The command LAVF(...) is for constructing a LAVF object. A valid LAVF object then has access to at least 60 methods which allow it to be manipulated and its contents queried. For more detail, see Overview of the LAVF object.

In the first calling sequence, the input argument vf must be a type VectorField whose components are indeterminant functionals (as infinitesimals), and dq must be a type LHPDE object whose dependent variables include all components of vf.

For convenience the second calling sequence is a special constructor for either a trivial LAVF object or a universal LAVF object. A trivial LAVF object means its determining system is trivial (i.e. only the zero solution). For example, let V be a VectorField object containing indeterminant infinitesimals, a call LAVF(V,"trivial") is equal to the call LAVF(V, LHPDE("trivial", dep = GetComponents(V), indep = GetSpace(V))). And a universal LAVF object has empty system (i.e. no restriction on solutions).

This command is part of the LieAlgebrasOfVectorFields package. For more detail, see Overview of the LieAlgebrasOfVectorFields package.

This command can be used in the form LAVF(...) only after executing the command with(LieAlgebrasOfVectorFields), but can always be used in the form :-LieAlgebrasOfVectorFields:-LAVF(...).

$\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right)\:$

$\mathrm{Typesetting}:-\mathrm{Settings}\left(\mathrm{userep}=\mathrm{true}\right)\:$

$\mathrm{Typesetting}:-\mathrm{Suppress}\left(\left[\mathrm{\xi }\left(x\,y\right)\,\mathrm{\eta }\left(x\,y\right)\right]\right)\:$

$V≔\mathrm{VectorField}\left(\left[\left[\mathrm{\xi }\left(x\,y\right)\,x\right]\,\left[\mathrm{\eta }\left(x\,y\right)\,y\right]\right]\,\mathrm{space}=\left[x\,y\right]\right)$

$V≔\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)$

$\mathrm{E2Sys}≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\,y\right)=0\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,x\right)+\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\right)=0\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,y\right)=0\,\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,x\right)=0\right]\,\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]\right)$

$\mathrm{E2Sys}≔\left[{\mathrm{\xi }}_{y,y}=0\,{\mathrm{\eta }}_x+{\mathrm{\xi }}_y=0\,{\mathrm{\eta }}_y=0\,{\mathrm{\xi }}_x=0\right],\mathrm{indep}=\left[x\,y\right],\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]$

$\mathrm{E2}≔\mathrm{LAVF}\left(V\,\mathrm{E2Sys}\right)$

$\mathrm{E2}≔\left[\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)\right]\&where\left\{\left[{\mathrm{\xi }}_{y,y}=0\,{\mathrm{\xi }}_x=0\,{\mathrm{\eta }}_x=-{\mathrm{\xi }}_y\,{\mathrm{\eta }}_y=0\right]\right\}$

$\mathrm{GetVectorField}\left(\mathrm{E2}\right)$

$\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)$

$\mathrm{GetDeterminingSystem}\left(\mathrm{E2}\right)$

$\left[{\mathrm{\xi }}_{y,y}=0\,{\mathrm{\xi }}_x=0\,{\mathrm{\eta }}_x=-{\mathrm{\xi }}_y\,{\mathrm{\eta }}_y=0\right],\mathrm{indep}=\left[x\,y\right],\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]$

$\mathrm{exports}\left(\mathrm{E2}\,'\mathrm{static}'\right)$

$\mathrm{indets},\mathrm{has},\mathrm{hastype},\mathrm{type},\mathrm{GetVectorField},\mathrm{GetDeterminingSystem},\mathrm{ImplicitForm},\mathrm{SolutionDimension},\mathrm{IsFiniteType},\mathrm{IsTrivial},\mathrm{ParametricDerivatives},\mathrm{GetRanking},\mathrm{SetIDBasis},\mathrm{GetIDBasis},\mathrm{GetSpace},\mathrm{IsFlat},\mathrm{OrbitDistribution},\mathrm{OrbitDimension},\mathrm{InvariantCount},\mathrm{IsTransitive},\mathrm{Invariants},\mathrm{IsLieAlgebra},\mathrm{IsPerfect},\mathrm{DerivedAlgebra},\mathrm{IsSolvable},\mathrm{IsSoluble},\mathrm{DerivedSeries},\mathrm{SolvableRadical},\mathrm{SolubleRadical},\mathrm{Radical},\mathrm{IsNilpotent},\mathrm{Hypercentre},\mathrm{Hypercenter},\mathrm{NilRadical},\mathrm{Nilradical},\mathrm{LowerCentralSeries},\mathrm{UpperCentralSeries},\mathrm{IsAbelian},\mathrm{IsCommutative},\mathrm{Centre},\mathrm{Center},\mathrm{IsSemiSimple},\mathrm{IsReductive},\mathrm{NilpotentRadical},\mathrm{StructureConstants},\mathrm{StructureCoefficients},\mathrm{KillingRadical},\mathrm{KillingPolynomial},\mathrm{KillingForm},\mathrm{KillingOrthogonal},\mathrm{AdjointMatrix},\mathrm{AreCommuting},\mathrm{AreSame},\mathrm{AreSameSpace},\mathrm{Centraliser},\mathrm{Centralizer},\mathrm{Normaliser},\mathrm{CleanDependencies},\mathrm{Copy},\mathrm{DChange},\mathrm{dchange},\mathrm{Intersection},\mathrm{IsIdeal},\mathrm{IsInvariant},\mathrm{IsotropyRepresentation},\mathrm{IsSubspace},\mathrm{LAVFSolve},\mathrm{VectorSpaceSum},\mathrm{LieProduct},\mathrm{ProjectToSpace},\mathrm{Transporter},\mathrm{ModuleCopy},\mathrm{ModulePrint},\mathrm{ModuleApply}$

$\mathrm{LAVF}\left(V\,''trivial''\right)$

$\left[\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)\right]\&where\left\{\left[\mathrm{\xi }=0\,\mathrm{\eta }=0\right]\right\}$

$\mathrm{LAVF}\left(V\,''universal''\right)$

$\left[\mathrm{\xi }\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(\frac{\ⅆ}{\ⅆy}\right)\right]\&where\left\{\left[\right]\right\}$

The LieAlgebrasOfVectorFields[LAVF] command was introduced in Maple 2020.

For more information on Maple 2020 changes, see Updates in Maple 2020.