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.

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

$X≔\mathrm{VectorField}\left(\mathrm{\xi }\left(x\,y\right)\mathrm{D}\left[x\right]+\mathrm{\eta }\left(x\,y\right)\mathrm{D}\left[y\right]\,\mathrm{space}=\left[x\,y\right]\right)$

$X≔\mathrm{\xi }\left(x\,y\right)\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(x\,y\right)\left(\frac{\ⅆ}{\ⅆy}\right)$

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

$S≔\left[\frac{\partial }{\partial x}\mathrm{\xi }\left(x\,y\right)=0\,\frac{{\partial }^2}{\partial y^2}\mathrm{\xi }\left(x\,y\right)=0\,\frac{\partial }{\partial x}\mathrm{\eta }\left(x\,y\right)=-\frac{\partial }{\partial y}\mathrm{\xi }\left(x\,y\right)\,\frac{\partial }{\partial y}\mathrm{\eta }\left(x\,y\right)=0\right],\mathrm{indep}=\left[x\,y\right],\mathrm{dep}=\left[\mathrm{\xi }\left(x\,y\right)\,\mathrm{\eta }\left(x\,y\right)\right]$

$L≔\mathrm{LAVF}\left(X\,S\right)$

$L≔\left[\mathrm{\xi }\left(x\,y\right)\left(\frac{\ⅆ}{\ⅆx}\right)+\mathrm{\eta }\left(x\,y\right)\left(\frac{\ⅆ}{\ⅆy}\right)\right]\&where\left\{\left[\frac{{\partial }^2}{\partial y^2}\mathrm{\xi }\left(x\,y\right)=0\,\frac{\partial }{\partial x}\mathrm{\xi }\left(x\,y\right)=0\,\frac{\partial }{\partial x}\mathrm{\eta }\left(x\,y\right)=-\frac{\partial }{\partial y}\mathrm{\xi }\left(x\,y\right)\,\frac{\partial }{\partial y}\mathrm{\eta }\left(x\,y\right)=0\right]\right\}$

$\mathrm{\Delta }≔\mathrm{VFPDO}\left(L\right)$

$\mathrm{\Delta }≔X\mapsto \left[\frac{{\partial }^2}{\partial y^2}X\left(x\right)\,\frac{\ⅆ}{\ⅆx}X\left(x\right)\,\frac{\partial }{\partial x}X\left(y\right)+\frac{\partial }{\partial y}X\left(x\right)\,\frac{\ⅆ}{\ⅆy}X\left(y\right)\right]$

$\mathrm{\Delta }\left(X\right)$

$\left[\frac{{\partial }^2}{\partial y^2}\mathrm{\xi }\left(x\,y\right)\,\frac{\partial }{\partial x}\mathrm{\xi }\left(x\,y\right)\,\frac{\partial }{\partial x}\mathrm{\eta }\left(x\,y\right)+\frac{\partial }{\partial y}\mathrm{\xi }\left(x\,y\right)\,\frac{\partial }{\partial y}\mathrm{\eta }\left(x\,y\right)\right]$

$R≔\mathrm{VectorField}\left(-\left(y-\mathrm{y0}\right)\mathrm{D}\left[x\right]+\left(x-\mathrm{x0}\right)\mathrm{D}\left[y\right]\,\mathrm{space}=\left[x\,y\right]\right)$

$R≔\left(-y+\mathrm{y0}\right)\left(\frac{\ⅆ}{\ⅆx}\right)+\left(x-\mathrm{x0}\right)\left(\frac{\ⅆ}{\ⅆy}\right)$

$\mathrm{\Delta }\left(R\right)$

$\left[0\,0\,0\,0\right]$

The VFPDO Object as Operator command was introduced in Maple 2020.

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