where

covariant components are identified by subscripts and contravariant components by superscripts and [C.sub.a], [C.sub.b], [C.sub.c], [D.sub.a], [D.sub.b], and [D.sub.c] are constants depending on material parameters and may involve metric coefficients.

Using relation (1), we can write the

covariant components of (77) in the form

Conjugation arises as a broader concept than that of principal directions (a particular case, where [[bar.[GAMMA]].sub.v] is collinear with V), but keeping the algebraic property of diagonalisation of tensors, if expressed in

covariant components.

[F.sub.i](x,p) being the

covariant components of external forces.

The inverse metric tensor [[??].sup.[mu]v] cannot be obtained directly from the embedding functions, but should be computed from the

covariant components as their inverse matrix.

Also, [f.sub.i] are components of the body force vector [??], per unit volume; [rho] is the mass density per unit volume; [[??].sub.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

Then, the

covariant components of Riemann tensors on S are defined by

The change from Cartesian components ([v.sub.x] ; [v.sub.y;] [v.sub.z]) of a vector [??] to

covariant components ([v.sub.x]' ; [v.sub.y]' ; [v.sub.z]') is given by [1]:

Regarding the electromagnetic field, with respect to (1.1), 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.sub.3], whose canonical form may be given as [alpha] [micro][micro]i ; k[micro] = 0 where k[ appa]is a parameter.