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)\:$

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

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

$X≔a\mathrm{D\_x}+b\mathrm{D\_y}+c\mathrm{D\_z}$

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

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

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

$X≔x\mathrm{D\_y}-z{y}^2\mathrm{D\_z}$

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

$Y≔y\mathrm{D\_x}+z^2\mathrm{D\_y}$

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

$x\mathrm{D\_x}-\left(2z^2{y}^2+y\right)\mathrm{D\_y}+2z^{3}y\mathrm{D\_z}$

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

$x\mathrm{D\_x}-\left(2z^2{y}^2+y\right)\mathrm{D\_y}+2z^{3}y\mathrm{D\_z}$

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

$X≔z^2\mathrm{D\_x}-y\mathrm{D\_z}$

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

$\mathrm{\omega }≔y\mathrm{dx}\bigwedge \mathrm{dz}$

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

$-y\mathrm{dx}\bigwedge \mathrm{dy}$

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

$-y\mathrm{dx}\bigwedge \mathrm{dy}$

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

$X≔z^2\mathrm{D\_x}-y\mathrm{D\_z}$

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

$T≔z\mathrm{D\_x}\mathrm{dy}\mathrm{dz}$

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

$-z\mathrm{D\_x}\mathrm{dy}\mathrm{dy}-y\mathrm{D\_x}\mathrm{dy}\mathrm{dz}$

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

$X≔z^2\mathrm{D\_x}-y^2\mathrm{D\_z}$

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

$T≔0\mathrm{D\_x}\mathrm{dx}\mathrm{dx}$

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

$-2\mathrm{D\_y}\mathrm{dz}\mathrm{dy}+2\mathrm{D\_z}\mathrm{dx}\mathrm{dz}$

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

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

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

$\left[0\mathrm{D\_x}\mathrm{D\_x}\,\mathrm{Dy}\mathrm{D\_x}\,2x\mathrm{Dz}\mathrm{D\_x}\right]$

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

$\left[\left[0\mathrm{D\_x}\,\mathrm{D\_x}\,0\mathrm{D\_x}\,y\mathrm{Dz}\right]\,\left[0\mathrm{D\_x}\,0\mathrm{D\_x}\,\mathrm{D\_x}\,x\mathrm{Dz}\right]\right]$

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

$T≔\left[\begin{array}{cc}{y}^2\mathrm{dx}\bigwedge \mathrm{dy}& x^2\mathrm{dy}\bigwedge \mathrm{dz}\\ xy\mathrm{dx}\bigwedge \mathrm{dz}& z^2\mathrm{dx}\bigwedge \mathrm{dy}\end{array}\right]$

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

$\left[\begin{array}{cc}4y^2\mathrm{dx}\bigwedge \mathrm{dy}& 4x^2\mathrm{dy}\bigwedge \mathrm{dz}\\ 4xy\mathrm{dx}\bigwedge \mathrm{dz}& 4z^2\mathrm{dx}\bigwedge \mathrm{dy}\end{array}\right]$

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

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

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

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

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

$-\frac{\mathrm{\σ2}}{yz}\bigwedge \mathrm{\σ3}$

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

$-\frac{\mathrm{U1}}{y}\mathrm{\σ3}$

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

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

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

$\mathrm{\ω3}$

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

${\mathrm{\iota }}_{2}d\mathrm{\β1}+d{\mathrm{\iota }}_2\mathrm{\β1}$

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

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

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

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

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

$\left[\begin{array}{ccccc}& |& \mathrm{e1}& \mathrm{e2}& \mathrm{e3}\\ & \mathrm{----}& \mathrm{----}& \mathrm{----}& \mathrm{----}\\ \mathrm{e1}& |& 0& \mathrm{e3}& -2\mathrm{e1}\\ \mathrm{e2}& |& -\mathrm{e3}& 0& 2\mathrm{e2}\\ \mathrm{e3}& |& 2\mathrm{e1}& -2\mathrm{e2}& 0\end{array}\right]$

$K≔\mathrm{KillingForm}\left(\right)$

$K≔4\mathrm{\θ1}\mathrm{\θ2}+4\mathrm{\θ2}\mathrm{\θ1}+8\mathrm{\θ3}\mathrm{\θ3}$

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

$\left[0\mathrm{\θ1}\mathrm{\θ1}\,0\mathrm{\θ1}\mathrm{\θ1}\,0\mathrm{\θ1}\mathrm{\θ1}\right]$