Covariant derivative
A generalization of the notion of a derivative to fields of different geometrical objects on manifolds, such as vectors, tensors, forms, etc. It is a linear operator acting on the module of tensor fields
of given valency and defined with respect to a vector field
on a manifold
and satisfying the following properties:
1) ,
2) , where
and
are differentiable functions on
. This mapping is trivially extended by linearity to the algebra of tensor fields and one additionally requires for the action on tensors
of different valency:
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where denotes the tensor product. Thus
is a derivation on the algebra of tensor fields (cf. Derivation in a ring); it has the additional properties of commuting with operations of contraction (cf. Contraction of a tensor), skew-symmetrization (cf. Alternation) and symmetrization of tensors (cf. Symmetrization (of tensors)).
Properties 1) and 2) of (for vector fields) allow one to introduce on
a linear connection (and the corresponding parallel displacement) and on the basis of this, to give a local definition of a covariant derivative which, when extended to the whole manifold, coincides with the operator
defined above; see also Covariant differentiation.
Comments
There is not much of a difference between the notions of a covariant derivative and covariant differentiation and both are used in the same context.
Covariant derivative. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Covariant_derivative&oldid=46543