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tensor[Lie_diff] - compute the Lie derivative of a tensor with respect to a contravariant vector field

Calling Sequence

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: [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](..).


Important: The tensor package has been deprecated. Use the superseding commands DifferentialGeometry[LieDerivative] and Physics[LieDerivative] instead.


Define a mixed rank 2 tensor type, T:





Define a contravariant vector field, V:




Define the coordinates:




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:




See Also

DifferentialGeometry[LieDerivative], Physics[LieDerivative], tensor(deprecated), tensor(deprecated)/Killing_eqns, tensor(deprecated)[commutator], tensor(deprecated)[simp]

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