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Lie_diff

  

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

 

Calling Sequence

Parameters

Description

Examples

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.

• 

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:

LvTa,b,c,...,l,m,n,...Ta,b,c,...,l,m,n,...,qVqTq,b,c,...,l,m,n,...Va,Ta,q,c,...,l,m,n,...Vb+q,Ta,b,q,...,l,m,n,...Vc+q,Ta,b,c,...,q,m,n,...Vq....+q,Ta,b,c,...,l,q,n,...Vq+l,Ta,b,c,...,l,m,q,...Vq+m,n+...

  

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](..).

Examples

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

withtensor:

Define a mixed rank 2 tensor type, T:

T_comptsarraysymmetric,1..3,1..3,rsinθ,φ3,0,φ3,rcosθ2,r3,0,r3,9:

Tcreate1,1,evalT_compts

T:=tableindex_char=1,1,compts=rsinθφ30φ3rcosθ2r30r39

(1)

Define a contravariant vector field, V:

Vcreate1,arrayr,rcosθ,rcosφ

V:=tableindex_char=1,compts=rrcosθrcosφ

(2)

Define the coordinates:

coordr,θ,φ

coord:=r,θ,φ

(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:

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

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

LvTtensor[Lie_diff]T,V,coord

LvT:=tableindex_char=1,1,compts=r2cosθ2+φ3cosθ+rsinθφ2sinθφr3rcosφ+φ0φ3rsinθ+3φ2rcosφ+r3cosφcosθ3rsinθcosθ+φ3+rcosθ2sinθcosθ2rφ3cosθ+rr3rsinθrsinφ+3rsinθcosφ+r3cosθ+9cosφ3r3φ3cosφ+r4sinφr4sinθ0

(4)

See Also

DifferentialGeometry[LieDerivative]

Physics[LieDerivative]

tensor(deprecated)

tensor(deprecated)/Killing_eqns

tensor(deprecated)[commutator]

tensor(deprecated)[simp]

 


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