Covariant derivative
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. In the special case of a manifold 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 changes of coordinate in physics: the covariant derivative transforms covariantly under a general coordinate transformation, that is, linearly via the Jacobian matrix of the transformation. 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. History. Historically, at the turn of the 20th century, the covariant derivative was introduced by Gregorio Ricci-Curbastro and Tullio Levi-Civita in the theory of Riemannian and pseudo-Riemannian geometry. Ricci and Levi-Civita (following ideas of Elwin Bruno Christoffel) observed that the Christoffel symbols used to define the curvature could also provide a notion of differentiation which generalized the classical directional derivative of vector fields on a  manifold. This new derivative – the Levi-Civita connection – was "covariant" in the sense that it satisfied Riemann's requirement that objects in geometry should be independent of their description in a particular coordinate system. It was soon noted by other mathematicians, prominent among these being Hermann Weyl, Jan Arnoldus Schouten, and Élie Cartan, that a covariant derivative could be defined abstractly without the presence of a metric. The crucial feature was not a particular dependence on the metric, but that the Christoffel symbols satisfied a certain precise second order transformation law. This transformation law could serve as a starting point  for defining the derivative in a covariant manner. Thus the theory of covariant differentiation forked off from the strictly Riemannian context to include a wider range of possible geometries. In the 1940s, practitioners of differential geometry began introducing other notions of covariant differentiation in general vector bundles which were, in contrast to the classical bundles of interest to geometers, not part of the tensor analysis of the manifold. By and large, these generalized covariant derivatives had to be specified "ad hoc" by some version of the connection concept. In 1950, Jean-Louis Koszul unified these new ideas of covariant differentiation  in a vector bundle by means of what is known today as a Koszul connection or a connection on a vector bundle. Using ideas from Lie algebra cohomology, Koszul successfully converted many of the analytic features of covariant differentiation into algebraic ones. In particular, Koszul connections eliminated the need for awkward manipulations of Christoffel symbols (and other analogous non-tensorial objects) in differential geometry. Thus they quickly supplanted the classical notion of covariant derivative in many post-1950 treatments of the subject. Motivation. The covariant derivative is a generalization of the directional derivative from vector calculus. As with the directional derivative , the covariant derivative is a rule, formula_1, which takes as its inputs: (1) a vector, u, defined at a point "P", and (2) a vector field v defined in a neighborhood of "P". The output is the vector formula_2, also at the point "P". The primary difference from the usual directional derivative is that formula_1 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, 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  translates one of the vectors to the origin of the other, keeping it parallel, then taking 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 Eulidean 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 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 moving along a curve in the Euclidean plane. In polar coordinates, may be written in terms of its radial and angular coordinates by . A vector at a particular time (for instance, the acceleration of the curve) is expressed in terms of formula_4, where formula_5 and formula_6 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. In a curved space, such as the surface of the Earth (regarded as a sphere), the 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 Q is directed to the north. Suppose we transport the vector  (keeping it parallel) first along the equator to the point P, then drag it along a meridian to the N pole, and finally 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 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, and can be defined in terms of the covariant derivative. Informal definition using an embedding into Euclidean space. Suppose an open subset formula_8 of a formula_9-dimensional Riemannian manifold formula_10 is embedded into Euclidean space formula_11 via a twice continuously-differentiable (C) mapping formula_12 such that the tangent space at formula_13 is spanned by the vectors formula_14 and the scalar product formula_15 on formula_16 is compatible with the metric on "M": formula_17 For a tangent vector field, one has formula_18 The last term is not tangential to "M", but can be expressed  as a linear combination of the tangent space base vectors using the Christoffel symbols as linear factors plus a vector orthogonal to the tangent space: formula_19 In the case of the Levi-Civita connection, the covariant derivative formula_20, also written is defined as the orthogonal projection of the usual derivative onto tangent space: formula_21 To obtain the relation between Christoffel symbols for the Levi-Civita connection and the metric, first we must note that, since formula_22 in previous equation is orthogonal to tangent space: formula_23 Second, the partial derivative of a component of the metric is: formula_24 implies for a  basis using the symmetry of the scalar product and swapping the order of partial differentiation: formula_25 adding first row to second and subtracting third one: formula_26 and yields the Christoffel symbols for the Levi-Civita connection in terms of the metric: formula_27 For a very simple example that captures the essence of the description above, draw a circle on a flat sheet of paper. Travel around the circle at a constant speed. The derivative of your velocity, your acceleration vector, always points radially inward. Roll this sheet of paper into a cylinder. Now the (Euclidean) derivative of your velocity has  a component that sometimes points inward toward the axis of the cylinder depending on whether you're near a solstice or an equinox. (At the point of the circle when you are moving parallel to the axis, there is no inward acceleration. Conversely, at a point (1/4 of a circle later) when the velocity is along the cylinder's bend, the inward acceleration is maximum.) This is the (Euclidean) normal component. The covariant derivative component is the component parallel to the cylinder's surface, and is the same as that before you rolled the sheet into a cylinder. Formal definition . A covariant derivative is a (Koszul) connection on the tangent bundle and other tensor bundles: it differentiates vector fields in a way analogous to the usual differential on functions. The definition extends to a differentiation on the dual of vector fields (i.e. covector fields) and to arbitrary tensor fields, in a unique way that ensures compatibility with the tensor product and trace operations (tensor contraction). Functions. Given a point formula_28 of the manifold formula_10, a real function formula_30 on the manifold and a tangent vector formula_31, the covariant derivative of at along is the scalar at , denoted formula_32, that represents  the principal part of the change in the value of "f" when the argument of is changed by the infinitesimal displacement vector . (This is the differential of evaluated against the vector .) Formally, there is a differentiable curve formula_33 such that formula_34 and formula_35, and the covariant derivative of "f" at "p" is defined by formula_36 When formula_37 is a vector field on formula_10, the covariant derivative formula_39 is the function that associates with each point "p" in the common domain of "f" and v the scalar formula_40. For a scalar function "f" and vector field v, the covariant derivative formula_41  coincides with the Lie derivative formula_42, and with the exterior derivative formula_43. Vector fields. Given a point formula_44 of the manifold formula_10, a vector field formula_46 defined in a neighborhood of "p" and a tangent vector formula_31, the covariant derivative of u at "p" along v is the tangent vector at "p", denoted formula_48, such that the following properties hold (for any tangent vectors v, x and y at "p", vector fields u and w defined in a neighborhood of "p", scalar values "g" and "h" at "p", and scalar function "f" defined in a neighborhood of "p"): Note that  formula_49 depends not only on the value of u at "p" but also on values of u in an infinitesimal neighborhood of "p" because of the last property, the product rule. If and are both vector fields defined over a common domain, then formula_59 denotes the vector field whose value at each point "p" of the domain is the tangent vector formula_60. Covector fields. Given a field of covectors (or one-form) formula_61 defined in a neighborhood of "p", its covariant derivative formula_62 is defined in a way to make the resulting operation compatible with tensor contraction and the product  rule. That is, formula_62 is defined as the unique one-form at "p" such that the following identity is satisfied for all vector fields u in a neighborhood of "p" formula_64 The covariant derivative of a covector field along a vector field is again a covector field. Tensor fields. Once the covariant derivative is defined for fields of vectors and covectors it can be defined for arbitrary tensor fields by imposing the following identities for every pair of tensor fields formula_65 and formula_66 in a neighborhood of the point "p": formula_67 and for formula_68 and formula_69 of the same valence  formula_70 The covariant derivative of a tensor field along a vector field v is again a tensor field of the same type. Explicitly, let "T" be a tensor field of type . Consider "T" to be a differentiable multilinear map of smooth sections "α", "α", …, "α" of the cotangent bundle "T""M" and of sections "X", "X", …, "X" of the tangent bundle "TM", written "T"("α", "α", …, "X", "X", …) into R. The covariant derivative of "T" along "Y" is given by the formula formula_71 Coordinate description. Given coordinate functions formula_72 any tangent vector can be described by its components in the basis formula_73  The covariant derivative of a basis vector along a basis vector is again a vector and so can be expressed as a linear combination formula_74. To specify the covariant derivative it is enough to specify the covariant derivative of each basis vector field formula_75 along formula_76. formula_77 the coefficients formula_78 are the components of the connection with respect to a system of local coordinates. In the theory of Riemannian and pseudo-Riemannian manifolds, the components of the Levi-Civita connection with respect to a system of local coordinates are called Christoffel symbols. Then using the rules in the definition, we  find that for general vector fields formula_79 and formula_80 we get formula_81 so formula_82 The first term in this formula is responsible for "twisting" the coordinate system with respect to the covariant derivative and the second for changes of components of the vector field "u". In particular formula_83 In words: the covariant derivative is the usual derivative along the coordinates with correction terms which tell how the coordinates change. For covectors similarly we have formula_84 where formula_85. The covariant derivative of a type tensor field along formula_86 is given by the expression: formula_87 Or, in words: take the partial derivative  of the tensor and add: formula_88 for every upper index formula_89, and formula_90 for every lower index formula_91. If instead of a tensor, one is trying to differentiate a "tensor density" (of weight +1), then you also add a term formula_92 If it is a tensor density of weight "W", then multiply that term by "W". For example, formula_93 is a scalar density (of weight +1), so we get: formula_94 where semicolon ";" indicates covariant differentiation and comma "," indicates partial differentiation. Incidentally, this particular expression is equal to zero, because the covariant derivative of a function solely of the metric is  always zero. Notation. In textbooks on physics, the covariant derivative is sometimes simply stated in terms of its components in this equation. Often a notation is used in which the covariant derivative is given with a semicolon, while a normal partial derivative is indicated by a comma. In this notation we write the same as: formula_95 In case two or more indexes appear after the semicolon, all of them must be understood as covariant derivatives: formula_96 In some older texts (notably Adler, Bazin & Schiffer, "Introduction to General Relativity"), the covariant derivative is denoted by a double pipe and the partial  derivative by single pipe: formula_97 Covariant derivative by field type. For a scalar field formula_98, covariant differentiation is simply partial differentiation: formula_99 For a contravariant vector field formula_100, we have: formula_101 For a covariant vector field formula_102, we have: formula_103 For a type (2,0) tensor field formula_104, we have: formula_105 For a type (0,2) tensor field formula_106, we have: formula_107 For a type (1,1) tensor field formula_108, we have: formula_109 The notation above is meant in the sense formula_110 Properties. In general, covariant derivatives do not commute. By example, the covariant derivatives of vector field formula_111. Riemann tensor formula_112 is  defined such that: formula_113 or, equivalently, formula_114 The covariant derivative of a (2,1)-tensor field fulfills: formula_115 The latter can be shown by taking (without loss of generality) that formula_116. Derivative along a curve. Since the covariant derivative formula_117 of a tensor field formula_118 at a point formula_44 depends only on the value of the vector field formula_120 at formula_44 one can define the covariant derivative along a smooth curve formula_122 in a manifold: formula_123 Note that the tensor field formula_118 only needs to be defined on the curve formula_122 for this definition to make sense. In particular, formula_126 is a  vector field along the curve formula_127 itself. If formula_128 vanishes then the curve is called a geodesic of the covariant derivative. If the covariant derivative is the Levi-Civita connection of a positive-definite metric then the geodesics for the connection are precisely the geodesics of the metric that are parametrized by arc length. The derivative along a curve is also used to define the parallel transport along the curve. Sometimes the covariant derivative along a curve is called absolute or intrinsic derivative. Relation to Lie derivative. A covariant derivative introduces an extra geometric structure on a manifold that allows  vectors in neighboring tangent spaces to be compared: there is no canonical way to compare vectors from different tangent spaces because there is no canonical coordinate system. There is however another generalization of directional derivatives which "is" canonical: the Lie derivative, which evaluates the change of one vector field along the flow of another vector field. Thus, one must know both vector fields in an open neighborhood, not merely at a single point. The covariant derivative on the other hand introduces its own change for vectors in a given direction, and it only depends on the vector direction at a  single point, rather than a vector field in an open neighborhood of a point. In other words, the covariant derivative is linear (over "C"("M")) in the direction argument, while the Lie derivative is linear in neither argument. Note that the antisymmetrized covariant derivative , and the Lie derivative differ by the torsion of the connection, so that if a connection is torsion free, then its antisymmetrization "is" the Lie derivative.