compute the Lie derivative of a tensor with respect to a contravariant vector field
Lie_diff( T, V, coord)
tensor whose Lie derivative is to be computed
contravariant vector field with respect to which the derivative is being taken
list of coordinate names
Important: The tensor package has been deprecated. Use the superseding commands DifferentialGeometry[LieDerivative] and Physics[LieDerivative] instead.
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:  (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](..).
Define a mixed rank 2 tensor type, T:
T_compts ≔ array⁡symmetric,1..3,1..3,r⁢sin⁡θ,φ3,0,φ3,r−cos⁡θ2,r3,0,r3,9:
T ≔ create⁡1,−1,eval⁡T_compts
Define a contravariant vector field, V:
V ≔ create⁡1,array⁡r,r⁢cos⁡θ,r⁢cos⁡φ
Define the coordinates:
coord ≔ r,θ,φ
Because the components of T and V involve trigonometric functions, customize the `tensor/Lie_diff/simp` routine so that it uses the `trig` option of the Maple simplify:
`tensor/Lie_diff/simp`:=proc(x) simplify(x,trig) end proc:
Now compute the Lie derivative of T with respect to the field V:
LvT ≔ tensorLie_diff⁡T,V,coord
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