Search results
Results From The WOW.Com Content Network
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.
In tensor analysis, a covariant vector varies more or less reciprocally to a corresponding contravariant vector. Expressions for lengths, areas and volumes of objects in the vector space can then be given in terms of tensors with covariant and contravariant indices.
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.
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 derivative can in turn be recovered from parallel transport. [25] In fact can be calculated at a point p, by taking a curve c through p with tangent X, using parallel transport to view the restriction of Y to c as a function in the tangent space at p and then taking the derivative.
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 ...