Covariant derivative
Encyclopedia
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
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 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 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.
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 bundle
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, 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-tensor
ial) objects in differential geometry. Thus they quickly supplanted the classical notion of covariant derivative in many post-1950 treatments of the subject.
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 field
, 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 system
.
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 basis
formula. Such a transformation law is known as a covariant transformation
. 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 space
, 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. 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 symbols
) serve to express this change.
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 transport
, 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 transport
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.
on the tangent bundle
and other tensor bundle
s. Thus it has a certain behavior on functions, on vector fields, on the duals of vector fields (i.e., covector
fields), and most generally of all, on arbitrary tensor field
s.
, the covariant derivative 
coincides with the normal differentiation of a real function in the direction of the vector v, usually denoted by 
and by 
.
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:
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.
(or one-form
)
, 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.
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.
any tangent vector
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 symbols
.
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:
Mathematics
Mathematics 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 derivative
Derivative
In 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 vector
Tangent vector
A 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 manifold
Manifold
In 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 connection
Connection (mathematics)
In 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 operator
Differential operator
In 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 connection
Connection (principal bundle)
In 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 connection
Affine connection
In 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 field
Vector field
In 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 system
Coordinate system
In 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 field
Tensor field
In 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 Ricci-CurbastroGregorio Ricci-Curbastro
Gregorio Ricci-Curbastro 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 Levi-Civita
Tullio Levi-Civita
Tullio Levi-Civita, 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 Ricci-Curbastro, the inventor of tensor calculus...
in the theory of Riemannian
Riemannian geometry
Riemannian 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 pseudo-Riemannian geometry
Pseudo-Riemannian manifold
In differential geometry, a pseudo-Riemannian 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 pseudo-Riemannian manifold is that on a pseudo-Riemannian manifold the...
. Ricci and Levi-Civita (following ideas of Elwin Bruno Christoffel
Elwin Bruno Christoffel
Elwin Bruno Christoffel was a German mathematician and physicist.-Life:...
) observed that the Christoffel symbols
Christoffel symbols
In 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 coordinate-space expressions for the...
used to define the curvature could also provide a notion of differentiation
Derivative
In 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 derivative
Directional derivative
In 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 Levi-Civita connection
Levi-Civita connection
In Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold. More specifically, it is the torsion-free metric connection, i.e., the torsion-free 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 Weyl
Hermann Weyl
Hermann 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 Schouten
Jan Arnoldus Schouten
Jan 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 Cartan
Élie Cartan
Élie 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 metric
Metric tensor
In 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 bundle
Vector bundle
In 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, Jean-Louis Koszul
Jean-Louis Koszul
Jean-Louis Koszul is a mathematician best known for studying geometry and discovering the Koszul complex.He was educated at the Lycée Fustel-de-Coulanges 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 connection
Connection (vector bundle)
In 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 cohomology
Lie algebra cohomology
In 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 symbols
Christoffel symbols
In 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 coordinate-space expressions for the...
(and other analogous non-tensor
Tensor
Tensors 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 multi-dimensional array of...
ial) 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 derivativeDirectional derivative
In 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 fieldVector field
In 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 systemCoordinate system
In 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 basis
Change of basis
In 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 transformation
Covariant transformation
In 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 space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional 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 translates
Translation (geometry)
In 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 symbolsChristoffel symbols
In 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 coordinate-space expressions for the...
) serve to express this change.
In a curved space, such as the surface of the Earth (regarded as a sphere), the translation
Translation (geometry)
In 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 transport
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...
, 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 transport
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...
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.
Remarks
- The definition of the covariant derivative does not use the metric in space. However, for each metric there is a unique torsionTorsion tensorIn 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 Frenet-Serret formulas, for instance, quantifies the twist of a curve about its tangent vector as the curve evolves In the...
-free covariant derivative called the Levi-Civita connectionLevi-Civita connectionIn Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold. More specifically, it is the torsion-free metric connection, i.e., the torsion-free 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 transportParallel 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...
of a vector. Also the curvatureCurvature of Riemannian manifoldsIn 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...
, torsionTorsion tensorIn 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 Frenet-Serret formulas, for instance, quantifies the twist of a curve about its tangent vector as the curve evolves In the...
, and geodesicGeodesicIn 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 connectionLinear 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) connectionConnection (vector bundle)
In 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 bundle
Tangent bundle
In 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 bundle
Tensor bundle
In 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., covector
Cotangent space
In 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 field
Tensor field
In 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 ruleProduct ruleIn 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 covectorsCotangent space
In 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 one-form
One-form
In linear algebra, a one-form on a vector space is the same as a linear functional on the space. The usage of one-form in this context usually distinguishes the one-forms from higher-degree 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 uThe 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 tensorTensor (intrinsic definition)
In mathematics, the modern component-free approach to the theory of a tensor views a tensor as an abstract object, expressing some definite type of multi-linear concept...
fields using the following identities where
and are any two tensors:and if
and are tensor fields of the same tensor bundle thenThe 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 vector
Tangent vector
A 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 symbolsChristoffel symbols
In 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 coordinate-space expressions for the...
.
Then using the rules in the definition, we find that for general vector fields
and we getthe 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 densityTensor 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 semicolonSemicolonThe 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 derivativePartial 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 commaCommaA 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 Levi-Civita connectionLevi-Civita connectionIn Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold. More specifically, it is the torsion-free metric connection, i.e., the torsion-free 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 metricMetric tensorIn 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 transportParallel 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 derivativeDirectional 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 derivativeLie 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 ∇uv − ∇vu, and the Lie derivative Luv differ by the torsion of the connection, so that if a connection is symmetric, then its antisymmetrization is the Lie derivative.
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