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  is used in computing the covariant components and a factor of  is used in computing the contravariant components.

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