For small values, a strain tensor [gamma], an inclination vector [psi] , a p-rotation scalar [OMEGA] , and the
permutation tensor E = [E.sub.[alpha][beta]] [a.sup.[alpha]] [cross product] [a.sup.[beta]] ([E.sub.11] = 0, [E.sub.12] = -[E.sub.21], [E.sub.22] = 0), the decomposition FN = N +[gamma] -[OMEGA] E + n[cross product][psi] is additive and [Q.sub.i] N [??] N + n[cross product][psi] , [Q.sub.p] [??] N -[OMEGA] E, V [??] N +[gamma], EE = ?N.
where [[epsilon].sub.[micro]v[rho][sigma]] are the components of the completely antisymmetric four-dimensional Levi-Civita permutation tensor density.
where [e.sub.i] = [e.sup.A.sub.i] [partial derivative]/[partial derivative][[bar.X].sup.A] are the elements of the basis vector spanning [[THETA].sub.3], [C.sup.p.sub.ik] are the spin coefficients, i= [square root of (-1)] [??] is a coupling constant, and [[member of].sub.ikl] = [square root of (det (h))][[epsilon].sub.ikl] (where [[epsilon].sub.ikl] are the components of the completely anti-symmetric three-dimensional Levi-Civita permutation tensor density).
Then the covariant and contravariant components of the totally anti-symmetric permutation tensor are given by
where [[epsilon].sub.ijk] are the components of the usual permutation tensor density.
In the same manner, we define the four-dimensional permutation tensor as one with components
[i.sub.p]/[square root of det(g) are the covariant and contravariant components of the completely anti-symmetric Levi-Civita
permutation tensor, respectively, with the ordinary permutation symbols being given as usual by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [[epsilon].sup.[i.sub.1][i.sub.2] ...
where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are the covariant and contravariant components of the completely anti-symmetric Levi-Civita permutation tensor, respectively, with the ordinary permutation symbols being given as usual by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
where [[member of].sub.abcd] = [square root of pdet (g)] [[epsilon].sub.abcd] are the components of the completely anti-symmetric four-dimensional Levi-Civita permutation tensor and [psi] is a vector field normal to a three-dimensional space (hypersurface) [summation](t) defined as the time section ct = [x.sup.0] = const.