The functionalities of some Maple builtin commands are extended for use on VFPDO object.

The following builtins have been overloaded for this purpose: indets, has, type, hastype

Let Delta be a VFPDO object.

(i) The call type(Delta, t) returns true if t is any of the following types: module, object, anything, appliable and VFPDO. See examples below.

(ii) The calls type(Delta, dependent(x)) and type(Delta, freeof(x)) respectively return true if the differential operator or the independent variables of Delta contain (respectively don't contain) x. See example below.

The indets, has, hastype builtin commands accept a VFPDO object and apply their methods onto the differential operator and the independent variables of the object.

These overloaded builtins are associated with the VFPDO object. For more detail, see Overview of the VFPDO object.

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

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

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

$\mathrm{\Delta }≔\mathrm{VFPDO}\left(\left[a\left[1\right]\mathrm{diff}\left(\mathrm{\xi }\left(x\,y\right)\,x\right)-a\left[2\right]\mathrm{\xi }\left(x\,y\right)\,\mathrm{diff}\left(\mathrm{\eta }\left(x\,y\right)\,x\,x\right)-\left({a\left[1\right]}^2+{a\left[2\right]}^2\right)\mathrm{\eta }\left(x\,y\right)\right]\,X\right)$

$\mathrm{\Delta }≔X\mapsto \left[a_1\cdot \left(\frac{\ⅆ}{\ⅆx}X\left(x\right)\right)-a_2\cdot X\left(x\right)\,\frac{{\partial }^2}{\partial x^2}X\left(y\right)+\left(-a_1^2-a_2^2\right)\cdot X\left(y\right)\right]$

The VFPDO Object Overloaded Builtins command was introduced in Maple 2020.

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