4 Substantial structure of covariant differentiation in [U.

represents covariant differentiation with respect to the symmetric connection [DELTA] (x, u) alone.

where a single line denotes

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

denotes

covariant differentiation with respect to the Christoffel symbols alone, and where

where the semicolon (;) denotes

covariant differentiation.

n], g) is said to be locally symmetric due to Cartan if its curvature tensor R satisfies the relation [nabla]R = 0, where [nabla] denotes the operator of

covariant differentiation with respect to the metric tensor g.

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.

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.

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.