# 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.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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Accordingly, from definition (10) of the covariant differentiation, we obtain
where the semicolon (;) denotes covariant differentiation. For small displacements, this expression can be linearized to give the symmetric tensor
where A is a non-zero 1-form such that g(X,[rho]) = A(X) for every vector field X and [nabla] is the operator of covariant differentiation with respect to the metric g.
[rho] and [bar.[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.
where the vertical bar (|) represents covariant differentiation. Also, [f.sub.i] are components of the body force vector [??], per unit volume; [rho] is the mass density per unit volume; [[??].sub.j] are the covariant components of the acceleration of the volume in the deformed body.
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.
where the semicolon (;) denotes covariant differentiation. Rearranging the dummy indices, this expression can be written
for all vector fields X,Y,Z,U,V [member of] [chi] ([M.sup.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.
where a single line denotes covariant differentiation (9-11) with respect to [Theta].
4 Substantial structure of covariant differentiation in [U.sub.4]

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