covariant derivative


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covariant derivative

[kō′ver·ē·ənt də′riv·əd·iv]
(mathematics)
For a tensor field at a point P of an affine space, a new tensor field equal to the difference between the derivative of the original field defined in the ordinary manner and the derivative of a field whose value at points close to P are parallel to the value of the original field at P as specified by the affine connection.
References in periodicals archive ?
From the definition of covariant derivative, applied to the contravariant vector, we have
The theory of scale relativity aims at describing a nondifferentiable continuous manifold by the building of new tools that implement Einstein's general relativity concepts in the new context (in particular, covariant derivative and geodesics equations).
mu]v] is the covariant derivative, defined in analogous way as in eqs.
mu]] denotes covariant derivative with respect to coordinates (in this case the coordinates [x.
His topics include the basics of geometry and relativity, affine connection and covariant derivative, the geodesic equation and its applications, curvature tensor and Einstein's equation, black holes, and cosmological models and the big bang theory.
In view of the above relations, the covariant derivative of (2.
19) Because the covariant derivative of the metric is zero ([[nabla.
m[xi]] is the bimetric covariant derivative with respect to [xi] and is given by [[nabla].
where D is the covariant derivative on (N, h) and [nabla] is the covariant derivative on (M,g).
In the parentheses we recognize the prototype of the covariant derivative.
By virtue of covariant derivative of (7) with respect to Z we get