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LieAlgebrasOfVectorFields

 IDBasis
 construct an IDBasis object

 Calling Sequence IDBasis(S, Vs)

Parameters

 S - a LHPDE object that is of finite type (see IsFiniteType) Vs - a list of Vectors, a Matrix, a list of linear combination of parametric derivatives of S, or the string "standardBasis", representing a change-of-basis for the initial data of S

Description

 • 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(...).

Examples

 > $\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right):$
 > $\mathrm{Typesetting}:-\mathrm{Suppress}\left(\left[\mathrm{\xi }\left(x,y\right),\mathrm{\eta }\left(x,y\right)\right]\right):$
 > $\mathrm{Typesetting}:-\mathrm{Settings}\left(\mathrm{userep}=\mathrm{true}\right):$
 > $\mathrm{E2}≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y,y\right)=0,\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),x\right)=-\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right),\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),y\right)=0,\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),x\right)=0\right],\mathrm{dep}=\left[\mathrm{\xi },\mathrm{\eta }\right],\mathrm{indep}=\left[x,y\right]\right)$
 ${\mathrm{E2}}{≔}\left[{{\mathrm{\xi }}}_{{y}{,}{y}}{=}{0}{,}{{\mathrm{\eta }}}_{{x}}{=}{-}{{\mathrm{\xi }}}_{{y}}{,}{{\mathrm{\eta }}}_{{y}}{=}{0}{,}{{\mathrm{\xi }}}_{{x}}{=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{,}{\mathrm{\eta }}\right]$ (1)
 > $\mathrm{ParametricDerivatives}\left(\mathrm{E2}\right)$
 $\left[{\mathrm{\xi }}{,}{{\mathrm{\xi }}}_{{y}}{,}{\mathrm{\eta }}\right]$ (2)

The solution dimension of E2 is 3, so we must have three vectors of length 3.

 > $B≔\mathrm{IDBasis}\left(\mathrm{E2},\left[⟨1,0,0⟩,⟨-y,-x,-1⟩,⟨0,1,0⟩\right]\right)$
 ${B}{≔}\left[{\mathrm{\xi }}{-}{y}{}\left({{\mathrm{\xi }}}_{{y}}\right){,}{-}{x}{}\left({{\mathrm{\xi }}}_{{y}}\right){+}{\mathrm{\eta }}{,}{-}{{\mathrm{\xi }}}_{{y}}\right]$ (3)

B looks like a list but it is really an IDBasis object.

 > $\mathrm{type}\left(B,'\mathrm{list}'\right)$
 ${\mathrm{false}}$ (4)
 > $\mathrm{type}\left(B,'\mathrm{IDBasis}'\right)$
 ${\mathrm{true}}$ (5)

The IDBasis can be constructed from a change-of-basis 3 by 3 matrix

 > $\mathrm{IDBasis}\left(\mathrm{E2},\mathrm{Matrix}\left(\left[\left[1,-y,0\right],\left[0,-x,1\right],\left[0,-1,0\right]\right]\right)\right)$
 $\left[{\mathrm{\xi }}{-}{y}{}\left({{\mathrm{\xi }}}_{{y}}\right){,}{-}{x}{}\left({{\mathrm{\xi }}}_{{y}}\right){+}{\mathrm{\eta }}{,}{-}{{\mathrm{\xi }}}_{{y}}\right]$ (6)

... or can be constructed from three linear combinations of the parametric derivatives of E2

 > $\mathrm{IDBasis}\left(\mathrm{E2},\left[\mathrm{\xi }\left(x,y\right)-y\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right),\mathrm{\eta }\left(x,y\right)-x\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right),-\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right)\right]\right)$
 $\left[{\mathrm{\xi }}{-}{y}{}\left({{\mathrm{\xi }}}_{{y}}\right){,}{-}{x}{}\left({{\mathrm{\xi }}}_{{y}}\right){+}{\mathrm{\eta }}{,}{-}{{\mathrm{\xi }}}_{{y}}\right]$ (7)

Or we can construct a standard IDBasis object whose change-of-basis is the identity matrix.

 > $\mathrm{IDBasis}\left(\mathrm{E2},"standardBasis"\right)$
 $\left[{\mathrm{\xi }}{,}{{\mathrm{\xi }}}_{{y}}{,}{\mathrm{\eta }}\right]$ (8)

Compatibility

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