If X is an algebraic expression then LieDerivative(X, vf) is the directional derivative vf(X) of X in the direction of the vector field vf.

If X is a VectorField object then LieDerivative(X, vf) is the VectorField object defined by the Lie bracket [vf, X] = LieBracket(vf, X). See LieBracket for more detail.

If X is a OneForm object then omega = LieDerivative(X, vf) is the OneForm object defined by omega(Y) = vf(omega(Y)) - omega([Y,vf]), where Y is any vector field on the same space as X.

This method is associated with the VectorField and OneForm objects. For more detail, see Overview of the VectorField object, Overview of the OneForm object.

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

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

$X≔\frac{\partial }{\partial x}$

$R≔\mathrm{VectorField}\left(-y\mathrm{D}\left[x\right]+x\mathrm{D}\left[y\right]\right)$

$R≔-y\frac{\partial }{\partial x}+x\frac{\partial }{\partial y}$

$\mathrm{LieDerivative}\left(a{x}^2-y^2\,X\right)$

$2ax$

$\mathrm{LieDerivative}\left(R\,X\right)$

$\frac{\partial }{\partial y}$

$\mathrm{LieBracket}\left(X\,R\right)$

$\frac{\partial }{\partial y}$

$\mathrm{\omega }≔\mathrm{OneForm}\left(xd\left[x\right]-yd\left[y\right]\right)$

$\mathrm{\omega }≔x\mathrm{dx}-y\mathrm{dy}$

$\mathrm{LieDerivative}\left(\mathrm{\omega }\,R\right)$

$-2y\mathrm{dx}-2x\mathrm{dy}$

The LieDerivative command was introduced in Maple 2020.

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