Since all the derivatives are

covariant, we can replace a passive coordinate change on an open set with an active transformation of the metric field [g.

The incidence algebra construction is functorial in both the

covariant and contravariant senses, and in Section 5, we prove a Beck-Chevalley theorem enabling the two to be unified.

algebra generated by a

covariant representation of (H, [phi]).

for all vector fields X, Y where [nabla] denotes the operator of

covariant differentiation with respect to the Lorentzian metric g, [eta] is 1-form associated to [xi],

Describing the matter to be also

covariant in the 5D spacetime as the fields are, however, this author analytically derived the fifteenth component of the 5D energy-momentum tensor without assuming any unknown parameter ([14] and references therein such as the early studies by the author [28-29]),

The corresponding Euler-Lagrange's equation and energy - momentum tensors are found on the basis of the

covariant Noether's identities.

256 Annotation: Covariance analysis was employed with blinking times as the

covariant In comparison with the control group, (*) p < 0.

Thus, this study observed that there was an apparent anti-knock benefit from HoV in DI engines, but with fuels for which changes in octane sensitivity and HoV are

covariant.

denotes the

covariant differentiation along [gamma] with respect to the Riemannian connection [nabla].

WISC-III was adopted in this study for selection of participants, and the full-scale IQ was used as a

covariant in the analysis of age effects on TLN-C performance.

In this section, we will consider the usual case that all charts (in the atlas of the space-time manifold) are allowed; that is, we will discuss generally

covariant theories.

i](y) constitute the

covariant basis at the point [theta](y), whereas the vectors [[?

Thus, an ACV deviates from a CKV field if there exists a second-order

covariant constant symmetric tensor K [not equal to] [bar.

To regain the local symmetry, the field theory replaces the usual derivatives with

covariant derivatives by adding a gauge term.

For the

covariant derivatives of the Levi-Civita connection of the left-invariant metric [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], defined above the following is true: