Accordingly, from definition (10) of the

covariant differentiation, we obtain

where the semicolon (;) denotes

covariant differentiation. For small displacements, this expression can be linearized to give the symmetric tensor

where A is a non-zero 1-form such that g(X,[rho]) = A(X) for every vector field X and [nabla] is the operator of

covariant differentiation with respect to the metric g.

[rho] and [bar.[rho]] being unit orthogonal vector fields, [nabla] denotes the operator of

covariant differentiation with respect to the metric tensor g.

for all vector fields X, Y, Z, U, V[member of] x(M), where [alpha], [beta], [gamma], [delta] and [sigma] are 1-forms (non zero simultaneously) and [nabla] is the operator of

covariant differentiation with respect to the Riemannian metric g.

where the vertical bar (|) represents

covariant differentiation. 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.

He begins with manifolds, tensors and exterior forms and progresses to such topics as the integration of differential forms and the Lie derivative, the Poincare Lemma and potentials, Monkowski space,

covariant differentiation and curvature, relativity, Betti numbers and De Rham's theorem, harmonic forms, the Aharonov-Bohm effect, and Yang- Mills fields.

where [rho] is the mass density, F is the body force vector per unit mass, and the first term implies

covariant differentiation.

where the semicolon (;) denotes

covariant differentiation. Rearranging the dummy indices, this expression can be written

for all vector fields X,Y,Z,U,V [member of] [chi] ([M.sup.n]), where A,B,C,D and E are 1-forms (not simultaneously zero) and [nabla] denotes the operator of

covariant differentiation with respect to the Riemannian metric g.

where a single line denotes

covariant differentiation (9-11) with respect to [Theta].

4 Substantial structure of

covariant differentiation in [U.sub.4]