The VFPDO object is designed and created to represent partial differential operators (PDO) that are applied to vector fields.

There is a collection of methods that are available for a VFPDO object, including (i) method allowing the VFPDO object to act as an operator / function  (ii) methods for exploring properties of VFPDO (e.g. specification of domain and codomain). Some existing Maple builtins are extended for allowing VFPDO object.

Methods of the VFPDO object become available only once a valid VFPDO object is constructed successfully. To construct a VFPDO object, see LieAlgebrasOfVectorFields[VFPDO].

The VFPDO object is exported by the LieAlgebrasOfVectorFields package. See Overview of the LieAlgebrasOfVectorFields package for more detail.

A VFPDO Delta acts as an operator on an $m$-tuple ($m\ge 1$) of scalar expressions, mapping it to an $s$-tuple ($s\ge 0$) of scalars.  Thus the input to Delta is a list of $m$ elements, and it returns a list of $s$ elements acting on a vector field of the same components.

The space where the vector field that the VFPDO object is operated on may be accessed via the GetSpace method.  The integers $m$, $s$ respectively are accessed via the GetSystemCount methods.

After a VFPDO object Delta is successfully constructed, each method of Delta can be accessed by either the short form method(Delta, arguments) or the long form Delta:-method(Delta, arguments).

The following is a list of available methods for a VFPDO object.

A VFPDO object can also act as a differential operator. See VFPDO Object as Operator for more detail.

The following Maple builtins are extended to allow VFPDO object: has, hastype, indets, type. See VFPDO Object Overloaded Builtins for more detail.

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

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

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

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

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

$L≔\mathrm{LAVF}\left(X\,S\right)$

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

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

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

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

$4$

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

$\left[x\,y\right]$

$\mathrm{AreSameSpace}\left(\mathrm{\Delta }\,L\,X\right)$

$\mathrm{true}$

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

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

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

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