If T is a Maple expression, then LieDerivative(X, T) is the directional derivative X(T) of T in the direction of the vector field X.

If T is a vector field, then LieDerivative(X, T) coincides with the Lie bracket [X, T] = LieBracket(X, T).

If T is a differential 1-form, then alpha = LieDerivative(X, T) is the 1-form defined by alpha(Y) = X(alpha(Y)) - alpha([X,Y]), where Y is any vector field on M.

The Lie derivative operator acts as a derivation with respect to both the wedge and tensor products.  If alpha and beta are differential forms and T and S are tensors, then LieDerivative(X, alpha &w beta) = LieDerivative(X, alpha) &w beta + alpha &w LieDerivative(X, beta), and LieDerivative(X, S &t T) = LieDerivative(X, S) &t T + S &w LieDerivative(X, T).

The Lie derivative of a differential form can also be calculated from the Cartan formula, LieDerivative(X, alpha) = ExteriorDerivative(Hook(X, alpha)) + Hook(X, ExteriorDerivative(alpha))

The Lie derivative of a connection nabla_Y(Z) is the type (1, 2) tensor field S = LieDerivative(X, nabla), defined (when viewed as mapping from pairs of vector fields to vector fields) by S(Y, Z) = LieDerivative(X, nabla_Y(Z)) - nabla_{LieDerivative(X, Y)}(Z) - nabla_X(LieDerivative(Y, Z)).

For the definition of the Lie derivative of these geometric objects in terms of the flow of the vector field X see, for example, Spivak page 207-208.

The Lie derivative of a tensor defined on a Lie algebra can also be computed.

The first argument also be a list of vectors. The second argument can be a list of a vectors, Maple  expressions, a differential forms or tensors.

This command is part of the DifferentialGeometry package, and so can be used in the form LieDerivative(...) only after executing the command with(DifferentialGeometry).  It can always be used in the long form DifferentialGeometry:-LieDerivative.

$\mathrm{with}\left(\mathrm{DifferentialGeometry}\right)\:$$\mathrm{with}\left(\mathrm{LieAlgebras}\right)\:$

$\mathrm{DGsetup}\left(\left[x\,y\,z\right]\,M\right)\:$

$X≔\mathrm{evalDG}\left(a\mathrm{D\_x}\+b\mathrm{D\_y}\+c\mathrm{D\_z}\right)$

$\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,a\right]\,\left[\left[2\right]\,b\right]\,\left[\left[3\right]\,c\right]\right]\right]\right)$

$\mathrm{LieDerivative}\left(X\,f\left(x\,y\,z\right)\right)$

$a\left(\frac{\partial }{\partial x}f\left(x\,y\,z\right)\right)+b\left(\frac{\partial }{\partial y}f\left(x\,y\,z\right)\right)+c\left(\frac{\partial }{\partial z}f\left(x\,y\,z\right)\right)$

$X≔\mathrm{evalDG}\left(x\mathrm{D\_y}-y^{2}z\mathrm{D\_z}\right)$

$\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[2\right]\,x\right]\,\left[\left[3\right]\,-z{y}^2\right]\right]\right]\right)$

$Y≔\mathrm{evalDG}\left(y\mathrm{D\_x}\+z^2\mathrm{D\_y}\right)$

$\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,y\right]\,\left[\left[2\right]\,z^2\right]\right]\right]\right)$

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

$\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,x\right]\,\left[\left[2\right]\,-2y^2{z}^2-y\right]\,\left[\left[3\right]\,2z^{3}y\right]\right]\right]\right)$

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

$\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,x\right]\,\left[\left[2\right]\,-2y^2{z}^2-y\right]\,\left[\left[3\right]\,2z^{3}y\right]\right]\right]\right)$

$X≔\mathrm{evalDG}\left(z^2\mathrm{D\_x}-y\mathrm{D\_z}\right)$

$\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,z^2\right]\,\left[\left[3\right]\,-y\right]\right]\right]\right)$

$\mathrm{\ω}≔\mathrm{evalDG}\left(y\mathrm{dx}\&w\mathrm{dz}\right)$

$\mathrm{\_DG}\left(\left[\left[''form''\,M\,2\right]\,\left[\left[\left[1\,3\right]\,y\right]\right]\right]\right)$

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

$\mathrm{\_DG}\left(\left[\left[''form''\,M\,2\right]\,\left[\left[\left[1\,2\right]\,-y\right]\right]\right]\right)$

$\left(\mathrm{Hook}\left(X\,\mathrm{ExteriorDerivative}\left(\mathrm{\ω}\right)\right)\right)\&plus\left(\mathrm{ExteriorDerivative}\left(\mathrm{Hook}\left(X\,\mathrm{\ω}\right)\right)\right)$

$\mathrm{\_DG}\left(\left[\left[''form''\,M\,2\right]\,\left[\left[\left[1\,2\right]\,-y\right]\right]\right]\right)$

$X≔\mathrm{evalDG}\left(z^2\mathrm{D\_x}-y\mathrm{D\_z}\right)$

$\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,z^2\right]\,\left[\left[3\right]\,-y\right]\right]\right]\right)$

$T≔\mathrm{evalDG}\left(z\left(\mathrm{D\_x}\&t\mathrm{dy}\right)\&t\mathrm{dz}\right)$

$\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,2\,3\right]\,z\right]\right]\right]\right)$

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

$\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,2\,2\right]\,-z\right]\,\left[\left[1\,2\,3\right]\,-y\right]\right]\right]\right)$

$X≔\mathrm{evalDG}\left(z^2\mathrm{D\_x}-y^2\mathrm{D\_z}\right)$

$\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,z^2\right]\,\left[\left[3\right]\,-y^2\right]\right]\right]\right)$

$T≔\mathrm{Tensor}:-\mathrm{Connection}\left(0\&mult\left(\left(\mathrm{D\_x}\&tensor\mathrm{dx}\right)\&tensor\mathrm{dx}\right)\right)$

$\mathrm{\_DG}\left(\left[\left[''connection''\,M\,\left[\left[''con\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\,1\right]\,0\right]\right]\right]\right)$

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

$\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_bas''\,''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[2\,3\,2\right]\,-2\right]\,\left[\left[3\,1\,3\right]\,2\right]\right]\right]\right)$

$\mathrm{LieDerivative}\left(\left[\mathrm{D\_x}\,\mathrm{D\_y}\,\mathrm{D\_z}\right]\,xyz\mathrm{Dz}\right)$

$\left[yz\mathrm{Dz}\,xz\mathrm{Dz}\,xy\mathrm{Dz}\right]$

$\mathrm{LieDerivative}\left(\mathrm{D\_x}\,\left[\mathrm{D\_x}\&t\mathrm{D\_x}\,x\mathrm{D\_x}\&t\mathrm{Dy}\,x^2\mathrm{D\_x}\&t\mathrm{Dz}\right]\right)$

$\left[\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_bas''\,''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,0\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\right]\,\mathrm{Dy}\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,M\,\left[\left[''con\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\right]\,2x\mathrm{Dz}\right]\right]\right]\right)\right]$

$\mathrm{LieDerivative}\left(\left[\mathrm{D\_x}\,\mathrm{D\_y}\right]\,\left[\mathrm{D\_x}\,x\mathrm{D\_x}\,y\mathrm{D\_x}\,xy\mathrm{Dz}\right]\right)$

$\left[\left[\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,0\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,1\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,0\right]\right]\right]\right)\,y\mathrm{Dz}\right]\,\left[\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,0\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,0\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''vector''\,M\,\left[\right]\right]\,\left[\left[\left[1\right]\,1\right]\right]\right]\right)\,x\mathrm{Dz}\right]\right]$

$T≔\mathrm{map}\left(\mathrm{evalDG}\,\mathrm{Matrix}\left(\left[\left[y^2\mathrm{dx}\&w\mathrm{dy}\,x^2\mathrm{dy}\&w\mathrm{dz}\right]\,\left[xy\mathrm{dx}\&w\mathrm{dz}\,z^2\mathrm{dx}\&w\mathrm{dy}\right]\right]\right)\right)$

$\left[\begin{array}{cc}\mathrm{\_DG}\left(\left[\left[''form''\,M\,2\right]\,\left[\left[\left[1\,2\right]\,y^2\right]\right]\right]\right)& \mathrm{\_DG}\left(\left[\left[''form''\,M\,2\right]\,\left[\left[\left[2\,3\right]\,x^2\right]\right]\right]\right)\\ \mathrm{\_DG}\left(\left[\left[''form''\,M\,2\right]\,\left[\left[\left[1\,3\right]\,xy\right]\right]\right]\right)& \mathrm{\_DG}\left(\left[\left[''form''\,M\,2\right]\,\left[\left[\left[1\,2\right]\,z^2\right]\right]\right]\right)\end{array}\right]$

$\mathrm{LieDerivative}\left(x\mathrm{D\_x}\+y\mathrm{D\_y}\+z\mathrm{D\_z}\,T\right)$

$\left[\begin{array}{cc}\mathrm{\_DG}\left(\left[\left[''form''\,M\,2\right]\,\left[\left[\left[1\,2\right]\,4y^2\right]\right]\right]\right)& \mathrm{\_DG}\left(\left[\left[''form''\,M\,2\right]\,\left[\left[\left[2\,3\right]\,4x^2\right]\right]\right]\right)\\ \mathrm{\_DG}\left(\left[\left[''form''\,M\,2\right]\,\left[\left[\left[1\,3\right]\,4xy\right]\right]\right]\right)& \mathrm{\_DG}\left(\left[\left[''form''\,M\,2\right]\,\left[\left[\left[1\,2\right]\,4z^2\right]\right]\right]\right)\end{array}\right]$

$\mathrm{FD}≔\mathrm{FrameData}\left(\left[y\mathrm{dx}\,\mathrm{dx}\+z\mathrm{dy}\,\mathrm{dz}\right]\,P\right)$

$\mathrm{FD}≔\left[d\mathrm{\Θ1}=-\frac{\mathrm{\Θ1}\bigwedge \mathrm{\Θ2}}{yz}\,d\mathrm{\Θ2}=\frac{\mathrm{\Θ1}\bigwedge \mathrm{\Θ3}}{yz}-\frac{\mathrm{\Θ2}\bigwedge \mathrm{\Θ3}}{z}\,d\mathrm{\Θ3}=0\right]$

$\mathrm{DGsetup}\left(\mathrm{FD}\,\left[U\right]\,\left[\mathrm{\σ}\right]\right)$

$\mathrm{frame\; name:\; P}$

$\mathrm{LieDerivative}\left(\mathrm{U1}\,\mathrm{\σ1}\&w\mathrm{\σ3}\right)$

$\mathrm{\_DG}\left(\left[\left[''form''\,P\,2\right]\,\left[\left[\left[2\,3\right]\,-\frac{1}{yz}\right]\right]\right]\right)$

$\mathrm{LieDerivative}\left(x\mathrm{U1}\,\mathrm{U1}\&t\mathrm{\σ3}\right)$

$\mathrm{\_DG}\left(\left[\left[''tensor''\,P\,\left[\left[''con\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,3\right]\,-\frac{1}{y}\right]\right]\right]\right)$

$\mathrm{DGsetup}\left(\left[\left[\mathrm{\ω1}\,\mathrm{\ω2}\,\mathrm{\ω3}\right]\,\mathrm{\β1}\=\mathrm{dgform}\left(2\right)\right]\,\left[d\left(\mathrm{\ω1}\right)\=\mathrm{\ω2}\&w\mathrm{\ω3}\,d\left(\mathrm{\ω2}\right)\=\mathrm{\β1}\right]\,N\right)$

$\mathrm{frame\; name:\; N}$

$\mathrm{LieDerivative}\left(\mathrm{D\_omega2}\,\mathrm{\ω1}\right)$

$\mathrm{\_DG}\left(\left[\left[''form''\,N\,1\right]\,\left[\left[\left[3\right]\,1\right]\right]\right]\right)$

$\mathrm{LieDerivative}\left(\mathrm{D\_omega2}\,\mathrm{\β1}\right)$

$\mathrm{\_DG}\left(\left[\left[''form''\,N\,2\right]\,\left[\left[\left[6\right]\,1\right]\,\left[\left[8\right]\,1\right]\right]\right]\right)$

$\mathrm{LD}≔\mathrm{LieAlgebraData}\left(\left[\left[\mathrm{x1}\,\mathrm{x2}\right]\=\mathrm{x3}\,\left[\mathrm{x3}\,\mathrm{x1}\right]\=2\mathrm{x1}\,\left[\mathrm{x3}\,\mathrm{x2}\right]\=-2\mathrm{x2}\right]\,\left[\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\right]\,\mathrm{alg}\right)$

$\mathrm{LD}≔\left[\left[\mathrm{e1}\,\mathrm{e2}\right]=\mathrm{e3}\,\left[\mathrm{e1}\,\mathrm{e3}\right]=-2\mathrm{e1}\,\left[\mathrm{e2}\,\mathrm{e3}\right]=2\mathrm{e2}\right]$

$\mathrm{DGsetup}\left(\mathrm{LD}\right)$

$\mathrm{Lie\; algebra:\; alg}$

$\mathrm{MultiplicationTable}\left(''LieTable''\right)$

$\left[\begin{array}{ccccc}& |& \mathrm{e1}& \mathrm{e2}& \mathrm{e3}\\ & \mathrm{----}& \mathrm{----}& \mathrm{----}& \mathrm{----}\\ \mathrm{e1}& |& 0& \mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{alg}\,\left[\right]\right]\,\left[\left[\left[3\right]\,1\right]\right]\right]\right)& \mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{alg}\,\left[\right]\right]\,\left[\left[\left[1\right]\,-2\right]\right]\right]\right)\\ \mathrm{e2}& |& \mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{alg}\,\left[\right]\right]\,\left[\left[\left[3\right]\,-1\right]\right]\right]\right)& 0& \mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{alg}\,\left[\right]\right]\,\left[\left[\left[2\right]\,2\right]\right]\right]\right)\\ \mathrm{e3}& |& \mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{alg}\,\left[\right]\right]\,\left[\left[\left[1\right]\,2\right]\right]\right]\right)& \mathrm{\_DG}\left(\left[\left[''vector''\,\mathrm{alg}\,\left[\right]\right]\,\left[\left[\left[2\right]\,-2\right]\right]\right]\right)& 0\end{array}\right]$

$K≔\mathrm{KillingForm}\left(\right)$

$\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{alg}\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,2\right]\,4\right]\,\left[\left[2\,1\right]\,4\right]\,\left[\left[3\,3\right]\,8\right]\right]\right]\right)$

$\mathrm{LieDerivative}\left(\left[\mathrm{e1}\,\mathrm{e2}\,\mathrm{e3}\right]\,K\right)$

$\left[\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{alg}\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,0\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{alg}\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,0\right]\right]\right]\right)\,\mathrm{\_DG}\left(\left[\left[''tensor''\,\mathrm{alg}\,\left[\left[''cov\_bas''\,''cov\_bas''\right]\,\left[\right]\right]\right]\,\left[\left[\left[1\,1\right]\,0\right]\right]\right]\right)\right]$