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| In [[mathematics]] and [[physics]], the '''Christoffel symbols''', named for [[Elwin Bruno Christoffel]] <ref name="christoffel" >{{citation|title=Ueber die Transformation der homogenen Differentialausdrücke zweiten Grades|last=Christoffel|first=E.B.|author-link=Elwin Bruno Christoffel|journal=Jour. für die reine und angewandte Mathematik|volume=B. 70|pages=46–70|year=1869|url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002153882&IDDOC=266356}}</ref> (1829–1900), are numerical arrays of real numbers that describe, in coordinates, the effects of [[parallel transport]] in curved [[surface]]s and, more generally, [[manifold]]s. As such, they are coordinate-space expressions for the [[Levi-Civita connection]] derived from the [[metric tensor]]. In a broader sense, the [[Covariant derivative#Coordinate description|connection coefficients]] of an arbitrary (not necessarily metric) [[affine connection]] in a coordinate basis are also called Christoffel symbols.<ref>See, for instance, {{harv|Spivak|1999}} and {{harv|Choquet-Bruhat|DeWitt-Morette|1977}}</ref> The Christoffel symbols may be used for performing practical calculations in [[differential geometry]]. For example, the [[Riemann curvature tensor]] can be expressed entirely in terms of the Christoffel symbols and their first [[partial derivative]]s.
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| At each point of the underlying ''n''-dimensional manifold, for any local coordinate system, the Christoffel symbol is an [[Matrix (mathematics)|array]] with three dimensions: ''n'' × ''n'' × ''n''. Each of the ''n''<sup>3</sup> components is a [[real number]].
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| Under ''linear'' [[coordinate transformations]] on the manifold, it behaves like a [[tensor]], but under general coordinate transformations, it does not. In many practical problems, most components of the Christoffel symbols are equal to [[0 (number)|zero]], provided the coordinate system and the metric tensor possess some common symmetries.
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| In [[general relativity]], the Christoffel symbol plays the role of the '''gravitational force field''' with the corresponding ''gravitational potential'' being the ''metric tensor''.
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| ==Preliminaries==
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| The definitions given below are valid for both [[Riemannian manifold]]s and [[pseudo-Riemannian manifold]]s, such as those of [[general relativity]], with careful distinction being made between upper and lower indices ([[covariance and contravariance of vectors|contra-variant and co-variant]] indices). The formulas hold for either [[sign convention]], unless otherwise noted.
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| [[Einstein notation|Einstein summation convention]] is used in this article. The connection coefficients of the [[Levi-Civita connection]] (or pseudo-Riemannian connection) expressed in a coordinate basis are called the '''Christoffel symbols'''.
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| ==Definition==
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| If ''x''<sup>''i''</sup>, ''i'' = 1,2,...,''n'', is a [[local coordinate system]] on a manifold ''M'' with [[metric tensor]] <math>g\,</math>, then the [[tangent space|tangent vectors]]
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| :<math>\mathrm{e}_i = \frac{\partial}{\partial x^i}=\partial_i , \quad i=1,2,\dots,n</math>
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| define a local coordinate [[basis of a vector space|basis]] of the tangent space to ''M'' at each point of its domain.
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| ===Christoffel symbols of the first kind===
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| The Christoffel symbols of the first kind can be derived either from the Christoffel symbols of the second kind and the metric,<ref name="ludvigsen">{{citation |last1=Ludvigsen |first1=Malcolm||title=General Relativity: A Geometrical Approach | year=1999|page=88}}</ref>
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| :<math>\Gamma_{cab} = g_{cd} \Gamma^{d}{}_{ab}\,,</math>
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| or from the metric alone,<ref name="ludvigsen">{{citation |last1=Ludvigsen |first1=Malcolm | title=General Relativity: A Geometrical Approach | year = 1999|page=88}}</ref>
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| :<math>\Gamma_{cab}
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| =\frac12 \left(\frac{\partial g_{ca}}{\partial x^b} + \frac{\partial g_{cb}}{\partial x^a} - \frac{\partial g_{ab}}{\partial x^c} \right)
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| = \frac12\, (g_{ca, b} + g_{cb, a} - g_{ab, c})
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| = \frac12\, \left(\partial_{b}g_{ca} + \partial_{a}g_{cb} - \partial_{c}g_{ab}\right) \,.
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| </math>
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| As an alternative notation one also finds<ref name="christoffel" >{{citation|title=Ueber die Transformation der homogenen Differentialausdrücke zweiten Grades|last=Christoffel|first=E.B.|author-link=Elwin Bruno Christoffel|journal=Jour. für die reine und angewandte Mathematik|volume=B. 70|pages=46–70|year=1869|url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002153882&IDDOC=266356}}</ref><ref name="chatterjee">{{cite book
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| |first1=U. |last1=Chatterjee
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| |first2=N. |last2=Chatterjee
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| |year=2010
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| |title=Vector & Tensor Analysis
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| |page=480}}</ref><ref name="dirkstruik">{{cite book
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| |first1=D.J. |last1=Struik
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| |title=Lectures on Classical Differential Geometry
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| |edition=first published in 1988 Dover
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| |year=1961
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| |page=114}}</ref>
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| :<math>\Gamma_{cab} = [ab, c].</math>
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| It is worth noting that <math>[ab, c] = [ba, c]</math>.<ref name="bishopgoldberg" >{{citation | last1=Bishop|first1=R.L.|last2=Goldberg|first2=| title = Tensor Analysis on Manifolds| year=1968|page=241}}</ref>
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| ===Christoffel symbols of the second kind (symmetric definition)===
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| The Christoffel symbols of the second kind are the connection coefficients —in a coordinate basis— of the [[Levi-Civita connection]], and since this connection has zero torsion, then in this basis the connection coefficients are symmetric,i.e., <math>\Gamma^k{}_{ij}=\Gamma^k{}_{ji}\,</math>.<ref name="chatterjee">{{cite book
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| |first1=U. |last1=Chatterjee
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| |first2=N. |last2= Chatterjee
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| |title=Vector & Tensor Analysis
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| |year=2010
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| |page=480}}</ref> For this reason a torsion-free connection is often called 'symmetric'.
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| In other words, the Christoffel symbols of the second kind <ref name="wolfram2ndkind" /><ref name="chatterjee">{{cite book
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| |first1=U. |last1=Chatterjee
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| |first2=N. |last2=Chatterjee
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| |title=Vector & Tensor Analysis
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| |year=2010
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| |page=480}}</ref> | |
| <math>\Gamma^k{}_{ij}</math> (sometimes <math>\Gamma^{k}_{ij}</math> or <math>\{\begin{smallmatrix} k\\ ij \end{smallmatrix}\}</math>)<ref name="christoffel" >{{citation|title=Ueber die Transformation der homogenen Differentialausdrücke zweiten Grades|last=Christoffel|first=E.B.|author-link=Elwin Bruno Christoffel|journal=Jour. für die reine und angewandte Mathematik|volume=B. 70|pages=46–70|year=1869|url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002153882&IDDOC=266356}}</ref><ref name="chatterjee"/> are defined as the unique coefficients such that the equation
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| :<math>\nabla_i \mathrm{e}_j = \Gamma^k{}_{ij}\mathrm{e}_k</math>
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| holds, where <math>\nabla_i</math> is the [[Levi-Civita connection]] on ''M'' taken in the coordinate direction <math>\mathrm{e}_{i}</math>, i.e., <math>\nabla_i\equiv \nabla_{\mathrm{e}_i}</math> and where <math>\mathrm{e}_i=\partial_i</math> is a local coordinate ([[holonomic basis|holonomic]]) [[basis of a vector space|basis]].
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| The Christoffel symbols can be derived from the vanishing of the [[covariant derivative]] of the [[metric tensor]] <math>g_{ik}\ </math>:
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| :<math>0 = \nabla_\ell g_{ik}=
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| \frac{\partial g_{ik}}{\partial x^\ell}- g_{mk}\Gamma^m{}_{i\ell} - g_{im}\Gamma^m{}_{k\ell}
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| =
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| \frac{\partial g_{ik}}{\partial x^\ell}- 2g_{m(k}\Gamma^m{}_{i)\ell}.
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| \
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| </math>
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| As a shorthand notation, the [[nabla symbol]] and the partial derivative symbols are frequently dropped, and instead a semi-colon and a comma are used to set off the index that is being used for the derivative. Thus, the above is sometimes written as
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| :<math>0 = \,g_{ik;\ell} = g_{ik,\ell} - g_{mk} \Gamma^m{}_{i\ell} - g_{im} \Gamma^m{}_{k\ell}. \ </math>
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| Using the fact that the symbols are symmetric in the lower two indices, one can solve explicitly for the Christoffel symbols as a function of the metric tensor by permuting the indices and resumming:<ref name="bishopgoldberg" >{{citation | last1=Bishop|first1=R.L.|last2=Goldberg|first2=|title=Lectures on Classical Differential Geometry|year=1968|page=241}}</ref>
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| :<math>\Gamma^i{}_{k\ell}=\frac{1}{2}g^{im} \left(\frac{\partial g_{mk}}{\partial x^\ell} + \frac{\partial g_{m\ell}}{\partial x^k} - \frac{\partial g_{k\ell}}{\partial x^m} \right) = {1 \over 2} g^{im} (g_{mk,\ell} + g_{m\ell,k} - g_{k\ell,m}), \ </math>
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| where <math>(g^{jk})\,</math> is the inverse of the [[matrix (mathematics)|matrix]] <math>(g_{jk})\,</math>, defined as (using the [[Kronecker delta]], and [[Einstein notation]] for summation)
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| <math>g^{j i} g_{i k}= \delta^j{}_k\ </math>.
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| Although the Christoffel symbols are written in the same notation as [[Classical treatment of tensors|tensors with index notation]], they are '''not''' [[tensor]]s,<ref>See, for example, {{harv|Kreyszig|1991}}, page 141</ref>
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| since they do not transform like tensors under a change of coordinates; see [[#Change of variable|below]].
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| ===Connection coefficients in a non holonomic basis===
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| The Christoffel symbols are most typically defined in a coordinate basis, which is the convention followed here. In other words, the name '''Christoffel symbols''' is reserved only for coordinate (i.e., holonomic) frames. However, the connection coefficients can also be defined in an arbitrary (i.e., non holonomic) basis of tangent vectors <math>\mathbf{u}_i</math> by
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| :<math>\nabla_{\mathbf{u}_i}\mathbf{u}_j = \omega^k{}_{ij}\mathbf{u}_k.</math> | |
| Explicitly, in terms of the metric tensor, this is<ref name="wolfram2ndkind">http://mathworld.wolfram.com/ChristoffelSymboloftheSecondKind.html.</ref>
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| :<math>\omega^i{}_{k\ell}=\frac{1}{2}g^{im} \left(
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| g_{mk,\ell} + g_{m\ell,k} - g_{k\ell,m} +
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| c_{mk\ell}+c_{m\ell k} - c_{k\ell m}
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| \right)\,</math>
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| where <math>c_{k\ell m}=g_{mp} {c_{k\ell}}^p\ </math> are the [[commutator#Ring theory|commutation coefficients]] of the basis; that is,
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| :<math>[\mathbf{u}_k,\mathbf{u}_\ell] = c_{k\ell}{}^m \mathbf{u}_m\,\ </math> | |
| where <math>\mathbf{u}_k</math> are the basis [[vector space|vector]]s and <math>[{\,},{\,}]\ </math> is the [[Lie derivative|Lie bracket]]. The standard unit vectors in [[vector fields in cylindrical and spherical coordinates|spherical and cylindrical coordinates]] furnish an example of a basis with non-vanishing commutation coefficients.
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| ===Ricci rotation coefficients (asymmetric definition)===
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| When we choose the basis <math>\mathrm{X}_i\equiv \mathbf{u}_i</math> orthonormal: <math>g_{ab} \equiv \eta_{ab} = \langle X_a, X_b\rangle</math> then <math>g_{mk,\ell}\equiv\eta_{mk,\ell}=0</math>. This implies that
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| :<math>\omega^i{}_{k\ell}=\frac{1}{2}\eta^{im} \left(
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| c_{mk\ell}+c_{m\ell k} - c_{k\ell m}
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| \right)\,</math>
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| and the connection coefficients become antisymmetric in the first two indices:
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| :<math>\omega_{abc} = - \omega_{bac}\, ,</math>
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| where <math>\omega_{abc} = \eta_{ad}\omega^d{}_{bc}\,</math>.
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| In this case, the connection coefficients <math>\omega^a{}_{bc}</math> are called the '''Ricci rotation coefficients'''.<ref>{{cite journal|title=Dei sistemi di congruenze ortogonali in una varietà qualunque|author=G. Ricci-Curbastro|journal=Mem. Acc. Lincei |volume=2|year=1896|pages=276–322|issue=5}}</ref><ref>{{cite journal|title=Ricci's coefficients of rotation|author=H. Levy|journal=Bull. Amer. Math. |volume=31|year=1925|pages=142–145|issue=3-4|url=http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183486405}}</ref>
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| Equivalently, one can define Ricci rotation coefficients as follows:<ref name="wolfram2ndkind" />
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| :<math>\omega^k{}_{ij} := {{\mathbf{u}}}^k \cdot \left( \nabla_j {{\mathbf{u}}}_i \right)\, ,</math>
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| where <math>\mathbf{u}_i</math> is an orthonormal non holonomic basis
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| and <math>\mathbf{u}^k=\eta^{k\ell}\mathbf{u}_{\ell}\,</math> its ''co-basis''.
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| ==Relationship to index-free notation==
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| Let ''X'' and ''Y'' be [[vector fields]] with components <math>X^i\ </math> and <math>Y^k\ </math>. Then the ''k''th component of the covariant derivative of ''Y'' with respect to ''X'' is given by
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| :<math>\left(\nabla_X Y\right)^k = X^i (\nabla_i Y)^k = X^i \left(\frac{\partial Y^k}{\partial x^i} + \Gamma^k{}_{im} Y^m\right).\ </math>
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| Here, the [[Einstein notation]] is used, so repeated indices indicate summation over indices and contraction with the metric tensor serves to raise and lower indices:
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| :<math>g(X,Y) = X^i Y_i = g_{ik}X^i Y^k = g^{ik}X_i Y_k.\ </math>
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| Keep in mind that <math>g_{ik}\neq g^{ik}\ </math> and that <math>g^i {}_k=\delta^i {}_k\ </math>, the [[Kronecker delta]]. The convention is that the metric tensor is the one with the lower indices; the correct way to obtain <math>g^{ik}\ </math> from <math>g_{ik}\ </math> is to solve the linear equations <math>g^{ij}g_{jk}=\delta^i {}_k\ </math>.
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| The statement that the connection is [[torsion tensor|torsion]]-free, namely that
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| :<math>\nabla_X Y - \nabla_Y X = [X,Y]\ </math>
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| is equivalent to the statement that —in a coordinate basis— the Christoffel symbol is symmetric in the lower two indices:
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| :<math>\Gamma^i{}_{jk}=\Gamma^i{}_{kj}.\ </math>
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| The index-less transformation properties of a tensor are given by [[pullback (differential geometry)|pullbacks]] for covariant indices, and [[pushforward (differential)|pushforwards]] for contravariant indices. The article on [[Covariant derivative#Coordinate description|covariant derivatives]] provides additional discussion of the correspondence between index-free notation and indexed notation.
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| ==Covariant derivatives of tensors==
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| The '''[[covariant derivative]]''' of a vector field <math>V^m\ </math> is
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| :<math>\nabla_\ell V^m = \frac{\partial V^m}{\partial x^\ell} + \Gamma^m{}_{k\ell} V^k.\ </math>
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| The covariant derivative of a scalar field <math>\varphi\ </math> is just
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| :<math>\nabla_i \varphi = \frac{\partial \varphi}{\partial x^i}\ </math>
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| and the covariant derivative of a [[covector]] field <math>\omega_m\ </math> is
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| :<math>\nabla_\ell \omega_m = \frac{\partial \omega_m}{\partial x^\ell} - \Gamma^k{}_{\ell m} \omega_k.\ </math>
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| The symmetry of the Christoffel symbol now implies
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| :<math>\nabla_i\nabla_j \varphi = \nabla_j\nabla_i \varphi\ </math>
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| for any scalar field, but in general the covariant derivatives of higher order tensor fields do not commute (see [[Riemann curvature tensor|curvature tensor]]).
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| The covariant derivative of a type (2,0) [[tensor]] field <math>A^{ik}\ </math> is
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| :<math>\nabla_\ell A^{ik}=\frac{\partial A^{ik}}{\partial x^\ell} + \Gamma^i{}_{m\ell} A^{mk} + \Gamma^k{}_{m\ell} A^{im}, \ </math>
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| that is,
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| :<math> A^{ik} {}_{;\ell} = A^{ik} {}_{,\ell} + A^{mk} \Gamma^i{}_{m\ell} + A^{im} \Gamma^k{}_{m\ell}. \ </math>
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| If the tensor field is [[mixed tensor|mixed]] then its covariant derivative is
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| :<math> A^i {}_{k;\ell} = A^i {}_{k,\ell} + A^{m} {}_k \Gamma^i{}_{m\ell} - A^i {}_m \Gamma^m{}_{k\ell}, \ </math>
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| and if the tensor field is of type (0,2) then its covariant derivative is
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| :<math> A_{ik;\ell} = A_{ik,\ell} - A_{mk} \Gamma^m{}_{i\ell} - A_{im} \Gamma^m{}_{k\ell}. \ </math>
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| ==Change of variable==
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| Under a change of variable from <math>(y^1,\dots,y^n)\ </math> to <math>(x^1,\dots,x^n)\ </math>, vectors transform as
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| :<math>\frac{\partial}{\partial y^i} = \frac{\partial x^k}{\partial y^i}\frac{\partial}{\partial x^k}\ </math> | |
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| and so
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| :<math>\overline{\Gamma^k{}_{ij}} =
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| \frac{\partial x^p}{\partial y^i}\,
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| \frac{\partial x^q}{\partial y^j}\,
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| \Gamma^r{}_{pq}\,
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| \frac{\partial y^k}{\partial x^r}
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| +
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| \frac{\partial y^k}{\partial x^m}\,
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| \frac{\partial^2 x^m}{\partial y^i \partial y^j}
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| \ </math>
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| where the overline denotes the Christoffel symbols in the ''y'' coordinate system. Note that the Christoffel symbol does '''not''' transform as a tensor, but rather as an object in the [[jet bundle]]. More precisely, the Christoffel symbols can be considered as functions on the jet bundle of the frame bundle of ''M'', independent of any local coordinate system. Choosing a local coordinate system determines a local section of this bundle, which can then be used to pull back the Christoffel symbols to functions on ''M'', though of course these functions then depend on the choice of local coordinate system.
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| At each point, there exist coordinate systems in which the Christoffel symbols vanish at the point.<ref>This is assuming that the connection is symmetric (e.g., the Levi-Civita connection). If the connection has [[torsion tensor|torsion]], then only the symmetric part of the Christoffel symbol can be made to vanish.</ref> These are called (geodesic) [[normal coordinates]], and are often used in [[Riemannian geometry]].
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| ==Applications to general relativity==
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| The Christoffel symbols find frequent use in Einstein's theory of [[general relativity]], where [[spacetime]] is represented by a curved 4-dimensional [[Lorentz manifold]] with a [[Levi-Civita connection]]. The [[Einstein field equations]]—which determine the geometry of spacetime in the presence of matter—contain the [[Ricci tensor]], and so calculating the Christoffel symbols is essential. Once the geometry is determined, the paths of particles and light beams are calculated by [[solving the geodesic equations]] in which the Christoffel symbols explicitly appear.
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| ==See also==
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| *[[Basic introduction to the mathematics of curved spacetime]]
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| *[[Proofs involving Christoffel symbols]]
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| *[[Differentiable manifold]]
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| *[[List of formulas in Riemannian geometry]]
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| *[[Ricci calculus]]
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| *[[Riemann–Christoffel tensor]]
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| *[[Gauss–Codazzi equations]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| *{{citation |last1=Abraham |first1=Ralph|authorlink1=Ralph Abraham| first2=Jerrold E.|last2=Marsden|title=Foundations of Mechanics|year=1978|publisher=Benjamin/Cummings Publishing|location=London|isbn=0-8053-0102-X|pages=See chapter 2, paragraph 2.7.1}}
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| *{{citation | last1=Bishop|first1=R.L.|last2=Goldberg|first2=S.I. | title = Tensor Analysis on Manifolds| publisher=The Macmillan Company | year=1968|edition=First Dover 1980|isbn=0-486-64039-6}}
| |
| *{{citation
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| |last = Choquet-Bruhat
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| |first = Yvonne
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| |authorlink = Yvonne Choquet-Bruhat
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| |first2 = Cécile |last2=DeWitt-Morette| title = Analysis, Manifolds and Physics| publisher = Elsevier| year= 1977| location = Amsterdam |isbn = 978-0-7204-0494-4}}
| |
| *{{citation |last1= Landau|first1=Lev Davidovich|authorlink1=Lev Davidovich Landau|first2=Evgeny Mikhailovich|last2=Lifshitz|authorlink2=Evgeny Mikhailovich Lifshitz|title=The Classical Theory of Fields|edition=Fourth Revised English|series=[[Course of Theoretical Physics]]|volume=Volume 2|year=1951|publisher=Pergamon Press|location=Oxford|isbn=0-08-025072-6|pages=See chapter 10, paragraphs 85, 86 and 87}}
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| *{{Citation
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| | last = Kreyszig
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| | first = Erwin
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| | authorlink = Erwin Kreyszig
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| | title = Differential Geometry
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| | publisher = [[Dover Publications]]
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| | year = 1991
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| | isbn = 978-0-486-66721-8 }}
| |
| *{{citation |last1=Misner |first1=Charles W.|first2=Kip S.|last2=Thorne
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| |first3=John Archibald|last3=Wheeler|title=Gravitation|year=1970|publisher=W.H. Freeman
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| |location=New York|isbn=0-7167-0344-0|pages=See chapter 8, paragraph 8.5}}
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| *{{citation |last1=Ludvigsen |first1=Malcolm|title=General Relativity: A Geometrical Approach|year=1999|publisher=Cambrigde University Press|isbn=0-521-63019-3}}
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| * {{citation|last=Spivak|first=Michael|authorlink=Michael Spivak|title=A Comprehensive introduction to differential geometry
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| |volume=Volume 2
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| |year=1999|publisher=Publish or Perish|isbn=0-914098-71-3}}
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| * {{cite book
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| |title=Vector & Tensor Analysis
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| |first1=U. |last1=Chatterjee
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| |first2=N. |last2=Chatterjee
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| |publisher=Academic Publishers
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| |isbn=978-93-8059-905-2
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| |year=2010
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| |page=
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| |url=http://books.google.nl/books?id=oTeGXkg0tn0C&pg=PA480}}
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| * {{cite book
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| |first1=D.J. |last1=Struik
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| |year=1961
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| |edition=first published in 1988 Dover
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| |title=Lectures on Classical Differential Geometry
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| |publisher=Dover
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| |isbn=0-486-65609-8
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| }}
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| {{tensors}}
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| {{DEFAULTSORT:Christoffel Symbols}}
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| [[Category:Riemannian geometry]]
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| [[Category:Lorentzian manifolds]]
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| [[Category:Mathematical notation]]
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| [[Category:Mathematical physics]]
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| [[Category:Connection (mathematics)]]
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