Given the coordinate variables, coord, a contravariant vector field V, and any tensor T, Lie_diff(T, V, coord) computes the Lie derivative of T with respect to the vector field V using the usual partial derivatives of T and V according to the standard formula:

where the comma denotes a partial derivative, a, b, c, ... are contravariant indices of T and l, m, n, ... are covariant indices of T, and * indicates an inner product on the repeated indices.

It is required that V be a tensor_type with character: [1] (that is, V is a contravariant vector field)

Note that the rank and index character of the result is identical to that of the input tensor, T.

Simplification:  This routine uses the routine `tensor/Lie_diff/simp` routine for simplification purposes.  The simplification routine is applied twice to each component: first, to the first term involving the inner product of the partial of T and the vector V, and second to the entire component once all of the subsequent terms have been added on. By default, this routine is initialized to the `tensor/simp` routine.  It is recommended that the `tensor/Lie_diff/simp` routine be customized to suit the needs of the particular problem.

This function is part of the tensor package, and so can be used in the form Lie_diff(..) only after performing the command with(tensor) or with(tensor, Lie_diff).  The function can always be accessed in the long form tensor[Lie_diff](..).

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

$\mathrm{T\_compts}≔\mathrm{array}\left(\mathrm{symmetric}\,1..3\,1..3\,\left[\left[r\sin \left(\mathrm{\theta }\right)\,{\mathrm{\phi }}^3\,0\right]\,\left[{\mathrm{\phi }}^3\,r-{\cos \left(\mathrm{\theta }\right)}^2\,r^3\right]\,\left[0\,r^3\,9\right]\right]\right)\:$

$T≔\mathrm{create}\left(\left[1\,-1\right]\,\mathrm{eval}\left(\mathrm{T\_compts}\right)\right)$

$T≔table\left(\left[\mathrm{compts}=\left[\begin{array}{ccc}r\sin \left(\mathrm{\theta }\right)& {\mathrm{\phi }}^3& 0\\ {\mathrm{\phi }}^3& r-{\cos \left(\mathrm{\theta }\right)}^2& r^3\\ 0& r^3& 9\end{array}\right]\,\mathrm{index\_char}=\left[1\,\mathrm{-1}\right]\right]\right)$

$V≔\mathrm{create}\left(\left[1\right]\,\mathrm{array}\left(\left[r\,r\cos \left(\mathrm{\theta }\right)\,r\cos \left(\mathrm{\phi }\right)\right]\right)\right)$

$V≔table\left(\left[\mathrm{compts}=\left[\begin{array}{ccc}r& r\cos \left(\mathrm{\theta }\right)& r\cos \left(\mathrm{\phi }\right)\end{array}\right]\,\mathrm{index\_char}=\left[1\right]\right]\right)$

$\mathrm{coord}≔\left[r\,\mathrm{\theta }\,\mathrm{\phi }\right]$

$\mathrm{coord}≔\left[r\,\mathrm{\theta }\,\mathrm{\phi }\right]$

`tensor/Lie_diff/simp`:=proc(x) simplify(x,trig) end proc:

$\mathrm{LvT}≔\mathrm{tensor}\left[\mathrm{Lie\_diff}\right]\left(T\,V\,\mathrm{coord}\right)$

$\mathrm{LvT}≔table\left(\left[\mathrm{compts}=\left[\begin{array}{ccc}{r}^2{\cos \left(\mathrm{\theta }\right)}^2+{\mathrm{\phi }}^3\cos \left(\mathrm{\theta }\right)+r\sin \left(\mathrm{\theta }\right)& 3{\mathrm{\phi }}^{2}r\cos \left(\mathrm{\phi }\right)-{\mathrm{\phi }}^3-{\mathrm{\phi }}^{3}r\sin \left(\mathrm{\theta }\right)& 0\\ -{\cos \left(\mathrm{\theta }\right)}^3-r\left(\sin \left(\mathrm{\theta }\right)-1\right)\cos \left(\mathrm{\theta }\right)+{\mathrm{\phi }}^{3}r\sin \left(\mathrm{\theta }\right)+\left(3{\mathrm{\phi }}^{2}r+r^3\right)\cos \left(\mathrm{\phi }\right)+{\mathrm{\phi }}^3& r+2\sin \left(\mathrm{\theta }\right){\cos \left(\mathrm{\theta }\right)}^{2}r-{\mathrm{\phi }}^3\cos \left(\mathrm{\theta }\right)& 3r^3+r^4\sin \left(\mathrm{\theta }\right)-r^4\sin \left(\mathrm{\phi }\right)\\ -r\sin \left(\mathrm{\theta }\right)\cos \left(\mathrm{\phi }\right)+r^3\cos \left(\mathrm{\theta }\right)+9\cos \left(\mathrm{\phi }\right)& 3r^3-{\mathrm{\phi }}^3\cos \left(\mathrm{\phi }\right)+r^4\sin \left(\mathrm{\phi }\right)-r^4\sin \left(\mathrm{\theta }\right)& 0\end{array}\right]\,\mathrm{index\_char}=\left[1\,\mathrm{-1}\right]\right]\right)$