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The tensor Package Indexing Functions

Description

Important: The tensor package has been deprecated. Use the superseding packages DifferentialGeometry and Physics instead.

 • The tensor package provides routines which calculate quantities that are specific to the Theory of Relativity.  Many of these quantities possess certain symmetries in the indices of their components.  For this reason, the tensor package provides and uses some specific indexing functions to implement these symmetries.  In some cases, the package also makes use of Maple symmetric and antisymmetric indexing functions.
 • The following indexing functions are implemented in the tensor package:

 cf1 - implements symmetry in the first and second indices of a quantity  with three indices - used for: Christoffel symbols of the first kind, first partials of the covariant metric tensor components cf2 - implements symmetry in the second and third indices of a quantity with three indices - used for: Christoffel symbols of the second kind cov_riemann -implements the symmetric / skew-symmetric properties of the covariant Riemann (and Weyl) tensor components -- that is, R[compts][c,d,a,b] =   R[compts][a,b,c,d] R[compts][b,a,c,d] = - R[compts][a,b,c,d] R[compts][a,b,d,c] = - R[compts][a,b,c,d] - used for: components of the covariant Riemann and Weyl tensors d2met - implements symmetry in the first and second indices and in the third and fourth indices of a quantity with 4 indices - used for: second partials of the covariant metric tensor components skew23 - implements skew-symmetry in the second and third indices of a 3-index quantity - used for: structural coefficients in the tensor[connexF] routine

Examples

Important: The tensor package has been deprecated. Use the superseding packages DifferentialGeometry and Physics instead.

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

Store the Christoffel symbols of the first kind in the array C1.  Use the cf1 indexing function to implement the symmetry in the last two indices.

 > $\mathrm{c1}≔\mathrm{array}\left(\mathrm{cf1},1..4,1..4,1..4\right)$
 ${\mathrm{c1}}{≔}{array}{}\left({\mathrm{cf1}}{,}{1}{..}{4}{,}{1}{..}{4}{,}{1}{..}{4}{,}\left[\left({1}{,}{1}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{1}{,}{1}}{,}\left({1}{,}{1}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{1}{,}{2}}{,}\left({1}{,}{1}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{1}{,}{3}}{,}\left({1}{,}{1}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{1}{,}{4}}{,}\left({1}{,}{2}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{1}}{,}\left({1}{,}{2}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{2}}{,}\left({1}{,}{2}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{3}}{,}\left({1}{,}{2}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{4}}{,}\left({1}{,}{3}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{1}}{,}\left({1}{,}{3}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{2}}{,}\left({1}{,}{3}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{3}}{,}\left({1}{,}{3}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{4}}{,}\left({1}{,}{4}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{4}{,}{1}}{,}\left({1}{,}{4}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{4}{,}{2}}{,}\left({1}{,}{4}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{4}{,}{3}}{,}\left({1}{,}{4}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{4}{,}{4}}{,}\left({2}{,}{1}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{1}}{,}\left({2}{,}{1}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{2}}{,}\left({2}{,}{1}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{3}}{,}\left({2}{,}{1}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{4}}{,}\left({2}{,}{2}{,}{1}\right){=}{{\mathrm{?}}}_{{2}{,}{2}{,}{1}}{,}\left({2}{,}{2}{,}{2}\right){=}{{\mathrm{?}}}_{{2}{,}{2}{,}{2}}{,}\left({2}{,}{2}{,}{3}\right){=}{{\mathrm{?}}}_{{2}{,}{2}{,}{3}}{,}\left({2}{,}{2}{,}{4}\right){=}{{\mathrm{?}}}_{{2}{,}{2}{,}{4}}{,}\left({2}{,}{3}{,}{1}\right){=}{{\mathrm{?}}}_{{2}{,}{3}{,}{1}}{,}\left({2}{,}{3}{,}{2}\right){=}{{\mathrm{?}}}_{{2}{,}{3}{,}{2}}{,}\left({2}{,}{3}{,}{3}\right){=}{{\mathrm{?}}}_{{2}{,}{3}{,}{3}}{,}\left({2}{,}{3}{,}{4}\right){=}{{\mathrm{?}}}_{{2}{,}{3}{,}{4}}{,}\left({2}{,}{4}{,}{1}\right){=}{{\mathrm{?}}}_{{2}{,}{4}{,}{1}}{,}\left({2}{,}{4}{,}{2}\right){=}{{\mathrm{?}}}_{{2}{,}{4}{,}{2}}{,}\left({2}{,}{4}{,}{3}\right){=}{{\mathrm{?}}}_{{2}{,}{4}{,}{3}}{,}\left({2}{,}{4}{,}{4}\right){=}{{\mathrm{?}}}_{{2}{,}{4}{,}{4}}{,}\left({3}{,}{1}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{1}}{,}\left({3}{,}{1}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{2}}{,}\left({3}{,}{1}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{3}}{,}\left({3}{,}{1}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{4}}{,}\left({3}{,}{2}{,}{1}\right){=}{{\mathrm{?}}}_{{2}{,}{3}{,}{1}}{,}\left({3}{,}{2}{,}{2}\right){=}{{\mathrm{?}}}_{{2}{,}{3}{,}{2}}{,}\left({3}{,}{2}{,}{3}\right){=}{{\mathrm{?}}}_{{2}{,}{3}{,}{3}}{,}\left({3}{,}{2}{,}{4}\right){=}{{\mathrm{?}}}_{{2}{,}{3}{,}{4}}{,}\left({3}{,}{3}{,}{1}\right){=}{{\mathrm{?}}}_{{3}{,}{3}{,}{1}}{,}\left({3}{,}{3}{,}{2}\right){=}{{\mathrm{?}}}_{{3}{,}{3}{,}{2}}{,}\left({3}{,}{3}{,}{3}\right){=}{{\mathrm{?}}}_{{3}{,}{3}{,}{3}}{,}\left({3}{,}{3}{,}{4}\right){=}{{\mathrm{?}}}_{{3}{,}{3}{,}{4}}{,}\left({3}{,}{4}{,}{1}\right){=}{{\mathrm{?}}}_{{3}{,}{4}{,}{1}}{,}\left({3}{,}{4}{,}{2}\right){=}{{\mathrm{?}}}_{{3}{,}{4}{,}{2}}{,}\left({3}{,}{4}{,}{3}\right){=}{{\mathrm{?}}}_{{3}{,}{4}{,}{3}}{,}\left({3}{,}{4}{,}{4}\right){=}{{\mathrm{?}}}_{{3}{,}{4}{,}{4}}{,}\left({4}{,}{1}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{4}{,}{1}}{,}\left({4}{,}{1}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{4}{,}{2}}{,}\left({4}{,}{1}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{4}{,}{3}}{,}\left({4}{,}{1}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{4}{,}{4}}{,}\left({4}{,}{2}{,}{1}\right){=}{{\mathrm{?}}}_{{2}{,}{4}{,}{1}}{,}\left({4}{,}{2}{,}{2}\right){=}{{\mathrm{?}}}_{{2}{,}{4}{,}{2}}{,}\left({4}{,}{2}{,}{3}\right){=}{{\mathrm{?}}}_{{2}{,}{4}{,}{3}}{,}\left({4}{,}{2}{,}{4}\right){=}{{\mathrm{?}}}_{{2}{,}{4}{,}{4}}{,}\left({4}{,}{3}{,}{1}\right){=}{{\mathrm{?}}}_{{3}{,}{4}{,}{1}}{,}\left({4}{,}{3}{,}{2}\right){=}{{\mathrm{?}}}_{{3}{,}{4}{,}{2}}{,}\left({4}{,}{3}{,}{3}\right){=}{{\mathrm{?}}}_{{3}{,}{4}{,}{3}}{,}\left({4}{,}{3}{,}{4}\right){=}{{\mathrm{?}}}_{{3}{,}{4}{,}{4}}{,}\left({4}{,}{4}{,}{1}\right){=}{{\mathrm{?}}}_{{4}{,}{4}{,}{1}}{,}\left({4}{,}{4}{,}{2}\right){=}{{\mathrm{?}}}_{{4}{,}{4}{,}{2}}{,}\left({4}{,}{4}{,}{3}\right){=}{{\mathrm{?}}}_{{4}{,}{4}{,}{3}}{,}\left({4}{,}{4}{,}{4}\right){=}{{\mathrm{?}}}_{{4}{,}{4}{,}{4}}\right]\right)$ (1)
 > ${\mathrm{c1}}_{1,2,1}≔\frac{m}{{r}^{2}}$
 ${{\mathrm{c1}}}_{{1}{,}{2}{,}{1}}{≔}\frac{{m}}{{{r}}^{{2}}}$ (2)
 > ${\mathrm{c1}}_{2,1,1}$
 $\frac{{m}}{{{r}}^{{2}}}$ (3)

Store the Christoffel symbols of the second kind in the array c2.  Use the cf2 indexing function to implement the symmetry in the first two indices.

 > $\mathrm{c2}≔\mathrm{array}\left(\mathrm{cf2},1..4,1..4,1..4\right)$
 ${\mathrm{c2}}{≔}{array}{}\left({\mathrm{cf2}}{,}{1}{..}{4}{,}{1}{..}{4}{,}{1}{..}{4}{,}\left[\left({1}{,}{1}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{1}{,}{1}}{,}\left({1}{,}{1}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{1}{,}{2}}{,}\left({1}{,}{1}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{1}{,}{3}}{,}\left({1}{,}{1}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{1}{,}{4}}{,}\left({1}{,}{2}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{1}{,}{2}}{,}\left({1}{,}{2}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{2}}{,}\left({1}{,}{2}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{3}}{,}\left({1}{,}{2}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{4}}{,}\left({1}{,}{3}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{1}{,}{3}}{,}\left({1}{,}{3}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{3}}{,}\left({1}{,}{3}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{3}}{,}\left({1}{,}{3}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{4}}{,}\left({1}{,}{4}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{1}{,}{4}}{,}\left({1}{,}{4}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{4}}{,}\left({1}{,}{4}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{4}}{,}\left({1}{,}{4}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{4}{,}{4}}{,}\left({2}{,}{1}{,}{1}\right){=}{{\mathrm{?}}}_{{2}{,}{1}{,}{1}}{,}\left({2}{,}{1}{,}{2}\right){=}{{\mathrm{?}}}_{{2}{,}{1}{,}{2}}{,}\left({2}{,}{1}{,}{3}\right){=}{{\mathrm{?}}}_{{2}{,}{1}{,}{3}}{,}\left({2}{,}{1}{,}{4}\right){=}{{\mathrm{?}}}_{{2}{,}{1}{,}{4}}{,}\left({2}{,}{2}{,}{1}\right){=}{{\mathrm{?}}}_{{2}{,}{1}{,}{2}}{,}\left({2}{,}{2}{,}{2}\right){=}{{\mathrm{?}}}_{{2}{,}{2}{,}{2}}{,}\left({2}{,}{2}{,}{3}\right){=}{{\mathrm{?}}}_{{2}{,}{2}{,}{3}}{,}\left({2}{,}{2}{,}{4}\right){=}{{\mathrm{?}}}_{{2}{,}{2}{,}{4}}{,}\left({2}{,}{3}{,}{1}\right){=}{{\mathrm{?}}}_{{2}{,}{1}{,}{3}}{,}\left({2}{,}{3}{,}{2}\right){=}{{\mathrm{?}}}_{{2}{,}{2}{,}{3}}{,}\left({2}{,}{3}{,}{3}\right){=}{{\mathrm{?}}}_{{2}{,}{3}{,}{3}}{,}\left({2}{,}{3}{,}{4}\right){=}{{\mathrm{?}}}_{{2}{,}{3}{,}{4}}{,}\left({2}{,}{4}{,}{1}\right){=}{{\mathrm{?}}}_{{2}{,}{1}{,}{4}}{,}\left({2}{,}{4}{,}{2}\right){=}{{\mathrm{?}}}_{{2}{,}{2}{,}{4}}{,}\left({2}{,}{4}{,}{3}\right){=}{{\mathrm{?}}}_{{2}{,}{3}{,}{4}}{,}\left({2}{,}{4}{,}{4}\right){=}{{\mathrm{?}}}_{{2}{,}{4}{,}{4}}{,}\left({3}{,}{1}{,}{1}\right){=}{{\mathrm{?}}}_{{3}{,}{1}{,}{1}}{,}\left({3}{,}{1}{,}{2}\right){=}{{\mathrm{?}}}_{{3}{,}{1}{,}{2}}{,}\left({3}{,}{1}{,}{3}\right){=}{{\mathrm{?}}}_{{3}{,}{1}{,}{3}}{,}\left({3}{,}{1}{,}{4}\right){=}{{\mathrm{?}}}_{{3}{,}{1}{,}{4}}{,}\left({3}{,}{2}{,}{1}\right){=}{{\mathrm{?}}}_{{3}{,}{1}{,}{2}}{,}\left({3}{,}{2}{,}{2}\right){=}{{\mathrm{?}}}_{{3}{,}{2}{,}{2}}{,}\left({3}{,}{2}{,}{3}\right){=}{{\mathrm{?}}}_{{3}{,}{2}{,}{3}}{,}\left({3}{,}{2}{,}{4}\right){=}{{\mathrm{?}}}_{{3}{,}{2}{,}{4}}{,}\left({3}{,}{3}{,}{1}\right){=}{{\mathrm{?}}}_{{3}{,}{1}{,}{3}}{,}\left({3}{,}{3}{,}{2}\right){=}{{\mathrm{?}}}_{{3}{,}{2}{,}{3}}{,}\left({3}{,}{3}{,}{3}\right){=}{{\mathrm{?}}}_{{3}{,}{3}{,}{3}}{,}\left({3}{,}{3}{,}{4}\right){=}{{\mathrm{?}}}_{{3}{,}{3}{,}{4}}{,}\left({3}{,}{4}{,}{1}\right){=}{{\mathrm{?}}}_{{3}{,}{1}{,}{4}}{,}\left({3}{,}{4}{,}{2}\right){=}{{\mathrm{?}}}_{{3}{,}{2}{,}{4}}{,}\left({3}{,}{4}{,}{3}\right){=}{{\mathrm{?}}}_{{3}{,}{3}{,}{4}}{,}\left({3}{,}{4}{,}{4}\right){=}{{\mathrm{?}}}_{{3}{,}{4}{,}{4}}{,}\left({4}{,}{1}{,}{1}\right){=}{{\mathrm{?}}}_{{4}{,}{1}{,}{1}}{,}\left({4}{,}{1}{,}{2}\right){=}{{\mathrm{?}}}_{{4}{,}{1}{,}{2}}{,}\left({4}{,}{1}{,}{3}\right){=}{{\mathrm{?}}}_{{4}{,}{1}{,}{3}}{,}\left({4}{,}{1}{,}{4}\right){=}{{\mathrm{?}}}_{{4}{,}{1}{,}{4}}{,}\left({4}{,}{2}{,}{1}\right){=}{{\mathrm{?}}}_{{4}{,}{1}{,}{2}}{,}\left({4}{,}{2}{,}{2}\right){=}{{\mathrm{?}}}_{{4}{,}{2}{,}{2}}{,}\left({4}{,}{2}{,}{3}\right){=}{{\mathrm{?}}}_{{4}{,}{2}{,}{3}}{,}\left({4}{,}{2}{,}{4}\right){=}{{\mathrm{?}}}_{{4}{,}{2}{,}{4}}{,}\left({4}{,}{3}{,}{1}\right){=}{{\mathrm{?}}}_{{4}{,}{1}{,}{3}}{,}\left({4}{,}{3}{,}{2}\right){=}{{\mathrm{?}}}_{{4}{,}{2}{,}{3}}{,}\left({4}{,}{3}{,}{3}\right){=}{{\mathrm{?}}}_{{4}{,}{3}{,}{3}}{,}\left({4}{,}{3}{,}{4}\right){=}{{\mathrm{?}}}_{{4}{,}{3}{,}{4}}{,}\left({4}{,}{4}{,}{1}\right){=}{{\mathrm{?}}}_{{4}{,}{1}{,}{4}}{,}\left({4}{,}{4}{,}{2}\right){=}{{\mathrm{?}}}_{{4}{,}{2}{,}{4}}{,}\left({4}{,}{4}{,}{3}\right){=}{{\mathrm{?}}}_{{4}{,}{3}{,}{4}}{,}\left({4}{,}{4}{,}{4}\right){=}{{\mathrm{?}}}_{{4}{,}{4}{,}{4}}\right]\right)$ (4)
 > ${\mathrm{c2}}_{4,3,4}≔\frac{\mathrm{cos}\left(\mathrm{θ}\right)}{\mathrm{sin}\left(\mathrm{θ}\right)}$
 ${{\mathrm{c2}}}_{{4}{,}{3}{,}{4}}{≔}\frac{{\mathrm{cos}}{}\left({\mathrm{\theta }}\right)}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}$ (5)
 > ${\mathrm{c2}}_{4,4,3}$
 $\frac{{\mathrm{cos}}{}\left({\mathrm{\theta }}\right)}{{\mathrm{sin}}{}\left({\mathrm{\theta }}\right)}$ (6)

Store the covariant Riemann tensor components in the array R.  Use the cov_riemann indexing function the implement the symmetries of the covariant Riemann tensor.

 > $R≔\mathrm{array}\left(\mathrm{cov_riemann},1..4,1..4,1..4,1..4\right)$
 ${R}{≔}{array}{}\left({\mathrm{cov_riemann}}{,}{1}{..}{4}{,}{1}{..}{4}{,}{1}{..}{4}{,}{1}{..}{4}{,}\left[\left({1}{,}{1}{,}{1}{,}{1}\right){=}{0}{,}\left({1}{,}{1}{,}{1}{,}{2}\right){=}{0}{,}\left({1}{,}{1}{,}{1}{,}{3}\right){=}{0}{,}\left({1}{,}{1}{,}{1}{,}{4}\right){=}{0}{,}\left({1}{,}{1}{,}{2}{,}{1}\right){=}{0}{,}\left({1}{,}{1}{,}{2}{,}{2}\right){=}{0}{,}\left({1}{,}{1}{,}{2}{,}{3}\right){=}{0}{,}\left({1}{,}{1}{,}{2}{,}{4}\right){=}{0}{,}\left({1}{,}{1}{,}{3}{,}{1}\right){=}{0}{,}\left({1}{,}{1}{,}{3}{,}{2}\right){=}{0}{,}\left({1}{,}{1}{,}{3}{,}{3}\right){=}{0}{,}\left({1}{,}{1}{,}{3}{,}{4}\right){=}{0}{,}\left({1}{,}{1}{,}{4}{,}{1}\right){=}{0}{,}\left({1}{,}{1}{,}{4}{,}{2}\right){=}{0}{,}\left({1}{,}{1}{,}{4}{,}{3}\right){=}{0}{,}\left({1}{,}{1}{,}{4}{,}{4}\right){=}{0}{,}\left({1}{,}{2}{,}{1}{,}{1}\right){=}{0}{,}\left({1}{,}{2}{,}{1}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{1}{,}{2}}{,}\left({1}{,}{2}{,}{1}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{1}{,}{3}}{,}\left({1}{,}{2}{,}{1}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{1}{,}{4}}{,}\left({1}{,}{2}{,}{2}{,}{1}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{2}{,}{1}{,}{2}}{,}\left({1}{,}{2}{,}{2}{,}{2}\right){=}{0}{,}\left({1}{,}{2}{,}{2}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{2}{,}{3}}{,}\left({1}{,}{2}{,}{2}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{2}{,}{4}}{,}\left({1}{,}{2}{,}{3}{,}{1}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{2}{,}{1}{,}{3}}{,}\left({1}{,}{2}{,}{3}{,}{2}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{2}{,}{2}{,}{3}}{,}\left({1}{,}{2}{,}{3}{,}{3}\right){=}{0}{,}\left({1}{,}{2}{,}{3}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{3}{,}{4}}{,}\left({1}{,}{2}{,}{4}{,}{1}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{2}{,}{1}{,}{4}}{,}\left({1}{,}{2}{,}{4}{,}{2}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{2}{,}{2}{,}{4}}{,}\left({1}{,}{2}{,}{4}{,}{3}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{2}{,}{3}{,}{4}}{,}\left({1}{,}{2}{,}{4}{,}{4}\right){=}{0}{,}\left({1}{,}{3}{,}{1}{,}{1}\right){=}{0}{,}\left({1}{,}{3}{,}{1}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{1}{,}{3}}{,}\left({1}{,}{3}{,}{1}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{1}{,}{3}}{,}\left({1}{,}{3}{,}{1}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{1}{,}{4}}{,}\left({1}{,}{3}{,}{2}{,}{1}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{2}{,}{1}{,}{3}}{,}\left({1}{,}{3}{,}{2}{,}{2}\right){=}{0}{,}\left({1}{,}{3}{,}{2}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{2}{,}{3}}{,}\left({1}{,}{3}{,}{2}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{2}{,}{4}}{,}\left({1}{,}{3}{,}{3}{,}{1}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{3}{,}{1}{,}{3}}{,}\left({1}{,}{3}{,}{3}{,}{2}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{3}{,}{2}{,}{3}}{,}\left({1}{,}{3}{,}{3}{,}{3}\right){=}{0}{,}\left({1}{,}{3}{,}{3}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{3}{,}{4}}{,}\left({1}{,}{3}{,}{4}{,}{1}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{3}{,}{1}{,}{4}}{,}\left({1}{,}{3}{,}{4}{,}{2}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{3}{,}{2}{,}{4}}{,}\left({1}{,}{3}{,}{4}{,}{3}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{3}{,}{3}{,}{4}}{,}\left({1}{,}{3}{,}{4}{,}{4}\right){=}{0}{,}\left({1}{,}{4}{,}{1}{,}{1}\right){=}{0}{,}\left({1}{,}{4}{,}{1}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{1}{,}{4}}{,}\left({1}{,}{4}{,}{1}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{1}{,}{4}}{,}\left({1}{,}{4}{,}{1}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{4}{,}{1}{,}{4}}{,}\left({1}{,}{4}{,}{2}{,}{1}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{2}{,}{1}{,}{4}}{,}\left({1}{,}{4}{,}{2}{,}{2}\right){=}{0}{,}\left({1}{,}{4}{,}{2}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{4}{,}{2}{,}{3}}{,}\left({1}{,}{4}{,}{2}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{4}{,}{2}{,}{4}}{,}\left({1}{,}{4}{,}{3}{,}{1}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{3}{,}{1}{,}{4}}{,}\left({1}{,}{4}{,}{3}{,}{2}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{4}{,}{2}{,}{3}}{,}\left({1}{,}{4}{,}{3}{,}{3}\right){=}{0}{,}\left({1}{,}{4}{,}{3}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{4}{,}{3}{,}{4}}{,}\left({1}{,}{4}{,}{4}{,}{1}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{4}{,}{1}{,}{4}}{,}\left({1}{,}{4}{,}{4}{,}{2}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{4}{,}{2}{,}{4}}{,}\left({1}{,}{4}{,}{4}{,}{3}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{4}{,}{3}{,}{4}}{,}\left({1}{,}{4}{,}{4}{,}{4}\right){=}{0}{,}\left({2}{,}{1}{,}{1}{,}{1}\right){=}{0}{,}\left({2}{,}{1}{,}{1}{,}{2}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{2}{,}{1}{,}{2}}{,}\left({2}{,}{1}{,}{1}{,}{3}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{2}{,}{1}{,}{3}}{,}\left({2}{,}{1}{,}{1}{,}{4}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{2}{,}{1}{,}{4}}{,}\left({2}{,}{1}{,}{2}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{1}{,}{2}}{,}\left({2}{,}{1}{,}{2}{,}{2}\right){=}{0}{,}\left({2}{,}{1}{,}{2}{,}{3}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{2}{,}{2}{,}{3}}{,}\left({2}{,}{1}{,}{2}{,}{4}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{2}{,}{2}{,}{4}}{,}\left({2}{,}{1}{,}{3}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{1}{,}{3}}{,}\left({2}{,}{1}{,}{3}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{2}{,}{3}}{,}\left({2}{,}{1}{,}{3}{,}{3}\right){=}{0}{,}\left({2}{,}{1}{,}{3}{,}{4}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{2}{,}{3}{,}{4}}{,}\left({2}{,}{1}{,}{4}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{1}{,}{4}}{,}\left({2}{,}{1}{,}{4}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{2}{,}{4}}{,}\left({2}{,}{1}{,}{4}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{3}{,}{4}}{,}\left({2}{,}{1}{,}{4}{,}{4}\right){=}{0}{,}\left({2}{,}{2}{,}{1}{,}{1}\right){=}{0}{,}\left({2}{,}{2}{,}{1}{,}{2}\right){=}{0}{,}\left({2}{,}{2}{,}{1}{,}{3}\right){=}{0}{,}\left({2}{,}{2}{,}{1}{,}{4}\right){=}{0}{,}\left({2}{,}{2}{,}{2}{,}{1}\right){=}{0}{,}\left({2}{,}{2}{,}{2}{,}{2}\right){=}{0}{,}\left({2}{,}{2}{,}{2}{,}{3}\right){=}{0}{,}\left({2}{,}{2}{,}{2}{,}{4}\right){=}{0}{,}\left({2}{,}{2}{,}{3}{,}{1}\right){=}{0}{,}\left({2}{,}{2}{,}{3}{,}{2}\right){=}{0}{,}\left({2}{,}{2}{,}{3}{,}{3}\right){=}{0}{,}\left({2}{,}{2}{,}{3}{,}{4}\right){=}{0}{,}\left({2}{,}{2}{,}{4}{,}{1}\right){=}{0}{,}\left({2}{,}{2}{,}{4}{,}{2}\right){=}{0}{,}\left({2}{,}{2}{,}{4}{,}{3}\right){=}{0}{,}\left({2}{,}{2}{,}{4}{,}{4}\right){=}{0}{,}\left({2}{,}{3}{,}{1}{,}{1}\right){=}{0}{,}\left({2}{,}{3}{,}{1}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{2}{,}{3}}{,}\left({2}{,}{3}{,}{1}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{2}{,}{3}}{,}\left({2}{,}{3}{,}{1}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{4}{,}{2}{,}{3}}{,}\left({2}{,}{3}{,}{2}{,}{1}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{2}{,}{2}{,}{3}}{,}\left({2}{,}{3}{,}{2}{,}{2}\right){=}{0}{,}\left({2}{,}{3}{,}{2}{,}{3}\right){=}{{\mathrm{?}}}_{{2}{,}{3}{,}{2}{,}{3}}{,}\left({2}{,}{3}{,}{2}{,}{4}\right){=}{{\mathrm{?}}}_{{2}{,}{3}{,}{2}{,}{4}}{,}\left({2}{,}{3}{,}{3}{,}{1}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{3}{,}{2}{,}{3}}{,}\left({2}{,}{3}{,}{3}{,}{2}\right){=}{-}{{\mathrm{?}}}_{{2}{,}{3}{,}{2}{,}{3}}{,}\left({2}{,}{3}{,}{3}{,}{3}\right){=}{0}{,}\left({2}{,}{3}{,}{3}{,}{4}\right){=}{{\mathrm{?}}}_{{2}{,}{3}{,}{3}{,}{4}}{,}\left({2}{,}{3}{,}{4}{,}{1}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{4}{,}{2}{,}{3}}{,}\left({2}{,}{3}{,}{4}{,}{2}\right){=}{-}{{\mathrm{?}}}_{{2}{,}{3}{,}{2}{,}{4}}{,}\left({2}{,}{3}{,}{4}{,}{3}\right){=}{-}{{\mathrm{?}}}_{{2}{,}{3}{,}{3}{,}{4}}{,}\left({2}{,}{3}{,}{4}{,}{4}\right){=}{0}{,}\left({2}{,}{4}{,}{1}{,}{1}\right){=}{0}{,}\left({2}{,}{4}{,}{1}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{2}{,}{4}}{,}\left({2}{,}{4}{,}{1}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{2}{,}{4}}{,}\left({2}{,}{4}{,}{1}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{4}{,}{2}{,}{4}}{,}\left({2}{,}{4}{,}{2}{,}{1}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{2}{,}{2}{,}{4}}{,}\left({2}{,}{4}{,}{2}{,}{2}\right){=}{0}{,}\left({2}{,}{4}{,}{2}{,}{3}\right){=}{{\mathrm{?}}}_{{2}{,}{3}{,}{2}{,}{4}}{,}\left({2}{,}{4}{,}{2}{,}{4}\right){=}{{\mathrm{?}}}_{{2}{,}{4}{,}{2}{,}{4}}{,}\left({2}{,}{4}{,}{3}{,}{1}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{3}{,}{2}{,}{4}}{,}\left({2}{,}{4}{,}{3}{,}{2}\right){=}{-}{{\mathrm{?}}}_{{2}{,}{3}{,}{2}{,}{4}}{,}\left({2}{,}{4}{,}{3}{,}{3}\right){=}{0}{,}\left({2}{,}{4}{,}{3}{,}{4}\right){=}{{\mathrm{?}}}_{{2}{,}{4}{,}{3}{,}{4}}{,}\left({2}{,}{4}{,}{4}{,}{1}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{4}{,}{2}{,}{4}}{,}\left({2}{,}{4}{,}{4}{,}{2}\right){=}{-}{{\mathrm{?}}}_{{2}{,}{4}{,}{2}{,}{4}}{,}\left({2}{,}{4}{,}{4}{,}{3}\right){=}{-}{{\mathrm{?}}}_{{2}{,}{4}{,}{3}{,}{4}}{,}\left({2}{,}{4}{,}{4}{,}{4}\right){=}{0}{,}\left({3}{,}{1}{,}{1}{,}{1}\right){=}{0}{,}\left({3}{,}{1}{,}{1}{,}{2}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{2}{,}{1}{,}{3}}{,}\left({3}{,}{1}{,}{1}{,}{3}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{3}{,}{1}{,}{3}}{,}\left({3}{,}{1}{,}{1}{,}{4}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{3}{,}{1}{,}{4}}{,}\left({3}{,}{1}{,}{2}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{1}{,}{3}}{,}\left({3}{,}{1}{,}{2}{,}{2}\right){=}{0}{,}\left({3}{,}{1}{,}{2}{,}{3}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{3}{,}{2}{,}{3}}{,}\left({3}{,}{1}{,}{2}{,}{4}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{3}{,}{2}{,}{4}}{,}\left({3}{,}{1}{,}{3}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{1}{,}{3}}{,}\left({3}{,}{1}{,}{3}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{2}{,}{3}}{,}\left({3}{,}{1}{,}{3}{,}{3}\right){=}{0}{,}\left({3}{,}{1}{,}{3}{,}{4}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{3}{,}{3}{,}{4}}{,}\left({3}{,}{1}{,}{4}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{1}{,}{4}}{,}\left({3}{,}{1}{,}{4}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{2}{,}{4}}{,}\left({3}{,}{1}{,}{4}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{3}{,}{4}}{,}\left({3}{,}{1}{,}{4}{,}{4}\right){=}{0}{,}\left({3}{,}{2}{,}{1}{,}{1}\right){=}{0}{,}\left({3}{,}{2}{,}{1}{,}{2}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{2}{,}{2}{,}{3}}{,}\left({3}{,}{2}{,}{1}{,}{3}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{3}{,}{2}{,}{3}}{,}\left({3}{,}{2}{,}{1}{,}{4}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{4}{,}{2}{,}{3}}{,}\left({3}{,}{2}{,}{2}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{2}{,}{3}}{,}\left({3}{,}{2}{,}{2}{,}{2}\right){=}{0}{,}\left({3}{,}{2}{,}{2}{,}{3}\right){=}{-}{{\mathrm{?}}}_{{2}{,}{3}{,}{2}{,}{3}}{,}\left({3}{,}{2}{,}{2}{,}{4}\right){=}{-}{{\mathrm{?}}}_{{2}{,}{3}{,}{2}{,}{4}}{,}\left({3}{,}{2}{,}{3}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{2}{,}{3}}{,}\left({3}{,}{2}{,}{3}{,}{2}\right){=}{{\mathrm{?}}}_{{2}{,}{3}{,}{2}{,}{3}}{,}\left({3}{,}{2}{,}{3}{,}{3}\right){=}{0}{,}\left({3}{,}{2}{,}{3}{,}{4}\right){=}{-}{{\mathrm{?}}}_{{2}{,}{3}{,}{3}{,}{4}}{,}\left({3}{,}{2}{,}{4}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{4}{,}{2}{,}{3}}{,}\left({3}{,}{2}{,}{4}{,}{2}\right){=}{{\mathrm{?}}}_{{2}{,}{3}{,}{2}{,}{4}}{,}\left({3}{,}{2}{,}{4}{,}{3}\right){=}{{\mathrm{?}}}_{{2}{,}{3}{,}{3}{,}{4}}{,}\left({3}{,}{2}{,}{4}{,}{4}\right){=}{0}{,}\left({3}{,}{3}{,}{1}{,}{1}\right){=}{0}{,}\left({3}{,}{3}{,}{1}{,}{2}\right){=}{0}{,}\left({3}{,}{3}{,}{1}{,}{3}\right){=}{0}{,}\left({3}{,}{3}{,}{1}{,}{4}\right){=}{0}{,}\left({3}{,}{3}{,}{2}{,}{1}\right){=}{0}{,}\left({3}{,}{3}{,}{2}{,}{2}\right){=}{0}{,}\left({3}{,}{3}{,}{2}{,}{3}\right){=}{0}{,}\left({3}{,}{3}{,}{2}{,}{4}\right){=}{0}{,}\left({3}{,}{3}{,}{3}{,}{1}\right){=}{0}{,}\left({3}{,}{3}{,}{3}{,}{2}\right){=}{0}{,}\left({3}{,}{3}{,}{3}{,}{3}\right){=}{0}{,}\left({3}{,}{3}{,}{3}{,}{4}\right){=}{0}{,}\left({3}{,}{3}{,}{4}{,}{1}\right){=}{0}{,}\left({3}{,}{3}{,}{4}{,}{2}\right){=}{0}{,}\left({3}{,}{3}{,}{4}{,}{3}\right){=}{0}{,}\left({3}{,}{3}{,}{4}{,}{4}\right){=}{0}{,}\left({3}{,}{4}{,}{1}{,}{1}\right){=}{0}{,}\left({3}{,}{4}{,}{1}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{3}{,}{4}}{,}\left({3}{,}{4}{,}{1}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{3}{,}{4}}{,}\left({3}{,}{4}{,}{1}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{4}{,}{3}{,}{4}}{,}\left({3}{,}{4}{,}{2}{,}{1}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{2}{,}{3}{,}{4}}{,}\left({3}{,}{4}{,}{2}{,}{2}\right){=}{0}{,}\left({3}{,}{4}{,}{2}{,}{3}\right){=}{{\mathrm{?}}}_{{2}{,}{3}{,}{3}{,}{4}}{,}\left({3}{,}{4}{,}{2}{,}{4}\right){=}{{\mathrm{?}}}_{{2}{,}{4}{,}{3}{,}{4}}{,}\left({3}{,}{4}{,}{3}{,}{1}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{3}{,}{3}{,}{4}}{,}\left({3}{,}{4}{,}{3}{,}{2}\right){=}{-}{{\mathrm{?}}}_{{2}{,}{3}{,}{3}{,}{4}}{,}\left({3}{,}{4}{,}{3}{,}{3}\right){=}{0}{,}\left({3}{,}{4}{,}{3}{,}{4}\right){=}{{\mathrm{?}}}_{{3}{,}{4}{,}{3}{,}{4}}{,}\left({3}{,}{4}{,}{4}{,}{1}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{4}{,}{3}{,}{4}}{,}\left({3}{,}{4}{,}{4}{,}{2}\right){=}{-}{{\mathrm{?}}}_{{2}{,}{4}{,}{3}{,}{4}}{,}\left({3}{,}{4}{,}{4}{,}{3}\right){=}{-}{{\mathrm{?}}}_{{3}{,}{4}{,}{3}{,}{4}}{,}\left({3}{,}{4}{,}{4}{,}{4}\right){=}{0}{,}\left({4}{,}{1}{,}{1}{,}{1}\right){=}{0}{,}\left({4}{,}{1}{,}{1}{,}{2}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{2}{,}{1}{,}{4}}{,}\left({4}{,}{1}{,}{1}{,}{3}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{3}{,}{1}{,}{4}}{,}\left({4}{,}{1}{,}{1}{,}{4}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{4}{,}{1}{,}{4}}{,}\left({4}{,}{1}{,}{2}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{1}{,}{4}}{,}\left({4}{,}{1}{,}{2}{,}{2}\right){=}{0}{,}\left({4}{,}{1}{,}{2}{,}{3}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{4}{,}{2}{,}{3}}{,}\left({4}{,}{1}{,}{2}{,}{4}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{4}{,}{2}{,}{4}}{,}\left({4}{,}{1}{,}{3}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{1}{,}{4}}{,}\left({4}{,}{1}{,}{3}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{4}{,}{2}{,}{3}}{,}\left({4}{,}{1}{,}{3}{,}{3}\right){=}{0}{,}\left({4}{,}{1}{,}{3}{,}{4}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{4}{,}{3}{,}{4}}{,}\left({4}{,}{1}{,}{4}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{4}{,}{1}{,}{4}}{,}\left({4}{,}{1}{,}{4}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{4}{,}{2}{,}{4}}{,}\left({4}{,}{1}{,}{4}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{4}{,}{3}{,}{4}}{,}\left({4}{,}{1}{,}{4}{,}{4}\right){=}{0}{,}\left({4}{,}{2}{,}{1}{,}{1}\right){=}{0}{,}\left({4}{,}{2}{,}{1}{,}{2}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{2}{,}{2}{,}{4}}{,}\left({4}{,}{2}{,}{1}{,}{3}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{3}{,}{2}{,}{4}}{,}\left({4}{,}{2}{,}{1}{,}{4}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{4}{,}{2}{,}{4}}{,}\left({4}{,}{2}{,}{2}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{2}{,}{4}}{,}\left({4}{,}{2}{,}{2}{,}{2}\right){=}{0}{,}\left({4}{,}{2}{,}{2}{,}{3}\right){=}{-}{{\mathrm{?}}}_{{2}{,}{3}{,}{2}{,}{4}}{,}\left({4}{,}{2}{,}{2}{,}{4}\right){=}{-}{{\mathrm{?}}}_{{2}{,}{4}{,}{2}{,}{4}}{,}\left({4}{,}{2}{,}{3}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{2}{,}{4}}{,}\left({4}{,}{2}{,}{3}{,}{2}\right){=}{{\mathrm{?}}}_{{2}{,}{3}{,}{2}{,}{4}}{,}\left({4}{,}{2}{,}{3}{,}{3}\right){=}{0}{,}\left({4}{,}{2}{,}{3}{,}{4}\right){=}{-}{{\mathrm{?}}}_{{2}{,}{4}{,}{3}{,}{4}}{,}\left({4}{,}{2}{,}{4}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{4}{,}{2}{,}{4}}{,}\left({4}{,}{2}{,}{4}{,}{2}\right){=}{{\mathrm{?}}}_{{2}{,}{4}{,}{2}{,}{4}}{,}\left({4}{,}{2}{,}{4}{,}{3}\right){=}{{\mathrm{?}}}_{{2}{,}{4}{,}{3}{,}{4}}{,}\left({4}{,}{2}{,}{4}{,}{4}\right){=}{0}{,}\left({4}{,}{3}{,}{1}{,}{1}\right){=}{0}{,}\left({4}{,}{3}{,}{1}{,}{2}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{2}{,}{3}{,}{4}}{,}\left({4}{,}{3}{,}{1}{,}{3}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{3}{,}{3}{,}{4}}{,}\left({4}{,}{3}{,}{1}{,}{4}\right){=}{-}{{\mathrm{?}}}_{{1}{,}{4}{,}{3}{,}{4}}{,}\left({4}{,}{3}{,}{2}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{3}{,}{4}}{,}\left({4}{,}{3}{,}{2}{,}{2}\right){=}{0}{,}\left({4}{,}{3}{,}{2}{,}{3}\right){=}{-}{{\mathrm{?}}}_{{2}{,}{3}{,}{3}{,}{4}}{,}\left({4}{,}{3}{,}{2}{,}{4}\right){=}{-}{{\mathrm{?}}}_{{2}{,}{4}{,}{3}{,}{4}}{,}\left({4}{,}{3}{,}{3}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{3}{,}{4}}{,}\left({4}{,}{3}{,}{3}{,}{2}\right){=}{{\mathrm{?}}}_{{2}{,}{3}{,}{3}{,}{4}}{,}\left({4}{,}{3}{,}{3}{,}{3}\right){=}{0}{,}\left({4}{,}{3}{,}{3}{,}{4}\right){=}{-}{{\mathrm{?}}}_{{3}{,}{4}{,}{3}{,}{4}}{,}\left({4}{,}{3}{,}{4}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{4}{,}{3}{,}{4}}{,}\left({4}{,}{3}{,}{4}{,}{2}\right){=}{{\mathrm{?}}}_{{2}{,}{4}{,}{3}{,}{4}}{,}\left({4}{,}{3}{,}{4}{,}{3}\right){=}{{\mathrm{?}}}_{{3}{,}{4}{,}{3}{,}{4}}{,}\left({4}{,}{3}{,}{4}{,}{4}\right){=}{0}{,}\left({4}{,}{4}{,}{1}{,}{1}\right){=}{0}{,}\left({4}{,}{4}{,}{1}{,}{2}\right){=}{0}{,}\left({4}{,}{4}{,}{1}{,}{3}\right){=}{0}{,}\left({4}{,}{4}{,}{1}{,}{4}\right){=}{0}{,}\left({4}{,}{4}{,}{2}{,}{1}\right){=}{0}{,}\left({4}{,}{4}{,}{2}{,}{2}\right){=}{0}{,}\left({4}{,}{4}{,}{2}{,}{3}\right){=}{0}{,}\left({4}{,}{4}{,}{2}{,}{4}\right){=}{0}{,}\left({4}{,}{4}{,}{3}{,}{1}\right){=}{0}{,}\left({4}{,}{4}{,}{3}{,}{2}\right){=}{0}{,}\left({4}{,}{4}{,}{3}{,}{3}\right){=}{0}{,}\left({4}{,}{4}{,}{3}{,}{4}\right){=}{0}{,}\left({4}{,}{4}{,}{4}{,}{1}\right){=}{0}{,}\left({4}{,}{4}{,}{4}{,}{2}\right){=}{0}{,}\left({4}{,}{4}{,}{4}{,}{3}\right){=}{0}{,}\left({4}{,}{4}{,}{4}{,}{4}\right){=}{0}\right]\right)$ (7)
 > ${R}_{1,2,3,4}≔\frac{\mathrm{cos}\left(\mathrm{θ}\right)}{r}$
 ${{R}}_{{1}{,}{2}{,}{3}{,}{4}}{≔}\frac{{\mathrm{cos}}{}\left({\mathrm{\theta }}\right)}{{r}}$ (8)
 > ${R}_{3,4,1,2}$
 $\frac{{\mathrm{cos}}{}\left({\mathrm{\theta }}\right)}{{r}}$ (9)
 > ${R}_{2,1,3,4}$
 ${-}\frac{{\mathrm{cos}}{}\left({\mathrm{\theta }}\right)}{{r}}$ (10)
 > ${R}_{1,2,4,3}$
 ${-}\frac{{\mathrm{cos}}{}\left({\mathrm{\theta }}\right)}{{r}}$ (11)
 > ${R}_{4,3,1,2}$
 ${-}\frac{{\mathrm{cos}}{}\left({\mathrm{\theta }}\right)}{{r}}$ (12)
 > ${R}_{3,4,2,1}$
 ${-}\frac{{\mathrm{cos}}{}\left({\mathrm{\theta }}\right)}{{r}}$ (13)
 > ${R}_{4,3,2,1}$
 $\frac{{\mathrm{cos}}{}\left({\mathrm{\theta }}\right)}{{r}}$ (14)
 > ${R}_{2,1,4,3}$
 $\frac{{\mathrm{cos}}{}\left({\mathrm{\theta }}\right)}{{r}}$ (15)

Store the second partials of the covariant metric tensor components in the array d2g. Use the d2met indexing function to implement the symmetries in the first and second pairs of indices:

 > $\mathrm{d2g}≔\mathrm{array}\left(\mathrm{d2met},1..4,1..4,1..4,1..4\right)$
 ${\mathrm{d2g}}{≔}{array}{}\left({\mathrm{d2met}}{,}{1}{..}{4}{,}{1}{..}{4}{,}{1}{..}{4}{,}{1}{..}{4}{,}\left[\left({1}{,}{1}{,}{1}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{1}{,}{1}{,}{1}}{,}\left({1}{,}{1}{,}{1}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{1}{,}{1}{,}{2}}{,}\left({1}{,}{1}{,}{1}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{1}{,}{1}{,}{3}}{,}\left({1}{,}{1}{,}{1}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{1}{,}{1}{,}{4}}{,}\left({1}{,}{1}{,}{2}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{1}{,}{1}{,}{2}}{,}\left({1}{,}{1}{,}{2}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{1}{,}{2}{,}{2}}{,}\left({1}{,}{1}{,}{2}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{1}{,}{2}{,}{3}}{,}\left({1}{,}{1}{,}{2}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{1}{,}{2}{,}{4}}{,}\left({1}{,}{1}{,}{3}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{1}{,}{1}{,}{3}}{,}\left({1}{,}{1}{,}{3}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{1}{,}{2}{,}{3}}{,}\left({1}{,}{1}{,}{3}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{1}{,}{3}{,}{3}}{,}\left({1}{,}{1}{,}{3}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{1}{,}{3}{,}{4}}{,}\left({1}{,}{1}{,}{4}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{1}{,}{1}{,}{4}}{,}\left({1}{,}{1}{,}{4}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{1}{,}{2}{,}{4}}{,}\left({1}{,}{1}{,}{4}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{1}{,}{3}{,}{4}}{,}\left({1}{,}{1}{,}{4}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{1}{,}{4}{,}{4}}{,}\left({1}{,}{2}{,}{1}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{1}{,}{1}}{,}\left({1}{,}{2}{,}{1}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{1}{,}{2}}{,}\left({1}{,}{2}{,}{1}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{1}{,}{3}}{,}\left({1}{,}{2}{,}{1}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{1}{,}{4}}{,}\left({1}{,}{2}{,}{2}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{1}{,}{2}}{,}\left({1}{,}{2}{,}{2}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{2}{,}{2}}{,}\left({1}{,}{2}{,}{2}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{2}{,}{3}}{,}\left({1}{,}{2}{,}{2}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{2}{,}{4}}{,}\left({1}{,}{2}{,}{3}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{1}{,}{3}}{,}\left({1}{,}{2}{,}{3}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{2}{,}{3}}{,}\left({1}{,}{2}{,}{3}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{3}{,}{3}}{,}\left({1}{,}{2}{,}{3}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{3}{,}{4}}{,}\left({1}{,}{2}{,}{4}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{1}{,}{4}}{,}\left({1}{,}{2}{,}{4}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{2}{,}{4}}{,}\left({1}{,}{2}{,}{4}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{3}{,}{4}}{,}\left({1}{,}{2}{,}{4}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{4}{,}{4}}{,}\left({1}{,}{3}{,}{1}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{1}{,}{1}}{,}\left({1}{,}{3}{,}{1}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{1}{,}{2}}{,}\left({1}{,}{3}{,}{1}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{1}{,}{3}}{,}\left({1}{,}{3}{,}{1}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{1}{,}{4}}{,}\left({1}{,}{3}{,}{2}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{1}{,}{2}}{,}\left({1}{,}{3}{,}{2}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{2}{,}{2}}{,}\left({1}{,}{3}{,}{2}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{2}{,}{3}}{,}\left({1}{,}{3}{,}{2}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{2}{,}{4}}{,}\left({1}{,}{3}{,}{3}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{1}{,}{3}}{,}\left({1}{,}{3}{,}{3}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{2}{,}{3}}{,}\left({1}{,}{3}{,}{3}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{3}{,}{3}}{,}\left({1}{,}{3}{,}{3}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{3}{,}{4}}{,}\left({1}{,}{3}{,}{4}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{1}{,}{4}}{,}\left({1}{,}{3}{,}{4}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{2}{,}{4}}{,}\left({1}{,}{3}{,}{4}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{3}{,}{4}}{,}\left({1}{,}{3}{,}{4}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{3}{,}{4}{,}{4}}{,}\left({1}{,}{4}{,}{1}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{4}{,}{1}{,}{1}}{,}\left({1}{,}{4}{,}{1}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{4}{,}{1}{,}{2}}{,}\left({1}{,}{4}{,}{1}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{4}{,}{1}{,}{3}}{,}\left({1}{,}{4}{,}{1}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{4}{,}{1}{,}{4}}{,}\left({1}{,}{4}{,}{2}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{4}{,}{1}{,}{2}}{,}\left({1}{,}{4}{,}{2}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{4}{,}{2}{,}{2}}{,}\left({1}{,}{4}{,}{2}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{4}{,}{2}{,}{3}}{,}\left({1}{,}{4}{,}{2}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{4}{,}{2}{,}{4}}{,}\left({1}{,}{4}{,}{3}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{4}{,}{1}{,}{3}}{,}\left({1}{,}{4}{,}{3}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{4}{,}{2}{,}{3}}{,}\left({1}{,}{4}{,}{3}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{4}{,}{3}{,}{3}}{,}\left({1}{,}{4}{,}{3}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{4}{,}{3}{,}{4}}{,}\left({1}{,}{4}{,}{4}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{4}{,}{1}{,}{4}}{,}\left({1}{,}{4}{,}{4}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{4}{,}{2}{,}{4}}{,}\left({1}{,}{4}{,}{4}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{4}{,}{3}{,}{4}}{,}\left({1}{,}{4}{,}{4}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{4}{,}{4}{,}{4}}{,}\left({2}{,}{1}{,}{1}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{1}{,}{1}}{,}\left({2}{,}{1}{,}{1}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{1}{,}{2}}{,}\left({2}{,}{1}{,}{1}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{1}{,}{3}}{,}\left({2}{,}{1}{,}{1}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{1}{,}{4}}{,}\left({2}{,}{1}{,}{2}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{1}{,}{2}}{,}\left({2}{,}{1}{,}{2}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{2}{,}{2}}{,}\left({2}{,}{1}{,}{2}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{2}{,}{3}}{,}\left({2}{,}{1}{,}{2}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{2}{,}{4}}{,}\left({2}{,}{1}{,}{3}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{1}{,}{3}}{,}\left({2}{,}{1}{,}{3}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{2}{,}{3}}{,}\left({2}{,}{1}{,}{3}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{3}{,}{3}}{,}\left({2}{,}{1}{,}{3}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{3}{,}{4}}{,}\left({2}{,}{1}{,}{4}{,}{1}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{1}{,}{4}}{,}\left({2}{,}{1}{,}{4}{,}{2}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{2}{,}{4}}{,}\left({2}{,}{1}{,}{4}{,}{3}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{3}{,}{4}}{,}\left({2}{,}{1}{,}{4}{,}{4}\right){=}{{\mathrm{?}}}_{{1}{,}{2}{,}{4}{,}{4}}{,}\left({2}{,}{2}{,}{1}{,}{1}\right){=}{{\mathrm{?}}}_{{2}{,}{2}{,}{1}{,}{1}}{,}\left({2}{,}{2}{,}{1}{,}{2}\right){=}{{\mathrm{?}}}_{{2}{,}{2}{,}{1}{,}{2}}{,}\left({2}{,}{2}{,}{1}{,}{3}\right){=}{{\mathrm{?}}}_{{2}{,}{2}{,}{1}{,}{3}}{,}\left({2}{,}{2}{,}{1}{,}{4}\right){=}{{\mathrm{?}}}_{{2}{,}{2}{,}{1}{,}{4}}{,}\left({2}{,}{2}{,}{2}{,}{1}\right){=}{{\mathrm{?}}}_{{2}{,}{2}{,}{1}{,}{2}}{,}\left({2}{,}{2}{,}{2}{,}{2}\right){=}{{\mathrm{?}}}_{{2}{,}{2}{,}{2}{,}{2}}{,}\left({2}{,}{2}{,}{2}{,}{3}\right){=}{{\mathrm{?}}}_{{2}{,}{2}{,}{2}{,}{3}}{,}\left({2}{,}{2}{,}{2}{,}{4}\right){=}{{\mathrm{?}}}_{{2}{,}{2}{,}{2}{,}{4}}{,}\left({2}{,}{2}{,}{3}{,}{1}\right){=}{{\mathrm{?}}}_{{2}{,}{2}{,}{1}{,}{3}}{,}\left({2}{,}{2}{,}{3}{,}{2}\right){=}{{\mathrm{?}}}_{{2}{,}{2}{,}{2}{,}{3}}{,}\left({2}{,}{2}{,}{3}{,}{3}\right){=}{{\mathrm{?}}}_{{2}{,}{2}{,}{3}{,}{3}}{,}\left({2}{,}{2}{,}{3}{,}{4}\right){=}{{\mathrm{?}}}_{{2}{,}{2}{,}{3}{,}{4}}{,}\left({2}{,}{2}{,}{4}{,}{1}\right){=}{{\mathrm{?}}}_{{2}{,}{2}{,}{1}{,}{4}}{,}\left({2}{,}{2}{,}{4}{,}{2}\right){=}{{\mathrm{?}}}_{{2}{,}{2}{,}{2}{,}{4}}{,}\left({2}{,}{2}{,}{4}{,}{3}\right){=}{{\mathrm{?}}}_{{2}{,}{2}{,}{3}{,}{4}}{,}\left({2}{,}{2}{,}{4}{,}{4}\right){=}{{\mathrm{?}}}_{{2}{,}{2}{,}{4}{,}{4}}{,}\left({2}{,}{3}{,}{1}{,}{1}\right){=}{{\mathrm{?}}}_{{2}{,}{3}{,}{1}{,}{1}}{,}\left({2}{,}{3}{,}{1}{,}{2}\right){=}{{\mathrm{?}}}_{{2}{,}{3}{,}{1}{,}{2}}{,}\left({2}{,}{3}{,}{1}{,}{3}\right){=}{{\mathrm{?}}}_{{2}{,}{3}{,}{1}{,}{3}}{,}\left({2}{,}{3}{,}{1}{,}{4}\right){=}{{\mathrm{?}}}_{{2}{,}{3}{,}{1}{,}{4}}{,}\left({2}{,}{3}{,}{2}{,}{1}\right){=}{{\mathrm{?}}}_{{2}{,}{3}{,}{1}{,}{2}}{,}\left({2}{,}{3}{,}{2}{,}{2}\right){=}{{\mathrm{?}}}_{{2}{,}{3}{,}{2}{,}{2}}{,}\left({2}{,}{3}{,}{2}{,}{3}\right){=}{{\mathrm{?}}}_{{2}{,}{3}{,}{2}{,}{3}}{,}\left({2}{,}{3}{,}{2}{,}{4}\right){=}{{\mathrm{?}}}_{{2}{,}{3}{,}{2}{,}{4}}{,}\left({2}{,}{3}{,}{3}{,}{1}\right){=}{{\mathrm{?}}}_{{2}{,}{3}{,}{1}{,}{3}}{,}\left({2}{,}{3}{,}{3}{,}{2}\right){=}{{\mathrm{?}}}_{{2}{,}{3}{,}{2}{,}{3}}{,}\left({2}{,}{3}{,}{3}{,}{3}\right){=}{{\mathrm{?}}}_{{2}{,}{3}{,}{3}{,}{3}}{,}\left({2}{,}{3}{,}{3}{,}{4}\right){=}{{\mathrm{?}}}_{{2}{,}{3}{,}{3}{,}{4}}{,}\left({2}{,}{3}{,}{4}{,}{1}\right){=}{{\mathrm{?}}}_{{2}{,}{3}{,}{1}{,}{4}}{,}\left({2}{,}{3}{,}{4}{,}{2}\right){=}{{\mathrm{?}}}_{{2}{,}{3}{,}{2}{,}{4}}{,}\left({2}{,}{3}{,}{4}{,}{3}\right){=}{{\mathrm{?}}}_{{2}{,}{3}{,}{3}{,}{4}}{,}\left({2}{,}{3}{,}{4}{,}{4}\right){=}{{\mathrm{?}}}_{{2}{,}{3}{,}{4}{,}{4}}{,}\left({2}{,}{4}{,}{1}{,}{1}\right){=}{{\mathrm{?}}}_{{2}{,}{4}{,}{1}{,}{1}}{,}\left({2}{,}{4}{,}{1}{,}{2}\right){=}{{\mathrm{?}}}_{{2}{,}{4}{,}{1}{,}{2}}{,}\left({2}{,}{4}{,}{1}{,}{3}\right){=}{{\mathrm{?}}}_{{2}{,}{4}{,}{1}{,}{3}}{,}\left({2}{,}{4}{,}{1}{,}{4}\right){=}{{\mathrm{?}}}_{{2}{,}{4}{,}{1}{,}{4}}{,}\left({2}{,}{4}{,}{2}{,}{1}\right){=}{{\mathrm{?}}}_{{2}{,}{4}{,}{1}{,}{2}}{,}\left({2}{,}{4}{,}{2}{,}{2}\right){=}{{\mathrm{?}}}_{{2}{,}{4}{,}{2}{,}{2}}{,}\left({2}{,}{4}{,}{2}{,}{3}\right){=}{{\mathrm{?}}}_{{2}{,}{4}{,}{2}{,}{3}}{,}\left({2}{,}\right)\right]\right)$