents}\&comma;\mathrm{ToSuperfields}\&comma;\mathrm{Trace}\&comma;\mathrm{TransformCoordinates}\&comma;\mathrm{Vectors}\&comma;\mathrm{Weyl}\&comma;\mathrm{`^`}\&comma;\mathrm{dAlembertian}\&comma;\mathrm{d\_}\&comma;\mathrm{diff}\&comma;\mathrm{g\_}\&comma;\mathrm{gamma\_}\right]$