Let S be a LHPDE object of finite type whose solution dimension is r. The IDBasis command constructs an IDBasis object representing a basis for the initial data of S.

The IDBasis object is managed with respect to a fixed standard basis, namely the parametric derivatives of the LHPDE object S.

The second input argument, Vs, represents a change-of-basis matrix to give a new basis.

If Vs is a list of Vectors, then there must be r Vectors of length r in the list.

If Vs is a Matrix, then it must be r x r invertible matrix.

If Vs is a list of linear combination of parametric derivatives of S, then there must be r items in the list.

The call IDBasis(S, "standardBasis") returns a IDBasis object containing the standard initial data basis (i.e. the change-of-basis matrix is the identity matrix).

Some methods become available once a valid IDBasis object is constructed. See Overview of the IDBasis object for more detail.

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 IDBasis(...) only after executing the command with(LieAlgebrasOfVectorFields), but can always be used by executing LieAlgebrasOfVectorFields:-IDBasis(...).

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

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

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

$\mathrm{E2}≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\,y\right)=0\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,x\right)=-\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\right)\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,y\right)=0\,\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,x\right)=0\right]\,\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]\,\mathrm{indep}=\left[x\,y\right]\right)$

$\mathrm{E2}≔\left[{\mathrm{\xi }}_{y,y}=0\,{\mathrm{\eta }}_x=-{\mathrm{\xi }}_y\,{\mathrm{\eta }}_y=0\,{\mathrm{\xi }}_x=0\right],\mathrm{indep}=\left[x\,y\right],\mathrm{dep}=\left[\mathrm{\xi }\,\mathrm{\eta }\right]$

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

$\left[\mathrm{\xi }\,{\mathrm{\xi }}_y\,\mathrm{\eta }\right]$

$B≔\mathrm{IDBasis}\left(\mathrm{E2}\,\left[⟨1\,0\,0⟩\,⟨-y\,-x\,-1⟩\,⟨0\,1\,0⟩\right]\right)$

$B≔\left[\mathrm{\xi }-y\left({\mathrm{\xi }}_y\right)\,-x\left({\mathrm{\xi }}_y\right)+\mathrm{\eta }\,-{\mathrm{\xi }}_y\right]$

$\mathrm{type}\left(B\,'\mathrm{list}'\right)$

$\mathrm{false}$

$\mathrm{type}\left(B\,'\mathrm{IDBasis}'\right)$

$\mathrm{true}$

$\mathrm{IDBasis}\left(\mathrm{E2}\,\mathrm{Matrix}\left(\left[\left[1\,-y\,0\right]\,\left[0\,-x\,1\right]\,\left[0\,-1\,0\right]\right]\right)\right)$

$\left[\mathrm{\xi }-y\left({\mathrm{\xi }}_y\right)\,-x\left({\mathrm{\xi }}_y\right)+\mathrm{\eta }\,-{\mathrm{\xi }}_y\right]$

$\mathrm{IDBasis}\left(\mathrm{E2}\,\left[\mathrm{\xi }\left(x\,y\right)-y\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\right)\,\mathrm{\eta }\left(x\,y\right)-x\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\right)\,-\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,y\right)\right]\right)$

$\left[\mathrm{\xi }-y\left({\mathrm{\xi }}_y\right)\,-x\left({\mathrm{\xi }}_y\right)+\mathrm{\eta }\,-{\mathrm{\xi }}_y\right]$

$\mathrm{IDBasis}\left(\mathrm{E2}\,''standardBasis''\right)$

$\left[\mathrm{\xi }\,{\mathrm{\xi }}_y\,\mathrm{\eta }\right]$

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

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