AsDerivation - Maple Help

VectorField Object as Derivation Operator

 Calling Sequence X( f)

Parameters

 X - a VectorField object f - scalar expression, or Vector, Matrix, list or table of scalar expressions

Description

 • A VectorField object X can act as derivation operator.
 • A derivation is an operator $X$ such that $X\left(f+g\right)=X\left(f\right)+X\left(g\right)$ and $X\left(\mathrm{fg}\right)=\mathrm{fX}\left(g\right)+X\left(f\right)g$
 • Because it can act as an operator, a VectorField object is of type appliable. See Overview of VectorField Overloaded Builtins for more detail.
 • When a vector field is acting as an operator, it will distribute itself over indexable types such as Vectors, Matrices, lists, and tables.
 • This method is associated with the VectorField object. For more detail, see Overview of the VectorField object.

Examples

 > $\mathrm{with}\left(\mathrm{LieAlgebrasOfVectorFields}\right):$
 > $X≔\mathrm{VectorField}\left(x\mathrm{D}\left[x\right]+y\mathrm{D}\left[y\right],\mathrm{space}=\left[x,y\right]\right)$
 ${X}{≔}{x}{}\frac{{\partial }}{{\partial }{x}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}{+}{y}{}\frac{{\partial }}{{\partial }{y}}\phantom{\rule[-0.0ex]{0.4em}{0.0ex}}$ (1)

 > $X\left({x}^{2}\right)$
 ${2}{}{{x}}^{{2}}$ (2)

 > $X\left(\left[x,{x}^{2},{x}^{3}\right]\right)$
 $\left[{x}{,}{2}{}{{x}}^{{2}}{,}{3}{}{{x}}^{{3}}\right]$ (3)
 > $A≔\mathrm{Matrix}\left(\left[\left[x,y\right],\left[{x}^{2},{y}^{2}\right]\right]\right)$
 ${A}{≔}\left[\begin{array}{cc}{x}& {y}\\ {{x}}^{{2}}& {{y}}^{{2}}\end{array}\right]$ (4)

 > $X\left(A\right)$
 $\left[\begin{array}{cc}{x}& {y}\\ {2}{}{{x}}^{{2}}& {2}{}{{y}}^{{2}}\end{array}\right]$ (5)

Compatibility

 • The VectorField Object as Derivation Operator command was introduced in Maple 2020.