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 ( 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
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
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: