Inmathematics, thecovariant derivative is a way of specifying aderivative alongtangent vectors of amanifold. Alternatively, the covariant derivative is a way of introducing and working with aconnection on a manifold by means of adifferential operator, to be contrasted with the approach given by aprincipal connection on theframe bundle – seeaffine connection. In the special case of a manifoldisometrically embedded into a higher-dimensionalEuclidean space, the covariant derivative can be viewed as theorthogonal projection of the Euclideandirectional derivative onto the manifold's tangent space. In this case the Euclidean derivative is broken into two parts, the extrinsic normal component (dependent on the embedding) and the intrinsic covariant derivative component.

The name is motivated by the importance ofchanges of coordinate inphysics: the covariant derivative transformscovariantly under a general coordinate transformation, that is, linearly via theJacobian matrix of the transformation.^[1]

This article presents an introduction to the covariant derivative of avector field with respect to a vector field, both in a coordinate-free language and using a localcoordinate system and the traditional index notation. The covariant derivative of atensor field is presented as an extension of the same concept. The covariant derivative generalizes straightforwardly to a notion of differentiation associated to aconnection on a vector bundle, also known as aKoszul connection.

Historically, at the turn of the 20th century, the covariant derivative was introduced byGregorio Ricci-Curbastro andTullio Levi-Civita in the theory ofRiemannian andpseudo-Riemannian geometry.^[2] Ricci and Levi-Civita (following ideas ofElwin Bruno Christoffel) observed that theChristoffel symbols used to define thecurvature could also provide a notion ofdifferentiation which generalized the classicaldirectional derivative ofvector fields on a manifold.^[3][4] This new derivative – theLevi-Civita connection – wascovariant 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 beingHermann Weyl,Jan Arnoldus Schouten, andÉlie Cartan,^[5] that a covariant derivative could be defined abstractly without the presence of ametric. 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 ofdifferential geometry began introducing other notions of covariant differentiation in generalvector bundles which were, in contrast to the classical bundles of interest to geometers, not part of thetensor analysis of the manifold. By and large, these generalized covariant derivatives had to be specifiedad 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 aKoszul connection or a connection on a vector bundle.^[6] Using ideas fromLie 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 ofChristoffel symbols (and other analogous non-tensorial objects) in differential geometry. Thus they quickly supplanted the classical notion of covariant derivative in many post-1950 treatments of the subject.

Thecovariant derivative is a generalization of thedirectional derivative fromvector calculus. As with the directional derivative, the covariant derivative is a rule,${\displaystyle {\mathrm{\nabla }}_{\mathbf{u}}\mathbf{v}}$, which takes as its inputs: (1) a vector,u, defined at a pointP, and (2) avector fieldv defined in aneighborhood ofP.^[7] The output is the vector${\displaystyle {\mathrm{\nabla }}_{\mathbf{u}}\mathbf{v}(P)}$, also at the pointP. The primary difference from the usual directional derivative is that${\displaystyle {\mathrm{\nabla }}_{\mathbf{u}}\mathbf{v}}$ must, in a certain precise sense, beindependent of the manner in which it is expressed in acoordinate system.

A vector may bedescribed as a list of numbers in terms of abasis, 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 achange of basis formula, with the coordinates undergoing acovariant transformation. The covariant derivative is required to transform, under a change in coordinates, by a covariant transformation in the same way as a basis does (hence the name).

In the case ofEuclidean space, one usually defines the directional derivative of a vector field in terms of the difference between two vectors at two nearby points.In such a system onetranslates one of the vectors to the origin of the other, keeping it parallel, then takes their difference within the same vector space. With a Cartesian (fixedorthonormal) 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" doesnot amount to keeping the polar components constant under translation, since the coordinate grid itself "rotates". Thus, the same covariant derivative written inpolar coordinates contains extra terms that describe how the coordinate grid itself rotates, or how in more general coordinates the grid expands, contracts, twists, interweaves, etc.

Consider the example of a particle moving along a curveγ(t) in the Euclidean plane. In polar coordinates,γ may be written in terms of its radial and angular coordinates byγ(t) = (r(t),θ(t)). A vector at a particular timet^[8] (for instance, a constant acceleration of the particle) is expressed in terms of${\displaystyle ({\mathbf{e}}_r,{\mathbf{e}}_{\theta })}$, where${\displaystyle {\mathbf{e}}_r}$ and${\displaystyle {\mathbf{e}}_{\theta }}$ are unit tangent vectors for the polar coordinates, serving as a basis to decompose a vector in terms of radial andtangential 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 (theChristoffel symbols) serve to express this change.

In a curved space, such as the surface of the Earth (regarded as a sphere), thetranslation of tangent vectors between different points is not well defined, and its analog,parallel transport, depends on the path along which the vector is translated. A vector on a globe on the equator at pointQ is directed to the north. Suppose we transport the vector (keeping it parallel) first along the equator to the pointP, then drag it along a meridian to theN pole, and finally transport it along another meridian back toQ. 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 thecurvature 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 thecurvature, and can be defined in terms of the covariant derivative.

• The definition of the covariant derivative does not use the metric in space. However, for each metric there is a uniquetorsion-free covariant derivative called theLevi-Civita connection such that the covariant derivative of the metric is zero.
• The properties of a derivative imply that${\displaystyle {\mathrm{\nabla }}_{\mathbf{v}}\mathbf{u}}$ depends on the values ofu in a neighborhood of a pointp in the same way as e.g. the derivative of a scalar functionf along a curve at a given pointp depends on the values off in a neighborhood ofp.
• The information in a neighborhood of a pointp in the covariant derivative can be used to defineparallel transport of a vector. Also thecurvature,torsion, andgeodesics may be defined only in terms of the covariant derivative or other related variation on the idea of alinear connection.
• Some equations involving covariant derivative can be locally solved using Chen's iterated integrals^[9] of using approach based on linear homotopy operator^[10].

Suppose an open subsetU of ad-dimensionalRiemannian manifoldM is embedded into Euclidean space${\displaystyle ({\mathbb{R}}^n,⟨\cdot ,\cdot ⟩)}$ via atwice continuously-differentiable (C²) mapping${\displaystyle \overrightarrow{\mathrm{\Psi }}:{\mathbb{R}}^d\supset U\to {\mathbb{R}}^n}$ such that the tangent space at${\displaystyle \overrightarrow{\mathrm{\Psi }}(p)}$ is spanned by the vectors$${\displaystyle \left\{{\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^i}|}_p:i\in \{1,\dots ,d\}\right\}}$$and the scalar product${\displaystyle ⟨\cdot ,\cdot ⟩}$ on${\displaystyle {\mathbb{R}}^n}$ is compatible with the metric onM:$${\displaystyle g_{ij}=⟨\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^i},\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^j}⟩.}$$

(Since the manifold metric is always assumed to be regular,^{[clarification needed]} the compatibility condition implies linear independence of the partial derivative tangent vectors.)

For a tangent vector field,${\displaystyle \overrightarrow{V}=v^j\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^j}}$, one has$${\displaystyle \frac{\mathrm{\partial }\overrightarrow{V}}{\mathrm{\partial }{x}^i}=\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^i}\left(v^j\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^j}\right)=\frac{\mathrm{\partial }{v}^j}{\mathrm{\partial }{x}^i}\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^j}+v^j\frac{{\mathrm{\partial }}^2\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^i\mathrm{\partial }{x}^j}.}$$

The last term is not tangential toM, but can be expressed as a linear combination of the tangent space base vectors using theChristoffel symbols as linear factors plus a vector orthogonal to the tangent space:$${\displaystyle v^j\frac{{\mathrm{\partial }}^2\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^i\mathrm{\partial }{x}^j}=v^j{{\mathrm{\Gamma }}^k}_{ij}\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^k}+\overrightarrow{n}.}$$

In the case of theLevi-Civita connection, the covariant derivative${\displaystyle {\mathrm{\nabla }}_{{\mathbf{e}}_i}\overrightarrow{V}}$, also written${\displaystyle {\mathrm{\nabla }}_i\overrightarrow{V}}$, is defined as the orthogonal projection of the usual derivative onto tangent space:$${\displaystyle {\mathrm{\nabla }}_{{\mathbf{e}}_i}\overrightarrow{V}:=\frac{\mathrm{\partial }\overrightarrow{V}}{\mathrm{\partial }{x}^i}-\overrightarrow{n}=\left(\frac{\mathrm{\partial }{v}^k}{\mathrm{\partial }{x}^i}+v^j{{\mathrm{\Gamma }}^k}_{ij}\right)\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^k}.}$$

From here it may be computationally convenient to obtain a relation between the Christoffel symbols for the Levi-Civita connection and the metric. To do this we first note that, since the vector${\displaystyle \overrightarrow{n}}$ in the previous equation is orthogonal to the tangent space,$${\displaystyle ⟨\frac{{\mathrm{\partial }}^2\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^i\mathrm{\partial }{x}^j},\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^l}⟩=⟨{{\mathrm{\Gamma }}^k}_{ij}\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^k}+\overrightarrow{n},\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^l}⟩=⟨\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^k},\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^l}⟩{{\mathrm{\Gamma }}^k}_{ij}=g_{kl}{{\mathrm{\Gamma }}^k}_{ij}.}$$

Then, since the partial derivative of a component${\displaystyle g_{ab}}$ of the metric with respect to a coordinate${\displaystyle x^c}$ is$${\displaystyle \frac{\mathrm{\partial }{g}_{ab}}{\mathrm{\partial }{x}^c}=\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^c}⟨\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^a},\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^b}⟩=⟨\frac{{\mathrm{\partial }}^2\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^c\mathrm{\partial }{x}^a},\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^b}⟩+⟨\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^a},\frac{{\mathrm{\partial }}^2\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^c\mathrm{\partial }{x}^b}⟩,}$$any tripleti,j,k of indices yields a system of equations(Here the symmetry of the scalar product has been used and the order of partial differentiations has been swapped.)

Adding the first two equations and subtracting the third, we obtain$${\displaystyle \frac{\mathrm{\partial }{g}_{jk}}{\mathrm{\partial }{x}^i}+\frac{\mathrm{\partial }{g}_{ki}}{\mathrm{\partial }{x}^j}-\frac{\mathrm{\partial }{g}_{ij}}{\mathrm{\partial }{x}^k}=2⟨\frac{\mathrm{\partial }\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^k},\frac{{\mathrm{\partial }}^2\overrightarrow{\mathrm{\Psi }}}{\mathrm{\partial }{x}^i\mathrm{\partial }{x}^j}⟩.}$$

Thus the Christoffel symbols for the Levi-Civita connection are related to the metric by$${\displaystyle g_{kl}{{\mathrm{\Gamma }}^k}_{ij}=\frac{1}{2}\left(\frac{\mathrm{\partial }{g}_{jl}}{\mathrm{\partial }{x}^i}+\frac{\mathrm{\partial }{g}_{li}}{\mathrm{\partial }{x}^j}-\frac{\mathrm{\partial }{g}_{ij}}{\mathrm{\partial }{x}^l}\right).}$$

Ifg is nondegenerate then${\displaystyle {{\mathrm{\Gamma }}^k}_{ij}}$ can be solved for directly as$${\displaystyle {{\mathrm{\Gamma }}^k}_{ij}=\frac{1}{2}{g}^{kl}\left(\frac{\mathrm{\partial }{g}_{jl}}{\mathrm{\partial }{x}^i}+\frac{\mathrm{\partial }{g}_{li}}{\mathrm{\partial }{x}^j}-\frac{\mathrm{\partial }{g}_{ij}}{\mathrm{\partial }{x}^l}\right).}$$

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.

A covariant derivative is a(Koszul) connection on thetangent bundle and othertensor bundles: it differentiates vector fields in a way analogous to the usual differential on functions. The definition extends to a differentiation on the dual of vector fields (i.e.covector fields) and to arbitrarytensor fields, in a unique way that ensures compatibility with the tensor product and trace operations (tensor contraction).

Given a point${\displaystyle p\in M}$ of the manifoldM, a real function${\displaystyle f:M\to \mathbb{R}}$ on the manifold and a tangent vector${\displaystyle \mathbf{v}\in T_{p}M}$, the covariant derivative off atp alongv is the scalar atp, denoted${\displaystyle {\left({\mathrm{\nabla }}_{\mathbf{v}}f\right)}_p}$, that represents theprincipal part of the change in the value off when the argument off is changed by the infinitesimal displacement vectorv. (This is thedifferential off evaluated against the vectorv.) Formally, there is a differentiable curve${\displaystyle \varphi :[-1,1]\to M}$ such that${\displaystyle \varphi (0)=p}$ and${\displaystyle {\varphi }^{\prime }(0)=\mathbf{v}}$, and the covariant derivative off atp is defined by$${\displaystyle {\left({\mathrm{\nabla }}_{\mathbf{v}}f\right)}_p={\left(f\circ \varphi \right)}^{\mathrm{\prime }}\left(0\right)=\underset{t\to 0}{lim}\frac{f(\varphi \left(t\right))-f(p)}{t}.}$$

When${\displaystyle \mathbf{v}:M\to T_{p}M}$ is a vector field onM, the covariant derivative${\displaystyle {\mathrm{\nabla }}_{\mathbf{v}}f:M\to \mathbb{R}}$ is the function that associates with each pointp in the common domain off andv the scalar${\displaystyle {\left({\mathrm{\nabla }}_{\mathbf{v}}f\right)}_p}$.

For a scalar functionf and vector fieldv, the covariant derivative${\displaystyle {\mathrm{\nabla }}_{\mathbf{v}}f}$ coincides with theLie derivative${\displaystyle L_{\mathbf{v}}(f)}$, and with theexterior derivative${\displaystyle df(\mathbf{v})}$.

Given a pointp of the manifoldM, a vector field${\displaystyle \mathbf{u}:M\to T_{p}M}$ defined in a neighborhood ofp and a tangent vector${\displaystyle \mathbf{v}\in T_{p}M}$, the covariant derivative ofu atp alongv is the tangent vector atp, denoted${\displaystyle ({\mathrm{\nabla }}_{\mathbf{v}}\mathbf{u}{)}_p}$, such that the following properties hold (for any tangent vectorsv,x andy atp, vector fieldsu andw defined in a neighborhood ofp, scalar valuesg andh atp, and scalar functionf defined in a neighborhood ofp):

1. ${\displaystyle {\left({\mathrm{\nabla }}_{\mathbf{v}}\mathbf{u}\right)}_p}$ is linear in${\displaystyle \mathbf{v}}$ so$${\displaystyle {\left({\mathrm{\nabla }}_{g\mathbf{x}+h\mathbf{y}}\mathbf{u}\right)}_p=g(p){\left({\mathrm{\nabla }}_{\mathbf{x}}\mathbf{u}\right)}_p+h(p){\left({\mathrm{\nabla }}_{\mathbf{y}}\mathbf{u}\right)}_p}$$
2. ${\displaystyle {\left({\mathrm{\nabla }}_{\mathbf{v}}\mathbf{u}\right)}_p}$ is additive in${\displaystyle \mathbf{u}}$ so:$${\displaystyle {\left({\mathrm{\nabla }}_{\mathbf{v}}\left[\mathbf{u}+\mathbf{w}\right]\right)}_p={\left({\mathrm{\nabla }}_{\mathbf{v}}\mathbf{u}\right)}_p+{\left({\mathrm{\nabla }}_{\mathbf{v}}\mathbf{w}\right)}_p}$$
3. ${\displaystyle ({\mathrm{\nabla }}_{\mathbf{v}}\mathbf{u}{)}_p}$ obeys theproduct rule; i.e., where${\displaystyle {\mathrm{\nabla }}_{\mathbf{v}}f}$ is defined above,$${\displaystyle {\left({\mathrm{\nabla }}_{\mathbf{v}}\left[f\mathbf{u}\right]\right)}_p=f(p){\left({\mathrm{\nabla }}_{\mathbf{v}}\mathbf{u}{)}_p+({\mathrm{\nabla }}_{\mathbf{v}}f\right)}_p{\mathbf{u}}_p.}$$

Note that${\displaystyle {\left({\mathrm{\nabla }}_{\mathbf{v}}\mathbf{u}\right)}_p}$ depends not only on the value ofu atp but also on values ofu in a neighborhood ofp, because the last property, the product rule, involves the directional derivative off (by the vectorv).

Ifu andv are both vector fields defined over a common domain, then${\displaystyle {\mathrm{\nabla }}_{\mathbf{v}}\mathbf{u}}$ denotes the vector field whose value at each pointp of the domain is the tangent vector${\displaystyle {\left({\mathrm{\nabla }}_{\mathbf{v}}\mathbf{u}\right)}_p}$.

Given a field ofcovectors (orone-form)${\displaystyle \alpha }$ defined in a neighborhood ofp, its covariant derivative${\displaystyle ({\mathrm{\nabla }}_{\mathbf{v}}\alpha {)}_p}$ is defined in a way to make the resulting operation compatible with tensor contraction and the product rule. That is,${\displaystyle ({\mathrm{\nabla }}_{\mathbf{v}}\alpha {)}_p}$ is defined as the unique one-form atp such that the following identity is satisfied for all vector fieldsu in a neighborhood ofp$${\displaystyle {\left({\mathrm{\nabla }}_{\mathbf{v}}\alpha \right)}_p\left({\mathbf{u}}_p\right)={\mathrm{\nabla }}_{\mathbf{v}}{\left[\alpha \left(\mathbf{u}\right)\right]}_p-{\alpha }_p\left[{\left({\mathrm{\nabla }}_{\mathbf{v}}\mathbf{u}\right)}_p\right].}$$

The covariant derivative of a covector field along a vector fieldv is again a covector field.

Once the covariant derivative is defined for fields of vectors and covectors it can be defined for arbitrarytensor fields by imposing the following identities for every pair of tensor fields${\displaystyle \phi }$ and${\displaystyle \psi }$ in a neighborhood of the pointp:$${\displaystyle {\mathrm{\nabla }}_{\mathbf{v}}{\left(\phi \otimes \psi \right)}_p={\left({\mathrm{\nabla }}_{\mathbf{v}}\phi \right)}_p\otimes \psi (p)+\phi (p)\otimes {\left({\mathrm{\nabla }}_{\mathbf{v}}\psi \right)}_p,}$$and for${\displaystyle \phi }$ and${\displaystyle \psi }$ of the same valence$${\displaystyle {\mathrm{\nabla }}_{\mathbf{v}}(\phi +\psi {)}_p=({\mathrm{\nabla }}_{\mathbf{v}}\phi {)}_p+({\mathrm{\nabla }}_{\mathbf{v}}\psi {)}_p.}$$The covariant derivative of a tensor field along a vector fieldv is again a tensor field of the same type.

Explicitly, letT be a tensor field of type(p,q). ConsiderT to be a differentiablemultilinear map ofsmoothsectionsα¹,α², ...,α^q of the cotangent bundleT^∗M and of sectionsX₁,X₂, ...,X_p of thetangent bundleTM, writtenT(α¹,α², ...,X₁,X₂, ...) intoR. The covariant derivative ofT alongY is given by the formula

Given coordinate functions$${\displaystyle x^i,i=0,1,2,\dots ,}$$ anytangent vector can be described by its components in the basis$${\displaystyle {\mathbf{e}}_i=\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^i}.}$$

The covariant derivative of a basis vector along a basis vector is again a vector and so can be expressed as a linear combination${\displaystyle {\mathrm{\Gamma }}^k{\mathbf{e}}_k}$.To specify the covariant derivative it is enough to specify the covariant derivative of each basis vector field${\displaystyle {\mathbf{e}}_i}$ along${\displaystyle {\mathbf{e}}_j}$.$${\displaystyle {\mathrm{\nabla }}_{{\mathbf{e}}_j}{\mathbf{e}}_i={{\mathrm{\Gamma }}^k}_{ij}{\mathbf{e}}_k,}$$

the coefficients${\displaystyle {\mathrm{\Gamma }}^k{}_{ij}}$ 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 calledChristoffel symbols.

Then using the rules in the definition, we find that for general vector fields${\displaystyle \mathbf{v}=v^j{\mathbf{e}}_j}$ and${\displaystyle \mathbf{u}=u^i{\mathbf{e}}_i}$ we getso$${\displaystyle {\mathrm{\nabla }}_{\mathbf{v}}\mathbf{u}=\left(v^j{u}^i{{\mathrm{\Gamma }}^k}_{ij}+v^j\frac{\mathrm{\partial }{u}^k}{\mathrm{\partial }{x}^j}\right){\mathbf{e}}_k.}$$

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 fieldu. In particular$${\displaystyle {\mathrm{\nabla }}_{{\mathbf{e}}_j}\mathbf{u}={\mathrm{\nabla }}_j\mathbf{u}=\left(\frac{\mathrm{\partial }{u}^i}{\mathrm{\partial }{x}^j}+u^k{{\mathrm{\Gamma }}^i}_{kj}\right){\mathbf{e}}_i}$$

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$${\displaystyle {\mathrm{\nabla }}_{{\mathbf{e}}_j}\theta =\left(\frac{\mathrm{\partial }{\theta }_i}{\mathrm{\partial }{x}^j}-{\theta }_k{{\mathrm{\Gamma }}^k}_{ij}\right){{\mathbf{e}}^{\ast }}^i,}$$where${\displaystyle {{\mathbf{e}}^{\ast }}^i({\mathbf{e}}_j)={{\delta }^i}_j}$.

The covariant derivative of a type(r,s) tensor field along${\displaystyle e_c}$ is given by the expression:

Or, in words: take the partial derivative of the tensor and add:${\displaystyle +{{\mathrm{\Gamma }}^{a_i}}_{dc}}$ for every upper index${\displaystyle a_i}$, and${\displaystyle -{{\mathrm{\Gamma }}^d}_{b_{i}c}}$ for every lower index${\displaystyle b_i}$.

If instead of a tensor, one is trying to differentiate atensor density (of weight +1), then one also adds a term$${\displaystyle -{{\mathrm{\Gamma }}^d}_{dc}{T^{a_1\dots a_r}}_{b_1\dots b_s}.}$$If it is a tensor density of weightW, then multiply that term byW.For example,$\sqrt{-g}$ is a scalar density (of weight +1), so we get:$${\displaystyle {\left(\sqrt{-g}\right)}_{;c}={\left(\sqrt{-g}\right)}_{,c}-\sqrt{-g}{{\mathrm{\Gamma }}^d}_{dc}}$$where the semicolon ";" indicates covariant differentiation and the 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.

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 asemicolon, while a normalpartial derivative is indicated by acomma. In this notation we write the same as:$${\displaystyle {\mathrm{\nabla }}_{e_j}\mathbf{v}\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{v^s}_{;j}{\mathbf{e}}_s{v^i}_{;j}={v^i}_{,j}+v^k{{\mathrm{\Gamma }}^i}_{kj}}$$In case two or more indexes appear after the semicolon, all of them must be understood as covariant derivatives:$${\displaystyle {\mathrm{\nabla }}_{e_k}\left({\mathrm{\nabla }}_{e_j}\mathbf{v}\right)\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{v^s}_{;jk}{\mathbf{e}}_s}$$

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:$${\displaystyle {\mathrm{\nabla }}_{e_j}\mathbf{v}\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{v^i}_{||j}={v^i}_{|j}+v^k{{\mathrm{\Gamma }}^i}_{kj}}$$

For a scalar field${\displaystyle \varphi }$, covariant differentiation is simply partial differentiation:$${\displaystyle {\varphi }_{;a}\equiv {\mathrm{\partial }}_a\varphi }$$

For a contravariant vector field${\displaystyle {\lambda }^a}$, we have:$${\displaystyle {{\lambda }^a}_{;b}\equiv {\mathrm{\partial }}_b{\lambda }^a+{{\mathrm{\Gamma }}^a}_{bc}{\lambda }^c}$$

For a covariant vector field${\displaystyle {\lambda }_a}$, we have:$${\displaystyle {\lambda }_{a;c}\equiv {\mathrm{\partial }}_c{\lambda }_a-{{\mathrm{\Gamma }}^b}_{ca}{\lambda }_b}$$

For a type (2,0) tensor field${\displaystyle {\tau }^{ab}}$, we have:$${\displaystyle {{\tau }^{ab}}_{;c}\equiv {\mathrm{\partial }}_c{\tau }^{ab}+{{\mathrm{\Gamma }}^a}_{cd}{\tau }^{db}+{{\mathrm{\Gamma }}^b}_{cd}{\tau }^{ad}}$$

For a type (0,2) tensor field${\displaystyle {\tau }_{ab}}$, we have:$${\displaystyle {\tau }_{ab;c}\equiv {\mathrm{\partial }}_c{\tau }_{ab}-{{\mathrm{\Gamma }}^d}_{ca}{\tau }_{db}-{{\mathrm{\Gamma }}^d}_{cb}{\tau }_{ad}}$$

For a type (1,1) tensor field${\displaystyle {{\tau }^a}_b}$, we have:$${\displaystyle {{\tau }^a}_{b;c}\equiv {\mathrm{\partial }}_c{{\tau }^a}_b+{{\mathrm{\Gamma }}^a}_{cd}{{\tau }^d}_b-{{\mathrm{\Gamma }}^d}_{cb}{{\tau }^a}_d}$$

The notation above is meant in the sense$${\displaystyle {{\tau }^{ab}}_{;c}\equiv {\left({\mathrm{\nabla }}_{{\mathbf{e}}_c}\tau \right)}^{ab}}$$

In general, covariant derivatives do not commute. By example, the covariant derivatives of vector field${\displaystyle {\lambda }_{a;bc}\ne {\lambda }_{a;cb}}$. TheRiemann tensor${\displaystyle {R^d}_{abc}}$ is defined such that:$${\displaystyle {\lambda }_{a;bc}-{\lambda }_{a;cb}={R^d}_{abc}{\lambda }_d}$$or, equivalently,$${\displaystyle {{\lambda }^a}_{;bc}-{{\lambda }^a}_{;cb}=-{R^a}_{dbc}{\lambda }^d}$$

The covariant derivative of a (2,0)-tensor field fulfills:$${\displaystyle {{\tau }^{ab}}_{;cd}-{{\tau }^{ab}}_{;dc}=-{R^a}_{ecd}{\tau }^{eb}-{R^b}_{ecd}{\tau }^{ae}}$$

The latter can be shown by taking (without loss of generality) that${\displaystyle {\tau }^{ab}={\lambda }^a{\mu }^b}$.

Since the covariant derivative${\displaystyle {\mathrm{\nabla }}_{X}T}$ of a tensor fieldT at a pointp depends only on the value of the vector fieldX atp one can define the covariant derivative along a smooth curve${\displaystyle \gamma (t)}$ in a manifold:$${\displaystyle D_{t}T={\mathrm{\nabla }}_{\dot{\gamma }(t)}T.}$$Note that the tensor fieldT only needs to be defined on the curve${\displaystyle \gamma (t)}$ for this definition to make sense.

In particular,${\displaystyle \dot{\gamma }(t)}$ is a vector field along the curve${\displaystyle \gamma }$ itself. If${\displaystyle {\mathrm{\nabla }}_{\dot{\gamma }(t)}\dot{\gamma }(t)}$ vanishes then the curve is called a geodesic of the covariant derivative. If the covariant derivative is theLevi-Civita connection of apositive-definite metric then the geodesics for the connection are precisely thegeodesics of the metric that are parametrized byarc length.

The derivative along a curve is also used to define theparallel transport along the curve.

Sometimes the covariant derivative along a curve is calledabsolute orintrinsic 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 whichis canonical: theLie derivative, which evaluates the change of one vector field along the flow of another vector field. Thus, one must know both vector fields in a 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 a neighborhood of a point. In other words, the covariant derivative is linear (overC^∞(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 derivativeL_uv differ by thetorsion of the connection, so that if a connection is torsion free, then its antisymmetrizationis the Lie derivative.

• Kobayashi, Shoshichi;Nomizu, Katsumi (1996).Foundations of Differential Geometry, Vol. 1 (New ed.).Wiley Interscience.ISBN 0-471-15733-3.
• I.Kh. Sabitov (2001) [1994],"Covariant differentiation",Encyclopedia of Mathematics,EMS Press
• Sternberg, Shlomo (1964).Lectures on Differential Geometry. Prentice-Hall.
• Spivak, Michael (1999).A Comprehensive Introduction to Differential Geometry (Volume Two). Publish or Perish, Inc.

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