Editing Covariant derivative (section)

{{Short description|Specification of a derivative along a tangent vector of a manifold}}
{{Use American English|date=March 2019}}
{{About||directional tensor derivatives in continuum mechanics|Tensor derivative (continuum mechanics)|the covariant derivative used in gauge theories|Gauge covariant derivative}}

In [[mathematics]], the '''covariant derivative''' is a way of specifying a [[derivative]] along [[tangent vector]]s of a [[manifold]]. Alternatively, the covariant derivative is a way of introducing and working with a [[connection (mathematics)|connection]] on a manifold by means of a [[differential operator]], to be contrasted with the approach given by a [[connection (principal bundle)|principal connection]] on the [[frame bundle]] – see [[affine connection]]. In the special case of a manifold [[isometry|isometrically]] embedded into a higher-dimensional [[Euclidean space]], the covariant derivative can be viewed as the [[orthogonal projection]] of the Euclidean [[directional derivative]] onto the manifold's tangent space. In this case the Euclidean derivative is broken into two parts, the extrinsic normal component (dependent on the embedding) and the intrinsic covariant derivative component.

The name is motivated by the importance of [[general covariance|changes of coordinate]] in [[physics]]: the covariant derivative transforms [[Covariant transformation|covariantly]] under a general coordinate transformation, that is, linearly via the [[Jacobian matrix and determinant|Jacobian matrix]] of the transformation.<ref>{{cite book|last=Einstein|first=Albert|title=The Meaning of Relativity|url=https://archive.org/details/meaningofrelativ00eins_0|chapter=The General Theory of Relativity | year=1922}}</ref>

This article presents an introduction to the covariant derivative of a [[vector field]] with respect to a vector field, both in a coordinate-free language and using a local [[coordinate system]] and the traditional index notation. The covariant derivative of a [[tensor field]] is presented as an extension of the same concept. The covariant derivative generalizes straightforwardly to a notion of differentiation associated to a [[connection on a vector bundle]], also known as a '''Koszul connection'''.