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tensor[Levi_Civita] - compute the covariant and contravariant Levi-Civita pseudo-tensors

Calling Sequence

Levi_Civita(detg, dim, cov_LC, con_LC)

Parameters

detg

-

determinant of the covariant metric tensor components

dim

-

dimension of the space

cov_LC

-

output parameter for the covariant Levi_Civita pseudo tensor

con_LC

-

output parameter for the contravariant Levi_Civita pseudo tensor

Description

Important: The tensor package has been deprecated. Use the superseding command Physics[LeviCivita] instead.

• 

The function Levi_Civita(detg, dim, cov_LC, con_LC) computes the Levi-Civita pseudo-tensor in the dimension dim using the metric determinant detg.  The covariant Levi-Civita tensor is output via the parameter cov_LC and the contravariant Levi-Civita tensor is output via the parameter con_LC.  The return value is NULL.

• 

detg must be an algebraic type.  It can be computed from the covariant metric tensor using tensor[invert].  Because the square root of detg is used in computing the components of the results, it is assumed that detg is positive (except in the case where dim=4, where it is assumed to be negative; see below).

• 

dim must be an integer greater than 1.

• 

cov_LC and con_LC must be unassigned names to be used as output parameters for the results.  Recall that the Levi-Civita pseudo-tensor is equal to the permutation symbol multiplied by a factor involving the square root of detg.  cov_LC is the covariant permutation symbol multiplied by square root of detg and con_LC is the contravariant permutation symbol multiplied by the reciprocal of the square root of detg (except in the case where dim=4; see below).

• 

In the case where the dimension is 4, it is assumed that the geometry is for Relativity applications, in which case, detg is assumed to be negative.  Thus, a factor of detg is used in computing the covariant components and a factor of 1detg is used in computing the contravariant components.

• 

Indexing Function: the results are completely anti-symmetric; their component arrays use Maple's antisymmetric indexing function.

Examples

Important: The tensor package has been deprecated. Use the superseding command Physics[LeviCivita] instead.

withtensor:

Compute the Levi-Civita pseudo-tensor in the Schwarzschild geometry of Relativity:

detg:=r4sinθ2

detg:=r4sinθ2

(1)

Levi_Civitadetg,4,cov_LC,con_LC

cov_LCcompts1,2,3,4

r4sinθ2

(2)

con_LCcompts1,2,3,4

1r4sinθ2

(3)

cov_LC and con_LC are totally antisymmetric:

op1,get_comptscov_LC

antisymmetric

(4)

op1,get_comptscon_LC

antisymmetric

(5)

The Levi-Civita components for the Poincare half-plane:

detg:=1v4

detg:=1v4

(6)

Levi_Civitadetg,2,Pcov_LC,Pcon_LC

evalPcov_LC

tablecompts=0csgn1v2v2csgn1v2v20,index_char=1,1

(7)

evalPcon_LC

tablecompts=0v2csgn1v2v2csgn1v20,index_char=1,1

(8)

See Also

DifferentialGeometry[Tensor][PermutationSymbol], linalg[det], Physics[LeviCivita], tensor(deprecated), tensor(deprecated)[invert]


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