Then, the

covariant components of Riemann tensors on S are defined by

v] is collinear with V), but keeping the algebraic property of diagonalisation of tensors, if expressed in

covariant components.

i](x,p) being the

covariant components of external forces.

mu]v] cannot be obtained directly from the embedding functions, but should be computed from the

covariant components as their inverse matrix.

j] are the

covariant components of the acceleration of the volume in the deformed body.

The Green strain tensor (E) may then be defined in terms of its

covariant components as

The

covariant components vx' and vy' are parallel to the coordinate surface z' = [z.

1), it is defined by a skew-symmetrical S[THETA](4)-invariant tensor field of degree 2 which may be expressed either by its

covariant componentsThe

covariant components of the metric tensor on the deformed middle surface are

Here 2 = [alpha] 1 and nffand nff respectively are the contravariant and

covariant components of the unit vector field n normal to the hypersurface of material coordinates [F.

Therefore, we can express the fully

covariant components of the generalized four-dimensional elasticity tensor in terms of the

covariant components of the symmetric metric tensor [g.

are the

covariant components of the tangent velocity vector field satisfying [u.