AsOperator - Maple Help
For the best experience, we recommend viewing online help using Google Chrome or Microsoft Edge.

VFPDO Object as Operator

 Calling Sequence Delta( obj )

Parameters

 Delta - a VFPDO object obj - a VectorField object

Description

 • An VFPDO object is appliable, and can act as partial differential operator.
 • An VFPDO object Delta is a function that acts on a vector field. See example below.
 • These methods are associated with the VFPDO object. For more detail, see Overview of the VFPDO object.

Examples

 > $\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right):$
 > $X≔\mathrm{VectorField}\left(\mathrm{ξ}\left(x,y\right){\mathrm{D}}_{x}+\mathrm{η}\left(x,y\right){\mathrm{D}}_{y},\mathrm{space}=\left[x,y\right]\right)$
 ${X}{≔}{\mathrm{\xi }}{}\left({x}{,}{y}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right){+}{\mathrm{\eta }}{}\left({x}{,}{y}\right){}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right)$ (1)
 > $S≔\mathrm{LHPDE}\left(\left[\frac{\partial }{\partial x}\mathrm{ξ}\left(x,y\right)=0,\frac{{\partial }^{2}}{\partial {y}^{2}}\mathrm{ξ}\left(x,y\right)=0,\frac{\partial }{\partial x}\mathrm{η}\left(x,y\right)=-\left(\frac{\partial }{\partial y}\mathrm{ξ}\left(x,y\right)\right),\frac{\partial }{\partial y}\mathrm{η}\left(x,y\right)=0\right],\mathrm{indep}=\left[x,y\right],\mathrm{dep}=\left[\mathrm{ξ},\mathrm{η}\right]\right)$
 ${S}{≔}\left[\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){=}{0}{,}\frac{{{\partial }}^{{2}}}{{\partial }{{y}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){=}{0}{,}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}{,}{y}\right){=}{-}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){,}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}{,}{y}\right){=}{0}\right]{,}{\mathrm{indep}}{=}\left[{x}{,}{y}\right]{,}{\mathrm{dep}}{=}\left[{\mathrm{\xi }}{}\left({x}{,}{y}\right){,}{\mathrm{\eta }}{}\left({x}{,}{y}\right)\right]$ (2)
 > $L≔\mathrm{LAVF}\left(X,S\right)$
 ${L}{≔}\left[{\mathrm{\xi }}{}\left({x}{,}{y}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right){+}{\mathrm{\eta }}{}\left({x}{,}{y}\right){}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right)\right]\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{&where}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left\{\left[\frac{{{\partial }}^{{2}}}{{\partial }{{y}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){=}{0}{,}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){=}{0}{,}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}{,}{y}\right){=}{-}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){,}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}{,}{y}\right){=}{0}\right]\right\}$ (3)
 > $\mathrm{Δ}≔\mathrm{VFPDO}\left(L\right)$
 ${\mathrm{\Delta }}{≔}{X}{→}\left[\frac{{\partial }}{{\partial }{y}}{}\left(\frac{{\partial }}{{\partial }{y}}{}{X}{}\left({x}\right)\right){,}\frac{{ⅆ}}{{ⅆ}{x}}{}{X}{}\left({x}\right){,}\frac{{\partial }}{{\partial }{x}}{}{X}{}\left({y}\right){+}\frac{{\partial }}{{\partial }{y}}{}{X}{}\left({x}\right){,}\frac{{ⅆ}}{{ⅆ}{y}}{}{X}{}\left({y}\right)\right]$ (4)

Check actions as operator on vector field

 > $\mathrm{Δ}\left(X\right)$
 $\left[\frac{{{\partial }}^{{2}}}{{\partial }{{y}}^{{2}}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){,}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){,}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}{,}{y}\right){+}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\xi }}{}\left({x}{,}{y}\right){,}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{\mathrm{\eta }}{}\left({x}{,}{y}\right)\right]$ (5)

it should kill rotations..

 > $R≔\mathrm{VectorField}\left(-\left(y-\mathrm{y0}\right){\mathrm{D}}_{x}+\left(x-\mathrm{x0}\right){\mathrm{D}}_{y},\mathrm{space}=\left[x,y\right]\right)$
 ${R}{≔}\left({-}{y}{+}{\mathrm{y0}}\right){}\left(\frac{{ⅆ}}{{ⅆ}{x}}\right){+}\left({x}{-}{\mathrm{x0}}\right){}\left(\frac{{ⅆ}}{{ⅆ}{y}}\right)$ (6)
 > $\mathrm{Δ}\left(R\right)$
 $\left[{0}{,}{0}{,}{0}{,}{0}\right]$ (7)

Compatibility

 • The VFPDO Object as Operator command was introduced in Maple 2020.
 • For more information on Maple 2020 changes, see Updates in Maple 2020.