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.

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

$\mathrm{detg}≔-r^4{\sin \left(\mathrm{\theta }\right)}^2$

$\mathrm{detg}≔-r^4{\sin \left(\mathrm{\theta }\right)}^2$

$\mathrm{Levi\_Civita}\left(\mathrm{detg}\,4\,\mathrm{cov\_LC}\,\mathrm{con\_LC}\right)$

$\mathrm{cov\_LC}\left[\mathrm{compts}\right]\left[1,2,3,4\right]$

$\sqrt{r^4{\sin \left(\mathrm{\theta }\right)}^2}$

$\mathrm{con\_LC}\left[\mathrm{compts}\right]\left[1,2,3,4\right]$

$-\frac{1}{\sqrt{r^4{\sin \left(\mathrm{\theta }\right)}^2}}$

$\mathrm{op}\left(1\,\mathrm{get\_compts}\left(\mathrm{cov\_LC}\right)\right)$

$\mathrm{antisymmetric}$

$\mathrm{op}\left(1\,\mathrm{get\_compts}\left(\mathrm{con\_LC}\right)\right)$

$\mathrm{antisymmetric}$

$\mathrm{detg}≔\frac{1}{v^4}$

$\mathrm{detg}≔\frac{1}{v^4}$

$\mathrm{Levi\_Civita}\left(\mathrm{detg}\,2\,\mathrm{Pcov\_LC}\,\mathrm{Pcon\_LC}\right)$

$\mathrm{eval}\left(\mathrm{Pcov\_LC}\right)$

$table\left(\left[\mathrm{compts}=\left[\begin{array}{cc}0& \frac{\mathrm{csgn}\left(\frac{1}{v^2}\right)}{v^2}\\ -\frac{\mathrm{csgn}\left(\frac{1}{v^2}\right)}{v^2}& 0\end{array}\right]\,\mathrm{index\_char}=\left[\mathrm{-1}\,\mathrm{-1}\right]\right]\right)$

$\mathrm{eval}\left(\mathrm{Pcon\_LC}\right)$

$table\left(\left[\mathrm{compts}=\left[\begin{array}{cc}0& \frac{v^2}{\mathrm{csgn}\left(\frac{1}{v^2}\right)}\\ -\frac{v^2}{\mathrm{csgn}\left(\frac{1}{v^2}\right)}& 0\end{array}\right]\,\mathrm{index\_char}=\left[1\,1\right]\right]\right)$