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In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold.Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection.
The covariant derivative in general relativity is a special example of the gauge covariant derivative. It corresponds to the Levi Civita connection (a special Riemannian connection ) on the tangent bundle (or the frame bundle ) i.e. it acts on tangent vector fields or more generally, tensors.
The Levi-Civita connection (like any affine connection) also defines a derivative along curves, sometimes denoted by D. Given a smooth curve γ on (M, g) and a vector field V along γ its derivative is defined by = ˙ (). Formally, D is the pullback connection γ*∇ on the pullback bundle γ*TM.
The covariant derivative is such a map for k = 0. The exterior covariant derivatives extends this map to general k. There are several equivalent ways to define this object: [3] Suppose that a vector-valued differential 2-form is regarded as assigning to each p a multilinear map s p: T p M × T p M → E p which is completely anti-symmetric.
The covariant derivatives (also called "tangential derivatives") of Tullio Levi-Civita and Gregorio Ricci-Curbastro provide a means of differentiating smooth tangential vector fields. Given a tangential vector field X and a tangent vector Y to S at p , the covariant derivative ∇ Y X is a certain tangent vector to S at p .
Contravariant vectors are often just called vectors and covariant vectors are called covectors or dual vectors. The terms covariant and contravariant were introduced by James Joseph Sylvester in 1851. [3] [4] Curvilinear coordinate systems, such as cylindrical or spherical coordinates, are often used in physical and geometric problems ...
The fact that the Ehresmann connection is linear implies that in addition it verifies for every function on the Leibniz rule, i.e. () = + (), and therefore is a covariant derivative of s. Conversely a covariant derivative ∇ on a vector bundle defines a linear Ehresmann connection by defining H e, for e ∈ E with x=π(e), to be the image ds x ...
In the math branches of differential geometry and vector calculus, the second covariant derivative, or the second order covariant derivative, of a vector field is the derivative of its derivative with respect to another two tangent vector fields.