Editing Covariant derivative (section)

==Motivation==
[[File:Ковариантная производная c.jpg|220x124px|thumb|right]]
The '''covariant derivative''' is a generalization of the [[directional derivative]] from [[vector calculus]]. As with the directional derivative, the covariant derivative is a rule, <math>\nabla_{\mathbf u}{\mathbf v}</math>, which takes as its inputs: (1) a vector, {{math|'''u'''}}, defined at a point {{mvar|P}}, and (2) a [[vector field]] {{math|'''v'''}} defined in a [[neighborhood (mathematics)|neighborhood]] of {{mvar|P}}.<ref>The covariant derivative is also denoted variously by '''∂{{sub|v}}u''', '''D{{sub|v}}u''', or other notations.</ref> The output is the vector <math>\nabla_{\mathbf u}{\mathbf v}(P)</math>, also at the point {{mvar|P}}. The primary difference from the usual directional derivative is that <math>\nabla_{\mathbf u}{\mathbf v}</math> must, in a certain precise sense, be ''independent'' of the manner in which it is expressed in a [[coordinate system]].

A vector may be ''described'' as a list of numbers in terms of a [[basis (mathematics)|basis]], but as a geometrical object the vector retains its identity regardless of how it is described. For a geometric vector written in components with respect to one basis, when the basis is changed the components transform according to a [[change of basis]] formula, with the coordinates undergoing a [[covariant transformation]]. The covariant derivative is required to transform, under a change in coordinates, by a covariant transformation in the same way as a basis does (hence the name).

In the case of [[Euclidean space]], one usually defines the directional derivative of a vector field in terms of the difference between two vectors at two nearby points.
In such a system one [[Translation (geometry)|translates]] one of the vectors to the origin of the other, keeping it parallel, then takes their difference within the same vector space. With a Cartesian (fixed [[orthonormal]]) coordinate system "keeping it parallel" amounts to keeping the components constant. This ordinary directional derivative on Euclidean space is the first example of a covariant derivative.

Next, one must take into account changes of the coordinate system. For example, if the Euclidean plane is described by polar coordinates, "keeping it parallel" does ''not'' amount to keeping the polar components constant under translation, since the coordinate grid itself "rotates". Thus, the same covariant derivative written in [[coordinates (elementary mathematics)|polar coordinates]] contains extra terms that describe how the coordinate grid itself rotates, or how in more general coordinates the grid expands, contracts, twists, interweaves, etc.

Consider the example of a particle moving along a curve {{math|''γ''(''t'')}} in the Euclidean plane. In polar coordinates, {{mvar|γ}} may be written in terms of its radial and angular coordinates by {{math|1=''γ''(''t'') = (''r''(''t''), ''θ''(''t''))}}. A vector at a particular time {{mvar|t}}<ref>In many applications, it may be better not to think of {{mvar|t}} as corresponding to time, at least for applications in [[general relativity]]. It is simply regarded as an abstract parameter varying smoothly and monotonically along the path.</ref> (for instance, a constant acceleration of the particle) is expressed in terms of <math>(\mathbf{e}_r, \mathbf{e}_{\theta})</math>, where <math>\mathbf{e}_r</math> and <math>\mathbf{e}_{\theta}</math> are unit tangent vectors for the polar coordinates, serving as a basis to decompose a vector in terms of radial and [[tangential component]]s. At a slightly later time, the new basis in polar coordinates appears slightly rotated with respect to the first set. The covariant derivative of the basis vectors (the [[Christoffel symbols]]) serve to express this change.
{{Clear}}

In a curved space, such as the surface of the Earth (regarded as a sphere), the [[Translation (geometry)|translation]] of tangent vectors between different points is not well defined, and its analog, [[parallel transport]], depends on the path along which the vector is translated. A vector on a globe on the equator at point {{mvar|Q}} is directed to the north. Suppose we transport the vector (keeping it parallel) first along the equator to the point {{mvar|P}}, then drag it along a meridian to the {{mvar|N}} pole, and finally transport it along another meridian back to {{mvar|Q}}. Then we notice that the parallel-transported vector along a closed circuit does not return as the same vector; instead, it has another orientation. This would not happen in Euclidean space and is caused by the ''curvature'' of the surface of the globe. The same effect occurs if we drag the vector along an infinitesimally small closed surface subsequently along two directions and then back. This infinitesimal change of the vector is a measure of the [[Curvature of Riemannian manifolds|curvature]], and can be defined in terms of the covariant derivative.{{Clear}}

===Remarks===
* The definition of the covariant derivative does not use the metric in space. However, for each metric there is a unique [[Torsion tensor|torsion]]-free covariant derivative called the [[Levi-Civita connection]] such that the covariant derivative of the metric is zero.
* The properties of a derivative imply that <math>\nabla_\mathbf{v} \mathbf{u}</math> depends on the values of {{mvar|u}} in a neighborhood of a point {{mvar|p}} in the same way as e.g. the derivative of a scalar function {{mvar|f}} along a curve at a given point {{mvar|p}} depends on the values of {{mvar|f}} in a neighborhood of {{mvar|p}}.
* The information in a neighborhood of a point {{mvar|p}} in the covariant derivative can be used to define [[parallel transport]] of a vector. Also the [[Curvature of Riemannian manifolds|curvature]], [[Torsion tensor|torsion]], and [[geodesic]]s may be defined only in terms of the covariant derivative or other related variation on the idea of a [[Connection (vector bundle)|linear connection]].