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There is no real difference between the covariant derivative and the connection concept except for the style in which they are introduced.
In the theory of Riemannian and pseudo-Riemannian manifolds the term covariant derivative is often used for the Levi-Civita connection.
Here we give a traditional index-notation introduction to the covariant derivative (also known as the tensor derivative) of a vector with respect to a vector field; the covariant derivative of a tensor is an extension of the same concept.
Everywhere in this article we use Einstein notation. It is assumed that the reader is familiar with concept of a differentiable manifold and in particular with tangent vectors.
The covariant derivative (also written as D) of a vector u in the direction of the vector v is a rule that defines a third vector called (also Dvu) which has the properties of a derivative, specified below. A vector is a geometrical object and independent of a chosen basis (coordinate system). Upon fixing a coordinate system, this derivative transforms under a change of coordinates "in the same way" as the vector itself ( covariant transformation), hence the name.
In the case of Euclidean space with an orthonormal coordinate system, one tends to define the derivative of a vector field in terms of the difference between two vectors at two nearby points. In such a system one translates one of the vectors to the origin of the other, keeping it parallel. The obtained covariant derivative on Euclidean space can simply be obtained by taking the derivative of the components.
In the general case, however, one must take into account the change of the coordinate system. In a curved space, such as the surface of the Earth (regarded as a sphere), the translation is not well defined and its analog, parallel transportIn mathematics, a parallel transport on a manifold M with specified connection is a way to transport vectors along smooth curves, in such a way that they stay "parallel" with respect to the given connection. A field on a smooth curve is called parallel if, depends on the path along which the vector is translated. For example, in polar coordinatesThis article describes some of the common coordinate systems that appear in elementary mathematics. For advanced topics, please refer to coordinate system. For more background, see Cartesian coordinate system. The coordinates of a point are the components in a two dimensional Euclidean plane, the derivative contains extra terms that describe how the coordinate grid itself "rotates". In other cases the extra terms describe how the coordinate grid expands, contracts, twists, interweaves, etc.
Here is an example of a curve in polar coordinates in a 2-dim Euclidean space. A vector at curve parameter t (say the acceleration, not shown) is expressed in a coordinate system , where and are unit tangent vectors for the polar coordinates, serving as a basis to decompose a vector in terms of radial and tangential components. 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.
(It is probably better not to think of t as a time parameter, at least for applications in general relativityGeneral relativity (GR or general relativity theory (GRT is the theory of gravitation published by Albert Einstein in 1915. The conceptual core of general relativity, from which its other consequences largely follow, is the Principle of Equivalence which. It is simply an arbitrary parameter varying smoothly and monotonically along the path.)
Another example: A vector e on a globe on the equator in Q is directed to the north. Suppose we parallel transportIn mathematics, a parallel transport on a manifold M with specified connection is a way to transport vectors along smooth curves, in such a way that they stay "parallel" with respect to the given connection. A field on a smooth curve is called parallel if the vector first along the equator until P and then (keeping it parallel to itself) drag it along a meridian to the pole N and (keeping the direction there) subsequently transport it along another meridian back to 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 can be noticed if we drag the vector along an infinitesimally small closed surface subsequently along two directions and then back. The infinitesimal change of the vector is a measure of the curvature.
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