In
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the
covariant derivative is a way of specifying a
derivativeIn calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
along
tangent vectorA tangent vector is a vector that is tangent to a curve or surface at a given point.Tangent vectors are described in the differential geometry of curves in the context of curves in Rn. More generally, tangent vectors are elements of a tangent space of a differentiable manifold....
s of a
manifoldIn mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
. Alternatively, the covariant derivative is a way of introducing and working with a
connectionIn geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner. There are a variety of kinds of connections in modern geometry, depending on what sort of data one wants to transport...
on a manifold by means of a
differential operatorIn mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...
, to be contrasted with the approach given by a
principal connectionIn mathematics, a connection is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points...
on the frame bundle – see
affine connectionIn the branch of mathematics called differential geometry, an affine connection is a geometrical object on a smooth manifold which connects nearby tangent spaces, and so permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space...
. In the special case of a manifold isometrically embedded into a higher dimensional Euclidean space, the covariant derivative can be viewed as the orthonormal projection of the Euclidean derivative along a tangent vector onto the manifold's tangent space. In this case the Euclidean derivative is broken into two parts, the extrinsic normal component and the intrinsic covariant derivative component.
This article presents an introduction to the covariant derivative of a
vector fieldIn vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
with respect to a vector field, both in a coordinate free language and using a local
coordinate systemIn geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...
and the traditional index notation. The covariant derivative of a
tensor fieldIn mathematics, physics and engineering, a tensor field assigns a tensor to each point of a mathematical space . Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical...
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.
Introduction and history
Historically, at the turn of the 20th century, the covariant derivative was introduced by
Gregorio RicciCurbastroGregorio RicciCurbastro was an Italian mathematician. He was born at Lugo di Romagna. He is most famous as the inventor of the tensor calculus but published important work in many fields....
and
Tullio LeviCivitaTullio LeviCivita, FRS was an Italian mathematician, most famous for his work on absolute differential calculus and its applications to the theory of relativity, but who also made significant contributions in other areas. He was a pupil of Gregorio RicciCurbastro, the inventor of tensor calculus...
in the theory of
RiemannianRiemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...
and
pseudoRiemannian geometryIn differential geometry, a pseudoRiemannian manifold is a generalization of a Riemannian manifold. It is one of many mathematical objects named after Bernhard Riemann. The key difference between a Riemannian manifold and a pseudoRiemannian manifold is that on a pseudoRiemannian manifold the...
. Ricci and LeviCivita (following ideas of
Elwin Bruno ChristoffelElwin Bruno Christoffel was a German mathematician and physicist.Life:...
) observed that the
Christoffel symbolsIn mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel , are numerical arrays of real numbers that describe, in coordinates, the effects of parallel transport in curved surfaces and, more generally, manifolds. As such, they are coordinatespace expressions for the...
used to define the curvature could also provide a notion of
differentiationIn calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
which generalized the classical
directional derivativeIn mathematics, the directional derivative of a multivariate differentiable function along a given vector V at a given point P intuitively represents the instantaneous rate of change of the function, moving through P in the direction of V...
of vector fields on a manifold. This new derivative – the
LeviCivita connectionIn Riemannian geometry, the LeviCivita connection is a specific connection on the tangent bundle of a manifold. More specifically, it is the torsionfree metric connection, i.e., the torsionfree connection on the tangent bundle preserving a given Riemannian metric.The fundamental theorem of...
– 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 WeylHermann Klaus Hugo Weyl was a German mathematician and theoretical physicist. Although much of his working life was spent in Zürich, Switzerland and then Princeton, he is associated with the University of Göttingen tradition of mathematics, represented by David Hilbert and Hermann Minkowski.His...
,
Jan Arnoldus SchoutenJan Arnoldus Schouten was a Dutch mathematician. He was an important contributor to the development of tensor calculus and was one of the founders of the Mathematisch Centrum in Amsterdam....
, and
Élie CartanÉlie Joseph Cartan was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications...
, that a covariant derivative could be defined abstractly without the presence of a
metricIn the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...
. 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 bundleIn mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
s 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,
JeanLouis KoszulJeanLouis Koszul is a mathematician best known for studying geometry and discovering the Koszul complex.He was educated at the Lycée FusteldeCoulanges in Strasbourg before studying at the Faculty of Science in Strasbourg and the Faculty of Science in Paris...
unified these new ideas of covariant differentiation in a vector bundle by means of what is known today as a
Koszul connectionIn mathematics, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. If the fiber bundle is a vector bundle, then the notion of parallel transport is required to be linear...
or a
connection on a vector bundle. Using ideas from
Lie algebra cohomologyIn mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras. It was defined by in order to give an algebraic construction of the cohomology of the underlying topological spaces of compact Lie groups...
, 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 symbolsIn mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel , are numerical arrays of real numbers that describe, in coordinates, the effects of parallel transport in curved surfaces and, more generally, manifolds. As such, they are coordinatespace expressions for the...
(and other analogous non
tensorTensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multidimensional array of...
ial) objects in differential geometry. Thus they quickly supplanted the classical notion of covariant derivative in many post1950 treatments of the subject.
Motivation
The
covariant derivative is a generalization of the
directional derivativeIn mathematics, the directional derivative of a multivariate differentiable function along a given vector V at a given point P intuitively represents the instantaneous rate of change of the function, moving through P in the direction of V...
from
vector calculus. As with the directional derivative, the covariant derivative is a rule,
, which takes as its inputs: (1) a vector,
u, defined at a point
P, and (2) a
vector fieldIn vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
,
v, defined in a neighborhood of
P. The output is the vector
, also at the point
P. The primary difference from the usual directional derivative is that
must, in a certain precise sense, be
independent of the manner in which it is expressed in a
coordinate systemIn geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...
.
A vector may be
described as a list of numbers in terms of a basis, but as a geometrical object a vector retains its own identity regardless of how one chooses to describe it in a basis. This persistence of identity is reflected in the fact that when a vector is written in one basis, and then the basis is changed, the vector transforms according to a
change of basisIn linear algebra, change of basis refers to the conversion of vectors and linear transformations between matrix representations which have different bases.Expression of a basis:...
formula. Such a transformation law is known as a
covariant transformationIn physics, a covariant transformation is a rule , that describes how certain physical entities change under a change of coordinate system....
. The covariant derivative is required to transform, under a change in coordinates, in the same way as a vector does: the covariant derivative must change by a covariant transformation (hence the name).
In the case of
Euclidean spaceIn mathematics, Euclidean space is the Euclidean plane and threedimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
, 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
translatesIn Euclidean geometry, a translation moves every point a constant distance in a specified direction. A translation can be described as a rigid motion, other rigid motions include rotations and reflections. A translation can also be interpreted as the addition of a constant vector to every point, or...
one of the vectors to the origin of the other, keeping it parallel. With a Cartesian (fixed orthonormal) coordinate system we thus obtain the simplest example: covariant derivative which is obtained by taking the derivative of the components.
In the general case, however, one must take into account the change of the coordinate system. For example, if the same covariant derivative is written in polar coordinates in a two dimensional Euclidean plane, then it 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.
Consider the example of moving along a curve γ(
t) in the Euclidean plane. In polar coordinates, γ may be written in terms of its radial and angular coordinates by γ(
t) = (
r(
t), θ(
t)). A vector at a particular time
t (for instance, the acceleration of the curve) is expressed in terms of
, 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 symbolsIn mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel , are numerical arrays of real numbers that describe, in coordinates, the effects of parallel transport in curved surfaces and, more generally, manifolds. As such, they are coordinatespace expressions for the...
) serve to express this change.
In a curved space, such as the surface of the Earth (regarded as a sphere), the
translationIn Euclidean geometry, a translation moves every point a constant distance in a specified direction. A translation can be described as a rigid motion, other rigid motions include rotations and reflections. A translation can also be interpreted as the addition of a constant vector to every point, or...
is not well defined and its analog,
parallel transportIn geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection , then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the...
, depends on the path along which the vector is translated.
A vector
e on a globe on the equator in Q is directed to the north. Suppose we
parallel transportIn geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection , then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the...
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 paralleltransported 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.
Remarks
 The definition of the covariant derivative does not use the metric in space. However, for each metric there is a unique torsion
In differential geometry, the notion of torsion is a manner of characterizing a twist or screw of a moving frame around a curve. The torsion of a curve, as it appears in the FrenetSerret formulas, for instance, quantifies the twist of a curve about its tangent vector as the curve evolves In the...
free covariant derivative called the LeviCivita connectionIn Riemannian geometry, the LeviCivita connection is a specific connection on the tangent bundle of a manifold. More specifically, it is the torsionfree metric connection, i.e., the torsionfree connection on the tangent bundle preserving a given Riemannian metric.The fundamental theorem of...
such that the covariant derivative of the metric is zero.
 The properties of a derivative imply that depends on an arbitrarily small neighborhood of a point p in the same way as e.g. the derivative of a scalar function along a curve at a given point p depends on a arbitrarily small neighborhood of p.
 The information on the neighborhood of a point p in the covariant derivative can be used to define parallel transport
In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection , then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the...
of a vector. Also the curvatureIn mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension at least 3 is too complicated to be described by a single number at a given point. Riemann introduced an abstract and rigorous way to define it, now known as the curvature tensor...
, torsionIn differential geometry, the notion of torsion is a manner of characterizing a twist or screw of a moving frame around a curve. The torsion of a curve, as it appears in the FrenetSerret formulas, for instance, quantifies the twist of a curve about its tangent vector as the curve evolves In the...
, and geodesicIn mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". In the presence of a Riemannian metric, geodesics are defined to be the shortest path between points in the space...
s may be defined only in terms of the covariant derivative or other related variation on the idea of a linear connectionIn the mathematical field of differential geometry, the term linear connection can refer to either of the following overlapping concepts:* a connection on a vector bundle, often viewed as a differential operator ;* a principal connection on the frame bundle of a manifold or the induced connection...
.
Formal definition
A covariant derivative is a
(Koszul) connectionIn mathematics, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. If the fiber bundle is a vector bundle, then the notion of parallel transport is required to be linear...
on the
tangent bundleIn differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...
and other
tensor bundleIn mathematics, the tensor bundle of a manifold is the direct sum of all tensor products of the tangent bundle and the cotangent bundle of that manifold. To do calculus on the tensor bundle a connection is needed....
s. Thus it has a certain behavior on functions, on vector fields, on the duals of vector fields (i.e.,
covectorIn differential geometry, one can attach to every point x of a smooth manifold a vector space called the cotangent space at x. Typically, the cotangent space is defined as the dual space of the tangent space at x, although there are more direct definitions...
fields), and most generally of all, on arbitrary
tensor fieldIn mathematics, physics and engineering, a tensor field assigns a tensor to each point of a mathematical space . Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical...
s.
Functions
Given a function
, the covariant derivative
coincides with the normal differentiation of a real function in the direction of the vector
v, usually denoted by
and by
.
Vector fields
A
covariant derivative of a vector field
in the direction of the vector
denoted
is defined by the following properties for any vector
v, vector fields
u, w and scalar functions
f and
g:
 is algebraically linear in so
 is additive in so
 obeys the product rule
In calculus, the product rule is a formula used to find the derivatives of products of two or more functions. It may be stated thus:'=f'\cdot g+f\cdot g' \,\! or in the Leibniz notation thus:...
, i.e. where is defined above.
Note that
at point
p depends on the value of
v at
p and on values of
u in a neighbourhood of
p because of the last property, the product rule.
Covector fields
Given a field of
covectorsIn differential geometry, one can attach to every point x of a smooth manifold a vector space called the cotangent space at x. Typically, the cotangent space is defined as the dual space of the tangent space at x, although there are more direct definitions...
(or
oneformIn linear algebra, a oneform on a vector space is the same as a linear functional on the space. The usage of oneform in this context usually distinguishes the oneforms from higherdegree multilinear functionals on the space. For details, see linear functional.In differential geometry, a...
)
, its covariant derivative
can be defined using the following identity which is satisfied for all vector fields
u
The covariant derivative of a covector field along a vector field
v 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
tensorIn mathematics, the modern componentfree approach to the theory of a tensor views a tensor as an abstract object, expressing some definite type of multilinear concept...
fields using the following identities where
and
are any two tensors:
and if
and
are tensor fields of the same tensor bundle then
The covariant derivative of a tensor field along a vector field
v is again a tensor field of the same type.
Coordinate description
Given coordinate functions
 ,
any
tangent vectorA tangent vector is a vector that is tangent to a curve or surface at a given point.Tangent vectors are described in the differential geometry of curves in the context of curves in Rn. More generally, tangent vectors are elements of a tangent space of a differentiable manifold....
can be described by its components in the basis
 .
The covariant derivative of a basis vector along a basis vector is again a vector and so can be expressed as a linear combination
.
To specify the covariant derivative it is enough to specify the covariant derivative of each basis vector field
along
.
the coefficients
are called
Christoffel symbolsIn mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel , are numerical arrays of real numbers that describe, in coordinates, the effects of parallel transport in curved surfaces and, more generally, manifolds. As such, they are coordinatespace expressions for the...
.
Then using the rules in the definition, we find that for general vector fields
and
we get
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
In words: the covariant derivative is the normal derivative along the coordinates with correction terms which tell how the coordinates change.
The covariant derivative of a type (
r,
s) tensor field along
is given by the expression:







Or, in words: take the partial derivative of the tensor and add: a for every upper index , and a for every lower index .
If instead of a tensor, one is trying to differentiate a tensor densityIn differential geometry, a tensor density or relative tensor is a generalization of the tensor concept. A tensor density transforms as a tensor when passing from one coordinate system to another , except that it is additionally multiplied or weighted by a power of the Jacobian determinant of the...
(of weight +1), then you also add a term
If it is a tensor density of weight W, then multiply that term by W.
For example, is a scalar density (of weight +1), so we get:
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.
Examples
For a scalar field , covariant differentiation is simply partial differentiation:
For a contravariant vector field , we have:
For a covariant vector field , we have:
For a type (2,0) tensor field , we have:
For a type (0,2) tensor field , we have:
For a type (1,1) tensor field , we have:
The notation above is meant in the sense
One must always remember that covariant derivatives do not commute, i.e. . It is actually easy to show that:
where is the Riemann tensor. Similarly,
and
The latter can be shown by taking (without loss of generality) that .
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 semicolonThe semicolon is a punctuation mark with several uses. The Italian printer Aldus Manutius the Elder established the practice of using the semicolon to separate words of opposed meaning and to indicate interdependent statements. "The first printed semicolon was the work of ... Aldus Manutius"...
, while a normal partial derivativeIn mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant...
is indicated by a commaA comma is a type of punctuation mark . The word comes from the Greek komma , which means something cut off or a short clause.Comma may also refer to:* Comma , a type of interval in music theory...
. In this notation we write the same as:
Once again this shows that the covariant derivative of a vector field is not just simply obtained by differentiating to the coordinates , but also depends on the vector v itself through .
In some older texts (notably Adler, Bazin & Schiffer, Introduction to General Relativity), the covariant derivative is denoted by a double pipe:
Derivative along curve
Since the covariant derivative of a tensor field at a point depends only on value of the vector field at one can define the covariant derivative along a smooth curve in a manifold:
Note that the tensor field only needs to be defined on the curve for this definition to make sense.
In particular, is a vector field along the curve itself. If vanishes then the curve is called a geodesic of the covariant derivative. If the covariant derivative is the LeviCivita connectionIn Riemannian geometry, the LeviCivita connection is a specific connection on the tangent bundle of a manifold. More specifically, it is the torsionfree metric connection, i.e., the torsionfree connection on the tangent bundle preserving a given Riemannian metric.The fundamental theorem of...
of a certain metric then the geodesics for the connection are precisely the geodesics of the metricIn the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...
that are parametrised by arc length.
The derivative along a curve is also used to define the parallel transportIn geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection , then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the...
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 which allows vectors in neighboring tangent spaces to be compared. This extra structure is necessary because there is no canonical way to compare vectors from different vector spaces, as is necessary for this generalization of the directional derivativeIn mathematics, the directional derivative of a multivariate differentiable function along a given vector V at a given point P intuitively represents the instantaneous rate of change of the function, moving through P in the direction of V...
. There is however another generalization of directional derivatives which is canonical: the Lie derivativeIn mathematics, the Lie derivative , named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a vector field or more generally a tensor field, along the flow of another vector field...
. The Lie derivative 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. 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 ∇_{u}v − ∇_{v}u, and the Lie derivative L_{u}v differ by the torsion of the connection, so that if a connection is symmetric, then its antisymmetrization is the Lie derivative.