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
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 :
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 components
The 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.