GetChangeBasis - Maple Help

GetChangeBasis

get the change of basis for an IDBasis object

 Calling Sequence GetChangeBasis( B, output = str) GetChangeBasis( B, "newToOld")

Parameters

 B - IDBasis object str - (optional) a string: either "matrix" or "expression"

Description

 • Let B be an IDBasis object associated with a LHPDE object S of finite type. The GetChangeBasis method returns the change-of-basis for initial data of S, as recorded in B.
 • By default, the method returns a list of expressions which are linear combinations of the standard initial data basis (i.e. of the parametric derivatives from the associated LHPDE object S).
 • The method can return the change-of-basis as a matrix by specifying the optional argument output = "matrix".
 • In the second calling sequence, the call GetChangeBasis(B, "newToOld") returns the inverse of this change-of-basis matrix.
 • This method is associated with the IDBasis object. For more detail, see Overview of the IDBasis object.

Examples

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

The parametric derivatives of E2 are used as the standard initial data basis.

 > $\mathrm{ParametricDerivatives}\left(\mathrm{E2}\right)$
 $\left[{\mathrm{\xi }}{,}{{\mathrm{\xi }}}_{{y}}{,}{\mathrm{\eta }}\right]$ (2)

The IDBasis object B is constructed by specifying linear combinations of the standard initial data basis (i.e. of the parametric derivatives of E2).

 > $B≔\mathrm{IDBasis}\left(\mathrm{E2},\left[\mathrm{\xi }\left(x,y\right)-y\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right),\mathrm{\eta }\left(x,y\right)-x\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right),-\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right)\right]\right)$
 ${B}{≔}\left[{\mathrm{\xi }}{-}{y}{}\left({{\mathrm{\xi }}}_{{y}}\right){,}{\mathrm{\eta }}{-}{x}{}\left({{\mathrm{\xi }}}_{{y}}\right){,}{-}{{\mathrm{\xi }}}_{{y}}\right]$ (3)
 > $\mathrm{GetChangeBasis}\left(B\right)$
 $\left[{\mathrm{\xi }}{-}{y}{}\left({{\mathrm{\xi }}}_{{y}}\right){,}{\mathrm{\eta }}{-}{x}{}\left({{\mathrm{\xi }}}_{{y}}\right){,}{-}{{\mathrm{\xi }}}_{{y}}\right]$ (4)

The change-of-basis matrix and the standard basis elements.

 > $\mathrm{GetChangeBasis}\left(B,'\mathrm{output}'="matrix"\right)$
 $\left[\begin{array}{ccc}{1}& {-}{y}& {0}\\ {0}& {-}{x}& {1}\\ {0}& {-1}& {0}\end{array}\right]$ (5)
 > $\mathrm{GetParametricDerivatives}\left(B\right)$
 $\left[{\mathrm{\xi }}{,}{{\mathrm{\xi }}}_{{y}}{,}{\mathrm{\eta }}\right]$ (6)

The inverse matrix of the change-of-basis from B.

 > $\mathrm{GetChangeBasis}\left(B,"newToOld"\right)$
 $\left[\begin{array}{ccc}{1}& {0}& {-}{y}\\ {0}& {0}& {-1}\\ {0}& {1}& {-}{x}\end{array}\right]$ (7)

Compatibility

 • The GetChangeBasis command was introduced in Maple 2020.