What is Covariant derivative: Definition and 171 Discussions

In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically embedded into a higher-dimensional Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean directional 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 of changes of coordinate in physics: the covariant derivative transforms covariantly under a general coordinate transformation, that is, linearly via the Jacobian matrix 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 and the traditional index notation. The covariant derivative of a tensor field is presented as an extension of the same concept. The covariant derivative generalizes straightforwardly to a notion of differentiation associated to a connection on a vector bundle, also known as a Koszul connection.

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  1. T

    Covariant Derivative derivation.

    Homework Statement Using the Leibniz rule and: \nabla_{c}X^{a}=\partial_{c}X^a+\Gamma_{bc}^{a}X^b \nabla_{a}\Phi=\partial\Phi Show that \nabla_c X_a = \partial_c X_a - \Gamma^{b}_{ac}X_{b} . The question is from Ray's Introducing Einsteins relativity, My attempt...
  2. G

    Weird version of covariant derivative on wikipedia

    http://en.wikipedia.org/wiki/Four-force At the bottom of that page, the author provides the generalization of four force in general relativity, where the partial derivative is replaced with the covariant derivative. However if you notice on the second term in the third equality, there is a...
  3. G

    Covariant Derivative and metric tensor

    Hi all, I am wondering if it is possible to derive the definition of a Christoffel symbols using the Covariant Derivative of the Metric Tensor. If yes, can I get a step-by-step solution? Thanks! Joe W.
  4. P

    Covariant derivative of a vector field

    Homework Statement Show that \nabla_a(\sqrt{-det\;h}S^a)=\partial_a(\sqrt{-det\;h}S^a) where h is the metric and S^a a vector. Homework Equations \nabla_a V^b = \partial_a V^b+\Gamma^b_{ac}V^c \Gamma^a_{ab} = \frac{1}{2det\;h}\partial_b\sqrt{det\;h} \nabla_a\sqrt{-det\;h} (is that...
  5. N

    Covariant derivative along a horizontal lift in an associated vector bundle

    I am trying to familiarize myself with the use of fibre bundles and associated bundles but am having some problems actually making calculations. I would like to show that the covariant derivative along the horizontal lift of a curve in the base space vanishes (which should be just a matter of...
  6. I

    Parallel propagator and covariant derivative of vector

    Hi all, I'm trying to figure out the link between the connection coefficients (Christoffel symbols), the propagator, and the coordinate description of the covariant derivative with the connection coefficients. As in...
  7. C

    Christoffel Symbol / Covariant derivative

    Homework Statement My teacher solved this in class but I'm not understanding some parts of tis solution. Show that \nabla_i V^i is scalar. Homework Equations \nabla_i V^i = \frac{\partial V^{i}}{\partial q^{i}} + \Gamma^{i}_{ik} V^{k} The Attempt at a Solution To start this...
  8. N

    Dual vector is the covariant derivative of a scalar?

    Homework Statement In Wald's text on General Relativity he makes an assertion that I'm not sure why it is allowed mathematically. Here's the basic setup: Let \omega_{b} be a dual vector, \nabla_{b} and \tilde{\nabla}_{b} be two covariant derivatives and f\in\mathscr{F}. Then we may let...
  9. F

    Whats the physical meaning of a covariant derivative?

    Hi there! I saw this exercise that we have to calculate the covariant derivative of a vector field (in polar coordinates). Most of them equals zero, but two of them are non-zero, sugesting that this vector field is not constant. What i want to understand is the physical meaning of this values...
  10. N

    Help Covariant Derivative of Ricci Tensor the hard way.

    I am trying to calculate the covariant derivative of the Ricci Tensor the way Einstein did it, but I keep coming up with \nabla_{μ}R_{αβ}=\frac{∂}{∂x^{μ}}R_{αβ}-2\Gamma^{α}_{μ\gamma}R_{αβ} or \nabla_{μ}R_{αβ}=\frac{∂}{∂x^{μ}}R_{αβ}-\Gamma^{α}_{μ\gamma}R_{αβ}-\Gamma^{β}_{μ\gamma}R_{αβ}...
  11. N

    Help Covariant Derivative of Ricci Tensor the hard way.

    I am trying to calculate the covariant derivative of the Ricci Tensor the way Einstein did it, but I keep coming up with \nabla_{μ}R_{αβ}=\frac{∂}{∂x^{μ}}R_{αβ}-2\Gamma^{α}_{μ\gamma}R_{αβ} or \nabla_{μ}R_{αβ}=\frac{∂}{∂x^{μ}}R_{αβ}-\Gamma^{α}_{μ\gamma}R_{αβ}-\Gamma^{β}_{μ\gamma}R_{αβ}...
  12. pellman

    Covariant derivative of connection coefficients?

    The connection \nabla is defined in terms of its action on tensor fields. For example, acting on a vector field Y with respect to another vector field X we get \nabla_X Y = X^\mu ({Y^\alpha}_{,\mu} + Y^\nu {\Gamma^\alpha}_{\mu\nu})e_\alpha = X^\mu {Y^\alpha}_{;\mu}e_\alpha and we call...
  13. V

    Calculating divergence using covariant derivative

    Homework Statement Using the definition of divergence d(i_{X}dV) = (div X)dV where X:M\rightarrow TM is a vector field, dV is a volume element and i_X is a contraction operator e.g. i_{X}T = X^{k}T^{i_{1}...i_{r}}_{kj_{2}...j_{s}}, prove that if we use Levi-Civita connection then the...
  14. D

    Commutation property of covariant derivative

    My book defines the covariant derivative of a tangent vector field as the directional derivative of each component, and then we subtract out the normal component to the surface. I am a little confused about proving some properties. One of them states: If x(u, v) is an orthogonal patch, x_u...
  15. jfy4

    Covariant derivative of coordinates

    Hi, I am familiar with the covariant derivative of the tangent vector to a path, \nabla_{\alpha}u^{\beta} and some interesting ways to use it. I am wondering about \nabla_{\alpha}x^{\beta}=\frac{\partial x^\beta}{\partial...
  16. H

    Covariant derivative of Lie-Bracket in normal orthonormal frame

    Hi there, I was doing some calculations with tensors and ran into a result which seems a bit odd to me. I hope someone can validate this or tell me where my mistake is. So I have a normal orthonormal frame field \{E_i\} in the neighbourhood of a point p in a Riemannian manifold (M,g), i.e...
  17. L

    Covariant derivative from connections

    On a 2 dimensional Riemannian manifold how does one derive the covariant derivative from the connection 1 form on the tangent unit circle bundle?
  18. T

    Covariant Derivative: Different for Vectors, Spinors & Matrices?

    The covariant derivative is different in form for different tensors, depending on their rank. What about other mathematical entities? The electromagnetic field A is a vector, but it has complex values. Is the covariant derivative different for complex valued vectors? And what about...
  19. S

    Covariant derivative of stress-energy tensor

    hi, I understand that Tab,b=0 because the change in density equals the negative divergence, but why do the christoffel symbols vanish for Tab;b=0?
  20. S

    Covariant derivative of riemann tensor

    what would Rabcd;e look like in terms of it's christoffels? or Rab;c
  21. R

    Covariant Derivative: Understanding $\partial_i e_j=\Gamma^{k}_{ij} e_k$

    How can the derivative of a basis vector at a point be the linear combination of tangent vectors at that point? For example, if you take a sphere, then the derivative of the polar basis vector with respect to the polar coordinate is in the radial direction. How can something in the radial...
  22. L

    Covariant derivative vs Gauge Covariant derivative

    As you may guess from the title this question is about the covariant derivatives, more precisely about the difference between the usual covariant derivative, the one used in General Relativity defined by:\nabla_{e_{\mu}}=\left(\frac{\partial v^{\beta}} {\partial...
  23. L

    Covariant Derivative and Gauge Covariant Derivative

    As you may guess from the title this question is about the covariant derivatives, more precisely about the difference between the usual covariant derivative, the one used in General Relativity defined by:\nabla_{e_{\mu}}=\left(\frac{\partial v^{\beta}} {\partial...
  24. 7

    Basis + covariant derivative question

    My apologies about lack of precision in nomenclature. So I wanted to know how to express a certain idea about choice of basis on a manifold... Let's suppose I am solving a reaction-diffusion equation with finite elements. If I consider a surface that is constrained to lie in a flat plane or...
  25. D

    Verifying identity involving covariant derivative

    i am trying to verify the following identity: 0 = ∂g_mn / ∂y^p + Γ ^s _pm g_sn + Γ ^r _pn g_mr where Γ is the christoffel symbol with ^ telling what is the upper index and _ telling what are the two lower indices. g_mn is the metric tensor with 2 lower indices and y^p is the component of y...
  26. TrickyDicky

    What is the physical meaning of metric compatibility and why is it important?

    What exactly is the physical meaning of the fact that the covariant derivative of the metric tensor vanishes?
  27. I

    Covariant derivative in gauge theory

    Is the following formula correct? Suppose we work in a 4D Euclidean space for a certain gauge theory, \int d^4x~ \text{tr}\Big(D_i(\phi X_i )\Big) = \oint d^3S_i~ \text{tr}(\phi X_i) and, \int d^4x~\partial_j \text{tr}(\phi F_{mn}\epsilon_{mnij}) = \oint d^2S_j~ \text{tr}(\phi...
  28. I

    Covariant derivative of an anti-symmetric tensor

    Given an antisymmetric tensor T^{ab}=-T^{ab} show that T_{ab;c} + T_{ca;b} + T_{bc;a} = 0 If I explicitly write out the covariant derivative, all terms with Christoffel symbols cancel pair-wise, and I'm left to demonstrate that T_{ab,c} + T_{ca,b} + T_{bc,a} = 0 and this I...
  29. L

    How Do I Delete a Thread on a Website or Forum?

    Edit: Solved Don't know how to delete thread though!
  30. T

    General Relativity - Double Covariant Derivative

    I know that for a scalar \nabla^2\phi=\nabla_a\nabla^a\phi=\nabla^a\nabla_a\phi. However what is \nabla^2 for a tensor? For example, is \nabla^2T_a=\nabla_b\nabla^bT_a or is it \nabla^2T_a=\nabla^b\nabla_bT_a? Because I don't think they're the same thing. Thanks.
  31. I

    Covariant derivative in spherical coordinate

    I am confused with the spherical coordinate. Say, in 2D, the polar coordinate (r, \theta) The mathworld website says that http://mathworld.wolfram.com/SphericalCoordinates.html D_k A_j = \frac{1}{g_{kk}} \frac{\partial A_j}{\partial x_k} - \Gamma^i_{ij}A_i I don't know why we...
  32. T

    Covariant derivative and vector functions

    So given this identity: [V,W] = \nablaVW-\nablaWV ^^I got the above identity from O'Neil 5.1 #9. From this I'm not sure how to make the jump with vector functions, or if it is even possible to apply that definition to a vector function [xu,xv].
  33. avorobey

    Covariant derivative and geometry of tensors

    I'm trying to teach myself GR from Wald's General Relativity, and it's very tough going. I do have basic knowledge of differential geometry, but I think my geometric intuition is next to nonexistent. I'd very much appreciate some help in understanding several basic questions, or pointers to...
  34. V

    Covariant derivative of the metric

    hello! just a quick question, does the covariant derivative of the metric give zero even when the indices(one of the indices) of the metric are(is) raised? also another question not entirely related, does the covariant deriv. of exp(2 phi) where phi is the field, also give zero or not...
  35. T

    Understanding Covariant Derivative: d Explained

    I get in essence what the covariant derivative is, and what it does, but I am having trouble with the definition, of all things.\nabla_{\alpha}T^{\beta\gamma}=\frac{\partial{T^{\beta\gamma}}}{\partial{x^{\alpha}}}+\Gamma^{\beta}_{d\alpha}T^{d\gamma}+\Gamma^{\gamma}_{d\alpha}T^{\beta d} Im good...
  36. Q

    Covariant derivative vs. Lie derivative

    Hey there, For quite some time I've been wondering now whether there's a well-understandable difference between the Lie and the covariant derivative. Although they're defined in fundamentally different ways, they're both (in a special case, at least) standing for the directional derivative of...
  37. R

    Covariant derivative of the Christoffel symbol

    Homework Statement Is the covariant derivative of a Christoffel symbol equal to zero? It seems like it would be since it is composed of nothing but metrics, and the covariant derivative of any metric is zero, right?
  38. R

    Lie derivative versus covariant derivative

    When calculating the derivative of a vector field X at a point p of a smooth manifold M, one uses the Lie derivative, which gives the derivative of X in the direction of another vector field Y at the same point p of the manifold. If the manifold is a Riemannian manifold (that is, equipped...
  39. R

    Interpretation of the covariant derivative

    Is this the right way to think about the covariant derivative, and if not, what improvements would you suggest to visualize the meaning of the covariant derivative? (1)\mbox{ }\vec{e}_i(x')=\vec{e}_i(x)+\frac{\partial \vec{e}_i(x)}{\partial x^j}dx^j...
  40. S

    Covariant Derivative: Definition & Meaning

    Hello everyone, While studying properties of Riemann and tensor and Killing vectors, I found this notation/concept that I'm not sure of it's meaning. What does it mean to have a covariant derivative, using semi-colon notation, showing in upper index position. Is it just a matter of raising...
  41. E

    Trying to understand covariant derivative, conceptually and visually. Help please?

    Hello! I registered here today because I'm quite curious about the covariant derivative, and although I've consulted several texts on the subject (and wikipedia, and other locations), I've found it somewhat difficult to piece together a visual understanding of the covariant derivative. The...
  42. T

    Metric and Covariant Derivative

    I've seen read a lot of books where they use different sign conventions for the metric and the covariant derivative. I'd like to ask the physics community the following questions: I've seen both, the (+, -, -, -) and (-, +, +, +), conventions used for the metric, and I've also seen both...
  43. A

    Covariant derivative in polar coordinates

    I calculated the christoffel symbols and know that I have them right. I want to take the covariant derivative of the basis vector field e_{r} on the curve s(t) = (a, t/a). I differentiate it and get s' = (0, 1/a) and according to the metric, this is a unit vector because a will always be equal...
  44. F

    Covariant derivative transformation

    Homework Statement The problem concerns how to transform a covariant differentiation. Using this formula for covariant differentiation and demanding that it is a (1,1) tensor: \nabla_cX^a=\partial_cX^a+\Gamma^a_{bc}X^b it should be proven that \Gamma'^a_{bc}=...
  45. T

    Help Needed: Rewriting Covariant Derivative to Killing Equations

    A little stuck while working through a derivation. Hope someone can help. Homework Statement Starting from -\xi^c(\Gamma^d_{ca}g_{db}+\Gamma^d_{cb}g_{ad})+\partial_b\xi^dg_{ad}+\partial_a\xi^cg_{cb}=0 I need to obtain the Killing equations, i.e...
  46. B

    Trouble understanding derivation of covariant derivative

    Hi, I'm having problems following a derivation for the covariant derivative. I've shown the line where I'm having trouble: http://img15.imageshack.us/img15/49/covariantderivative.jpg The general argument being used is that if the covariant derivative must follow the product rule it can...
  47. M

    Newtonian limit of covariant derivative of stress-energy tensor(schutz ch7)

    Homework Statement For a perfect fluid verify that the spatial components of T^{\mu \nu};_{\nu} = 0 in the Newtonian limit reduce to (\rho v^{i}),_{t} + (\rho v^{i} v^{j}),_{j} + P,_{i} + \rho \phi ,_{i} Homework Equations Metric ds^{2} = -(1+2 \phi )dt^{2} + (1-2 \phi) (dx^{2} +...
  48. N

    Covariant derivative repect to connection?

    Could anyone tell me about the covariant derivative with respect to the connection?
  49. snoopies622

    Directional covariant derivative

    Is this correct? \nabla _{\vec{p}} \vec{p} = (\nabla_a \vec{p} ) p^a =< (\nabla_a p^0 ) p^a, (\nabla_a p^1 ) p^a , (\nabla_a p^2 ) p^a, (\nabla_a p^3 ) p^a > (where the a's are summed from 0 to 3)
  50. Phrak

    Yang-Mills covariant derivative

    In developing the Yang-Mills Lagrangian, Wikipedia defines the covariant derivative as \ D_ \mu = \partial _\mu + A _\mu (x) . Is A_mu to be taken as a 1-form, so that \ D _\mu \Phi = \partial _\mu \Phi + A _\mu (x) or an operator on \Phi, such that \ D _\mu \Phi = \partial...
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