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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 $\sqrt{-\mathrm{detg}}$ is used in computing the covariant components and a factor of $-\frac{1}{\sqrt{-\mathrm{detg}}}$ 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.

 > $\mathrm{with}\left(\mathrm{tensor}\right):$

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

 > $\mathrm{detg}≔-{r}^{4}{\mathrm{sin}\left(\mathrm{\theta }\right)}^{2}$
 ${\mathrm{detg}}{≔}{-}{{r}}^{{4}}{}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}^{{2}}$ (1)
 > $\mathrm{Levi_Civita}\left(\mathrm{detg},4,\mathrm{cov_LC},\mathrm{con_LC}\right)$
 > $\mathrm{cov_LC}\left[\mathrm{compts}\right]\left[1,2,3,4\right]$
 $\sqrt{{{r}}^{{4}}{}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}^{{2}}}$ (2)
 > $\mathrm{con_LC}\left[\mathrm{compts}\right]\left[1,2,3,4\right]$
 ${-}\frac{{1}}{\sqrt{{{r}}^{{4}}{}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}^{{2}}}}$ (3)

cov_LC and con_LC are totally antisymmetric:

 > $\mathrm{op}\left(1,\mathrm{get_compts}\left(\mathrm{cov_LC}\right)\right)$
 ${\mathrm{antisymmetric}}$ (4)
 > $\mathrm{op}\left(1,\mathrm{get_compts}\left(\mathrm{con_LC}\right)\right)$
 ${\mathrm{antisymmetric}}$ (5)

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

 > $\mathrm{detg}≔\frac{1}{{v}^{4}}$
 ${\mathrm{detg}}{≔}\frac{{1}}{{{v}}^{{4}}}$ (6)
 > $\mathrm{Levi_Civita}\left(\mathrm{detg},2,\mathrm{Pcov_LC},\mathrm{Pcon_LC}\right)$
 > $\mathrm{eval}\left(\mathrm{Pcov_LC}\right)$
 ${table}{}\left(\left[{\mathrm{compts}}{=}\left[\begin{array}{cc}{0}& \frac{{\mathrm{csgn}}{}\left(\frac{{1}}{{{v}}^{{2}}}\right)}{{{v}}^{{2}}}\\ {-}\frac{{\mathrm{csgn}}{}\left(\frac{{1}}{{{v}}^{{2}}}\right)}{{{v}}^{{2}}}& {0}\end{array}\right]{,}{\mathrm{index_char}}{=}\left[{-1}{,}{-1}\right]\right]\right)$ (7)
 > $\mathrm{eval}\left(\mathrm{Pcon_LC}\right)$
 ${table}{}\left(\left[{\mathrm{compts}}{=}\left[\begin{array}{cc}{0}& \frac{{{v}}^{{2}}}{{\mathrm{csgn}}{}\left(\frac{{1}}{{{v}}^{{2}}}\right)}\\ {-}\frac{{{v}}^{{2}}}{{\mathrm{csgn}}{}\left(\frac{{1}}{{{v}}^{{2}}}\right)}& {0}\end{array}\right]{,}{\mathrm{index_char}}{=}\left[{1}{,}{1}\right]\right]\right)$ (8)