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 ?
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.
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.