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

Calling Sequence


Lie_diff( T, V, coord)


Parameters


T



tensor whose Lie derivative is to be computed

V



contravariant vector field with respect to which the derivative is being taken

coord



list of coordinate names





Description


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

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

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

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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.

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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](..).



Examples


Important: The tensor package has been deprecated. Use the superseding commands DifferentialGeometry[LieDerivative] and Physics[LieDerivative] instead.
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Define a mixed rank 2 tensor type, T:
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 (1) 
Define a contravariant vector field, V:
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 (2) 
Define the coordinates:
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 (3) 
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:
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`tensor/Lie_diff/simp`:=proc(x) simplify(x,trig) end proc:

Now compute the Lie derivative of T with respect to the field V:
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 (4) 

