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LieAlgebrasOfVectorFields

 VFPDO
 construct a VFPDO object

 Calling Sequence VFPDO( trivial, V) VFPDO( p, V ) VFPDO( L)

Parameters

 V - an VectorField object L - an LAVF object p - a list or set of linear homogeneous differential expressions or a LHPDE object whose dependent variables are the components of the VectorField V.

Description

 • The command VFPDO(...) is for constructing a VFPDO object. It returns a VFPDO object if successful. A valid VFPDO object has access to various methods which allow it to be manipulated and its contents queried. For more detail, see Overview of the VFPDO object.
 • The VFPDO constructor command works by taking a list of linear homogeneous differential expressions and return essentially the same thing, but as a differential operator on the vector field V.
 • In the first calling sequence VFPDO("trivial", V) returns the zero VFPDO operator on the vector field V.
 • In the second calling sequence, the first input argument is a list or set of scalar expressions or equations, or alternatively a table as returned by DEtools[rifsimp] or a LHPDE object, and their dependant variables must be components of the vector field V.
 • In the third sequence, it builds a VFPDO from LAVF L. Note here if the L is integrated, the VFPDO will be built from the implicit form of L.
 • 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 VFPDO(...) only after executing the command with(LieAlgebrasOfVectorFields), but can always be used in the form LieAlgebrasOfVectorFields:-VFPDO(...).

Examples

 > $\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right):$
 > $X≔\mathrm{VectorField}\left(\mathrm{\xi }\left(x,y\right)\mathrm{D}\left[x\right]+\mathrm{\eta }\left(x,y\right)\mathrm{D}\left[y\right],\mathrm{space}=\left[x,y\right]\right)$
 ${X}{≔}{\mathrm{\xi }}{}\left({x}{,}{y}\right){}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\left({x}{,}{y}\right){}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (1)
 > $S≔\mathrm{LHPDE}\left(\left[\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),x\right)=0,\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y,y\right)=0,\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),x\right)=-\mathrm{diff}\left(\mathrm{\xi }\left(x,y\right),y\right),\mathrm{diff}\left(\mathrm{\eta }\left(x,y\right),y\right)=0\right],\mathrm{indep}=\left[x,y\right],\mathrm{dep}=\left[\mathrm{\xi },\mathrm{\eta }\right]\right)$
 ${S}{≔}\left[\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){=}{0}{,}\frac{{{\partial }}^{{2}}}{{\partial }{{y}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){=}{0}{,}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}{,}{y}\right){=}{-}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){,}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}{,}{y}\right){=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{}\left({x}{,}{y}\right){,}{\mathrm{\eta }}{}\left({x}{,}{y}\right)\right]$ (2)
 > $L≔\mathrm{LAVF}\left(X,S\right)$
 ${L}{≔}\left[{\mathrm{\xi }}{}\left({x}{,}{y}\right){}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{\mathrm{\eta }}{}\left({x}{,}{y}\right){}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[\frac{{{\partial }}^{{2}}}{{\partial }{{y}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){=}{0}{,}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){=}{0}{,}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}{,}{y}\right){=}{-}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){,}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}{,}{y}\right){=}{0}\right]\right\}$ (3)

First we build a trivial VFPDO

 > $\mathrm{VFPDO}\left("trivial",X\right)$
 ${X}{↦}\left[{X}{}\left({x}\right){,}{X}{}\left({y}\right)\right]$ (4)

or we can build a VFPDO directly from a LAVF..

 > $\mathrm{VFPDO}\left(L\right)$
 ${X}{→}\left[\frac{{\partial }}{{\partial }{y}}{}\left(\frac{{\partial }}{{\partial }{y}}{}{X}{}\left({x}\right)\right){,}\frac{{ⅆ}}{{ⅆ}{x}}{}{X}{}\left({x}\right){,}\frac{{\partial }}{{\partial }{x}}{}{X}{}\left({y}\right){+}\frac{{\partial }}{{\partial }{y}}{}{X}{}\left({x}\right){,}\frac{{ⅆ}}{{ⅆ}{y}}{}{X}{}\left({y}\right)\right]$ (5)

Compatibility

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