mu]] with indices raised as a contravariant object is opposite that of the

covariant form that [A.

They cover the general relativistic two-body problem, Hamiltonian dynamics of spinning compact binaries through high post-Newtonian approximations, the

covariant theory of the post-Newtonian equations of motion of extended bodies, the DSX-framework, the general relativistic theory of light propagation in multipolar gravitational fields, the back-reaction problem in cosmology, and post-Newtonian approximations in cosmology, ([umlaut] Ringgold, Inc.

The key is the

covariant functoriality of the incidence algebra construction.

Understanding of classical field theory underlies understanding of quantum field theory, and this text covers the subject beginning with a chapter on differential calculus on fiber bundles and proceeding with chapters on Lagrangian field theory, Grassmann-graded Lagrangian field theory, Lagrangian BRST theory, gauge theory on principal bundles, gravitation theory on natural bundles, spinor fields, topological field theories, and

covariant Hamiltonian field theory.

perpendicular to]] = {a [member of] A | ab = 0 for all b [member of] ker [phi]}, and we say that a representation ([pi], t) of (H,[phi]) is

covariant if

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.

The body weight changes (g) were calculated as initial body weight (before hibernation) minus final body weight (after hibernation), and then ANCOVA (Analysis of Covariance) was used to evaluate the effects of water depth and sand bed, using the initial body weight as a

covariant.

The second half of the book addresses multi-dimensional modified metric theories of gravity, calculating conserved charges in Einstein-Gauss-Bonnet gravity and generic gravity, and canonical conserved quantities in

covariant field theories possessing the intrinsic symmetries of the field Lagrangian.

Recently, the author, in terms of his 5D fully

covariant theory of gravitation, has quantitatively determined the dielectric constant of the polarized vacuum in accordance with the charge-mass ratio of a charged object [14].

Manoff [5] applied the method of lagrangian with

covariant derivative to special type of lagrangian density depending on scalar and vector fields.

Because blinking could affect the free-view task more, blinking time was used as a

covariant in the

covariant analysis to compare the means.

Since the metric g is parallel with respect to the Levi-Civita connection [nabla], then by taking the

covariant differentiation of (1.