tensor derivative
   HOME

TheInfoList



OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the covariant derivative is a way of specifying a
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
along
tangent vector In mathematics, 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 R''n''. More generally, tangent vectors are e ...
s of a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a
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 that accepts a function and retur ...
, to be contrasted with the approach given by a
principal connection In mathematics, and especially differential geometry and gauge theory, 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. A principal ''G''-connect ...
on the
frame bundle In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts nat ...
– see
affine connection In differential geometry, an affine connection is a geometric object on a smooth manifold which ''connects'' nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values i ...
. In the special case of a manifold isometrically embedded into a higher-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
, the covariant derivative can be viewed as the
orthogonal projection In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it wer ...
of the Euclidean
directional derivative In mathematics, the directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity ...
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 Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
: the covariant derivative transforms covariantly under a general coordinate transformation, that is, linearly via the
Jacobian matrix In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variable ...
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 In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
and the traditional index notation. The covariant derivative of a
tensor field In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...
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 In mathematics, and especially differential geometry and gauge theory, 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. T ...
, also known as a Koszul connection.


History

Historically, at the turn of the 20th century, the covariant derivative was introduced by
Gregorio Ricci-Curbastro Gregorio Ricci-Curbastro (; 12January 1925) was an Italian mathematician. He is most famous as the discoverer of tensor calculus. With his former student Tullio Levi-Civita, he wrote his most famous single publication, a pioneering work on th ...
and
Tullio Levi-Civita Tullio Levi-Civita, (, ; 29 March 1873 – 29 December 1941) was an Italian mathematician, most famous for his work on absolute differential calculus ( tensor calculus) and its applications to the theory of relativity, but who also made signi ...
in the theory of Riemannian and
pseudo-Riemannian geometry In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which th ...
. Ricci and Levi-Civita (following ideas of
Elwin Bruno Christoffel Elwin Bruno Christoffel (; 10 November 1829 – 15 March 1900) was a German mathematician and physicist. He introduced fundamental concepts of differential geometry, opening the way for the development of tensor calculus, which would later provi ...
) observed that the Christoffel symbols used to define the
curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the can ...
could also provide a notion of differentiation which generalized the classical
directional derivative In mathematics, the directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity ...
of vector fields on a manifold. This new derivative – the
Levi-Civita connection In Riemannian or pseudo Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold (i.e. affine connection) that preserves ...
– 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 Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is asso ...
,
Jan Arnoldus Schouten Jan Arnoldus Schouten (28 August 1883 – 20 January 1971) was a Dutch mathematician and Professor at the Delft University of Technology. He was an important contributor to the development of tensor calculus and Ricci calculus, and was one of the ...
, and
Élie Cartan Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometr ...
, that a covariant derivative could be defined abstractly without the presence of a
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathe ...
. 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 Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and mult ...
began introducing other notions of covariant differentiation in general
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 (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
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 (; January 3, 1921 – January 12, 2018) was a French mathematician, best known for studying geometry and discovering the Koszul complex. He was a second generation member of Bourbaki. Biography Koszul was educated at the in ...
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 In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
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 derivative In mathematics, the directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity ...
from
vector calculus Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subjec ...
. As with the directional derivative, the covariant derivative is a rule, \nabla_, 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 \nabla_(P), also at the point ''P''. The primary difference from the usual directional derivative is that \nabla_ must, in a certain precise sense, be ''independent'' of the manner in which it is expressed in a
coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sig ...
. 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 In mathematics, an ordered basis of a vector space of finite dimension allows representing uniquely any element of the vector space by a coordinate vector, which is a sequence of scalars called coordinates. If two different bases are consider ...
formula, with the coordinates undergoing a
covariant transformation In physics, a covariant transformation is a rule that specifies how certain entities, such as vectors or tensors, change under a change of basis. The transformation that describes the new basis vectors as a linear combination of the old basis ve ...
. 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 Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
, 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 In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of ...
) coordinate system "keeping it parallel" amounts to keeping the components constant. This ordinary directional derivative on Euclidean space is the first example of a covariant derivative . Next, one must take into account changes of the coordinate system. For example, if the Euclidean plane is described by polar coordinates, "keeping it parallel" does ''not'' amount to keeping the polar components constant under translation, since the coordinate grid itself "rotates". Thus, the same covariant derivative written in
polar coordinates In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to th ...
contains extra terms that describe how the coordinate grid itself rotates, or how in more general coordinates the grid expands, contracts, twists, interweaves, etc. Consider the example of a particle moving along a curve 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 In many applications, it may be better not to think of as corresponding to time, at least for applications in
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. It is simply regarded as an abstract parameter varying smoothly and monotonically along the path.
(for instance, a constant acceleration of the particle) is expressed in terms of (\mathbf_r, \mathbf_), where \mathbf_r and \mathbf_ 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 Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transla ...
of tangent vectors between different points is not well defined, and its analog,
parallel transport In geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on the tangent b ...
, 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 In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the can ...
, and can be defined in terms of the covariant derivative.


Remarks

* The definition of the covariant derivative does not use the metric in space. However, for each metric there is a unique torsion-free covariant derivative called the
Levi-Civita connection In Riemannian or pseudo Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold (i.e. affine connection) that preserves ...
such that the covariant derivative of the metric is zero. * The properties of a derivative imply that \nabla_\mathbf \mathbf depends on the values of ''u'' on an arbitrarily small neighborhood of a point ''p'' in the same way as e.g. the derivative of a scalar function ''f'' along a curve at a given point ''p'' depends on the values of ''f'' in an 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 (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on the tangent b ...
of a vector. Also the
curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the can ...
, torsion, and
geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connecti ...
s may be defined only in terms of the covariant derivative or other related variation on the idea of a linear connection.


Informal definition using an embedding into Euclidean space

Suppose an open subset U of a d-dimensional Riemannian manifold M is embedded into Euclidean space (\R^n, \langle\cdot, \cdot\rangle) via a twice continuously-differentiable (C) mapping \vec\Psi : \R^d \supset U \to \R^n such that the tangent space at \vec\Psi(p) \in M is spanned by the vectors \left\ and the scalar product \left \langle \cdot, \cdot \right \rangle on \R^n is compatible with the metric on ''M'': g_ = \left\langle \frac, \frac \right\rangle. (Since the manifold metric is always assumed to be regular, the compatibility condition implies linear independence of the partial derivative tangent vectors.) For a tangent vector field, one has \frac = \frac \left( v^j \frac \right)= \frac \frac + v^j \frac . 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: \frac = _ \frac + \vec n . In the case of the
Levi-Civita connection In Riemannian or pseudo Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold (i.e. affine connection) that preserves ...
, the covariant derivative \nabla_ \vec V, also written is defined as the orthogonal projection of the usual derivative onto tangent space: \nabla_ \vec V := \frac - \vec n = \left( \frac + v^j _ \right) \frac. To obtain the relation between Christoffel symbols for the Levi-Civita connection and the metric, first we must note that, since \vec n in previous equation is orthogonal to tangent space: \left\langle \frac, \frac \right\rangle = \left\langle _ \frac + \vec n, \frac \right\rangle = _ \left\langle \frac, \frac \right\rangle = _ \, g_ . Second, the partial derivative of a component of the metric is: \frac = \frac \left\langle \frac, \frac \right\rangle = \left\langle \frac, \frac \right\rangle + \left\langle \frac, \frac \right\rangle implies for a basis using the symmetry of the scalar product and swapping the order of partial differentiation: \begin \frac \\ \frac \\ \frac \end = \begin 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end \begin \left\langle \frac, \frac \right\rangle \\ \left\langle \frac, \frac \right\rangle \\ \left\langle \frac, \frac \right\rangle \end adding first row to second and subtracting third one: \frac + \frac - \frac = 2\left\langle \frac, \frac \right\rangle and yields the Christoffel symbols for the Levi-Civita connection in terms of the metric: g_ _ = \frac \left( \frac + \frac- \frac\right). 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 In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
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 In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the ...
fields) and to arbitrary
tensor field In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...
s, in a unique way that ensures compatibility with the tensor product and trace operations (tensor contraction).


Functions

Given a point p \in M of the manifold M, a real function f : M \to \R on the manifold and a tangent vector \mathbf \in T_pM, the covariant derivative of at along is the scalar at , denoted \left(\nabla_\mathbf f\right)_p, that represents the
principal part In mathematics, the principal part has several independent meanings, but usually refers to the negative-power portion of the Laurent series of a function. Laurent series definition The principal part at z=a of a function : f(z) = \sum_^\infty a_ ...
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 \phi:
1, 1 Onekama ( ) is a village in Manistee County in the U.S. state of Michigan. The population was 411 at the 2010 census. The village is located on the shores of Portage Lake and is surrounded by Onekama Township. The town's name is derived from "On ...
to M such that \phi(0) = p and \phi'(0) = \mathbf, and the covariant derivative of ''f'' at ''p'' is defined by \left(\nabla_\mathbf f\right)_p = \left(f \circ \phi\right)'\left(0\right) = \lim_ \frac. When \mathbf : M \to T_pM is a vector field on M, the covariant derivative \nabla_\mathbff : M \to \R is the function that associates with each point ''p'' in the common domain of ''f'' and v the scalar \left(\nabla_\mathbff\right)_p. For a scalar function ''f'' and vector field v, the covariant derivative \nabla_\mathbf f coincides with the Lie derivative L_v(f), and with the
exterior derivative On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The re ...
df(v).


Vector fields

Given a point p of the manifold M, a vector field \mathbf : M \to T_p M defined in a neighborhood of ''p'' and a tangent vector \mathbf \in T_pM, the covariant derivative of ''u'' at ''p'' along v is the tangent vector at ''p'', denoted (\nabla_\mathbf \mathbf)_p, 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''): # \left(\nabla_\mathbf \mathbf\right)_p is linear in \mathbf so \left(\nabla_ \mathbf\right)_p = \left(\nabla_\mathbf \mathbf\right)_p g + \left(\nabla_\mathbf \mathbf\right)_p h # \left(\nabla_\mathbf \mathbf\right)_p is additive in \mathbf so: \left(\nabla_\mathbf\left mathbf + \mathbf\rightright)_p = \left(\nabla_\mathbf \mathbf\right)_p + \left(\nabla_\mathbf \mathbf\right)_p # (\nabla_\mathbf \mathbf)_p obeys the
product rule In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v ...
; i.e., where \nabla_\mathbff is defined above, \left(\nabla_\mathbf \left \mathbf\rightright)_p = f(p)\left(\nabla_\mathbf \mathbf)_p + (\nabla_\mathbff\right)_p\mathbf_p. Note that \left(\nabla_\mathbf \mathbf\right)_p 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 \nabla_\mathbf\mathbf u denotes the vector field whose value at each point ''p'' of the domain is the tangent vector \left(\nabla_\mathbf\mathbf u\right)_p.


Covector fields

Given a field of covectors (or
one-form In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to \R whose restriction to e ...
) \alpha defined in a neighborhood of ''p'', its covariant derivative (\nabla_\mathbf\alpha)_p is defined in a way to make the resulting operation compatible with tensor contraction and the product rule. That is, (\nabla_\mathbf\alpha)_p 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'' \left(\nabla_\mathbf\alpha\right)_p \left(\mathbf_p\right) = \nabla_\mathbf\left alpha\left(\mathbf\right)\rightp - \alpha_p\left left(\nabla_\mathbf\mathbf\right)_p\right 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 In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
fields by imposing the following identities for every pair of tensor fields \varphi and \psi in a neighborhood of the point ''p'': \nabla_\mathbf\left(\varphi \otimes \psi\right)_p = \left(\nabla_\mathbf\varphi\right)_p \otimes \psi(p) + \varphi(p) \otimes \left(\nabla_\mathbf\psi\right)_p, and for \varphi and \psi of the same valence \nabla_\mathbf(\varphi + \psi)_p = (\nabla_\mathbf\varphi)_p + (\nabla_\mathbf\psi)_p. 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 Section, Sectioning or Sectioned may refer to: Arts, entertainment and media * Section (music), a complete, but not independent, musical idea * Section (typography), a subdivision, especially of a chapter, in books and documents ** Section sig ...
''α''1, ''α''2, …, ''α''''q'' of the cotangent bundle ''T''''M'' and of sections ''X''1, ''X''2, …, ''X''''p'' of the
tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
''TM'', written ''T''(''α''1, ''α''2, …, ''X''1, ''X''2, …) into R. The covariant derivative of ''T'' along ''Y'' is given by the formula \begin (\nabla_Y T)\left(\alpha_1, \alpha_2, \ldots, X_1, X_2, \ldots\right) = & \nabla_Y\left(T\left(\alpha_1,\alpha_2, \ldots, X_1, X_2, \ldots\right)\right) \\ &- T\left(\nabla_Y\alpha_1, \alpha_2, \ldots, X_1, X_2, \ldots\right) - T\left(\alpha_1, \nabla_Y\alpha_2, \ldots, X_1, X_2, \ldots\right) - \cdots \\ &- T\left(\alpha_1, \alpha_2, \ldots, \nabla_YX_1, X_2, \ldots\right) - T\left(\alpha_1, \alpha_2, \ldots, X_1, \nabla_Y X_2, \ldots\right) - \cdots \end


Coordinate description

Given coordinate functions x^i,\ i=0,1,2,\dots , any
tangent vector In mathematics, 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 R''n''. More generally, tangent vectors are e ...
can be described by its components in the basis \mathbf_i = \frac . The covariant derivative of a basis vector along a basis vector is again a vector and so can be expressed as a linear combination \Gamma^k \mathbf_k. To specify the covariant derivative it is enough to specify the covariant derivative of each basis vector field \mathbf_i along \mathbf_j. \nabla_ \mathbf_i = _ \mathbf_k, the coefficients \Gamma^k_ 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 \mathbf = v^j \mathbf_j and \mathbf = u^i \mathbf_i we get \begin \nabla_\mathbf \mathbf &= \nabla_ u^i \mathbf_i \\ &= v^j \nabla_ u^i \mathbf_i \\ &= v^j u^i \nabla_ \mathbf_i + v^j \mathbf_i \nabla_ u^i \\ &= v^j u^i _\mathbf_k + v^j \mathbf_i \end so \nabla_\mathbf \mathbf = \left(v^j u^i _ + v^j \right)\mathbf_k . 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 \nabla_ \mathbf = \nabla_j \mathbf = \left( \frac + u^k _ \right) \mathbf_i 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 \nabla_ = \left( \frac - \theta_k _ \right) ^i where ^i (\mathbf_j) = _j. The covariant derivative of a type tensor field along e_c is given by the expression: \begin _ = &\frac_ \\ &+ \,_ _ + \cdots + _ _ \\ &-\,_ _ - \cdots - _ _. \end Or, in words: take the partial derivative of the tensor and add: +_ for every upper index a_i, and -_ for every lower index b_i. If instead of a tensor, one is trying to differentiate a ''
tensor density In differential geometry, a tensor density or relative tensor is a generalization of the tensor field concept. A tensor density transforms as a tensor field when passing from one coordinate system to another (see tensor field), except that it is ...
'' (of weight +1), then one also adds a term -_ _. If it is a tensor density of weight ''W'', then multiply that term by ''W''. For example, \sqrt is a scalar density (of weight +1), so we get: \left(\sqrt\right)_ = \left(\sqrt\right)_ - \sqrt\,_ 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 The semicolon or semi-colon is a symbol commonly used as orthographic punctuation. In the English language, a semicolon is most commonly used to link (in a single sentence) two independent clauses that are closely related in thought. When a ...
, while a normal
partial derivative In 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 (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
is indicated by a
comma The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...
. In this notation we write the same as: \nabla_ \mathbf \ \stackrel\ _\mathbf_s \;\;\;\;\;\; _ = _ + v^k _ In case two or more indexes appear after the semicolon, all of them must be understood as covariant derivatives: \nabla_ \left( \nabla_ \mathbf \right) \ \stackrel\ _\mathbf_s 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: \nabla_ \mathbf \ \stackrel\ _ = _ + v^k _


Covariant derivative by field type

For a scalar field \phi\,, covariant differentiation is simply partial differentiation: \phi_ \equiv \partial_a \phi For a contravariant vector field \lambda^a, we have: _ \equiv \partial_b \lambda^a + _\lambda^c For a covariant vector field \lambda_a, we have: \lambda_ \equiv \partial_c \lambda_a - _\lambda_b For a type (2,0) tensor field \tau^, we have: _ \equiv \partial_c \tau^ + _\tau^ + _\tau^ For a type (0,2) tensor field \tau_, we have: \tau_ \equiv \partial_c \tau_ - _\tau_ - _\tau_ For a type (1,1) tensor field _, we have: _\equiv \partial_c _b + __b - _ _d The notation above is meant in the sense _ \equiv \left(\nabla_\tau\right)^


Properties

In general, covariant derivatives do not commute. By example, the covariant derivatives of vector field \lambda_ \neq \lambda_.
Riemann tensor In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds. ...
_ is defined such that: \lambda_ - \lambda_ = _\lambda_d or, equivalently, _ - _ = -_\lambda^d The covariant derivative of a (2,0)-tensor field fulfills: _ - _ = -_\tau^ - _\tau^ The latter can be shown by taking (without loss of generality) that \tau^ = \lambda^a \mu^b .


Derivative along a curve

Since the covariant derivative \nabla_X T of a tensor field T at a point p depends only on the value of the vector field X at p one can define the covariant derivative along a smooth curve \gamma(t) in a manifold: D_tT=\nabla_T. Note that the tensor field T only needs to be defined on the curve \gamma(t) for this definition to make sense. In particular, \dot(t) is a vector field along the curve \gamma itself. If \nabla_\dot\gamma(t) vanishes then the curve is called a geodesic of the covariant derivative. If the covariant derivative is the
Levi-Civita connection In Riemannian or pseudo Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold (i.e. affine connection) that preserves ...
of a positive-definite metric then the geodesics for the connection are precisely the
geodesics In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
of the metric that are parametrized by
arc length ARC may refer to: Business * Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s * Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services * ...
. The derivative along a curve is also used to define the
parallel transport In geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on the tangent b ...
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.


See also

*
Affine connection In differential geometry, an affine connection is a geometric object on a smooth manifold which ''connects'' nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values i ...
* Christoffel symbols * Connection (algebraic framework) * Connection (mathematics) * Connection (vector bundle) * Connection form * Exterior covariant derivative *
Gauge covariant derivative The gauge covariant derivative is a variation of the covariant derivative used in general relativity, quantum field theory and fluid dynamics. If a theory has gauge transformations, it means that some physical properties of certain equations are ...
* Introduction to the mathematics of general relativity *
Levi-Civita connection In Riemannian or pseudo Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold (i.e. affine connection) that preserves ...
*
Parallel transport In geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on the tangent b ...
*
Ricci calculus In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern name for what used to be ...
* Tensor derivative (continuum mechanics) *
List of formulas in Riemannian geometry This is a list of formulas encountered in Riemannian geometry. Einstein notation is used throughout this article. This article uses the "analyst's" sign convention for Laplacians, except when noted otherwhise. Christoffel symbols, covariant deriva ...


Notes


References

* * * * {{DEFAULTSORT:Covariant Derivative Connection (mathematics) Differential geometry Mathematical methods in general relativity Riemannian geometry Solid mechanics