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. mertcan

    A Covariant derivative only for tensor

    Hi initially I am aware that christoffel symbols are not tensor so their covariant derivatives are meaningless, but my question is why do we have to use covariant derivative only with tensors? ?? Is there a logic of this situation? ?
  2. tomdodd4598

    I Problem with Commutator of Gauge Covariant Derivatives?

    Hi there, I have just read that the gauge field term Fμν is proportional to the commutator of covariant derivatives [Dμ,Dν]. However, when I try to calculate this commatator, taking the symmetry group to be U(1), I get the following: \left[ { D }_{ \mu },{ D }_{ \nu } \right] =\left( {...
  3. J

    A Evaluate Covariant Derivative on Tensors

    Hello there, Recently I encountered a type of covariant derivative problem that I never before encountered: $$ \nabla_\mu (k^\sigma \partial_\sigma l_\nu) $$ My goal: to evaluate this term According to Carroll, the covariant derivative statisfies ##\nabla_\mu ({T^\lambda}_{\lambda \rho}) =...
  4. P

    I Product rule for exterior covariant derivative

    It is well known that the product rule for the exterior derivative reads d(a\wedge b)=(da)\wedge b +(-1)^p a\wedge (db),where a is a p-form. In gauge theory we then introduce the exterior covariant derivative D=d+A\wedge. What is then D(a ∧ b) and how do you prove it? I obtain D(a\wedge...
  5. joneall

    I Understanding parallel transfer

    I've read Collier's book on General Relativity and consulted parts of Schutz, Hartle and Carroll. In the terms they use, i have yet to gain anything resembling an intuitive understanding of parallel transport. In fact, it seems to me it is usually presented backwards, saying that the geodesic...
  6. MattRob

    Covariant Derivative Homework: Solve ∇_c ({∂}_b X^a)

    Homework Statement Take the Covariant Derivative ∇_{c} ({∂}_b X^a) Homework Equations ∇_{c} (X^a) = ∂_c X^a + Γ_{bc}^a X^b ∇_{c} (X^a_b) = ∂_c X^a_b + Γ_{dc}^a X^d_b - Γ^d_{bc} X^a_d The Attempt at a Solution Looking straight at ∇_{c} ({∂}_b X^a) I'm seeing two indices. However, the b is...
  7. F

    I Conservation of dot product with parallel transport

    Hello, I have 2 questions regarding similar issues : 1*) Why does one say that parallel transport preserves the value of dot product (scalar product) between the transported vector and the tangent vector ? Is it due to the fact that angle between the tangent vector and transported vector is...
  8. Q

    I Christoffel Symbol vs. Vector Potential

    As far as I can tell, in GR, the Chirstoffel symbol in the expression of the Connection is analogous to the vector potential, A, in the definition of the Covariant Derivative. The Chirstoffel symbol compensates for changes in curvature and helps define what it means for a tensor to remain...
  9. D

    I Covariant derivative of field strength tensor

    Hi, I am struggling to derive the relations on the right hand column of eq.(4) in https://arxiv.org/pdf/1008.4884.pdfEven the easy abelian one (third row) which is $$D_\rho B_{\mu\nu}=\partial_\rho B_{\mu\nu}$$ doesn't match my calculation Since $$D_\rho B_{\mu\nu}=(\partial_\rho+i g...
  10. L

    A Interpretation of Derivative of Metric = 0 in GR - Learning from Wald

    I am trying to learn GR, primarily from Wald. I understand that, given a metric, a unique covariant derivative is picked out which preserves inner products of vectors which are parallel transported. What I don't understand is the interpretation of the fact that, using this definition of the...
  11. rezkyputra

    Covariant Derivatives (1st, 2nd) of a Scalar Field

    Homework Statement Suppose we have a covariant derivative of covariant derivative of a scalar field. My lecturer said that it should be equal to zero. but I seem to not get it Homework Equations Suppose we have $$X^{AB} = \nabla^A \phi \nabla^B \phi - \frac{1}{2} g^{AB} \nabla_C \phi \nabla^C...
  12. S

    A Gauge invariance and covariant derivative

    Consider the covariant derivative ##D_{\mu}=\partial_{\mu}+ieA_{\mu}## of scalar QED. I understand that ##D_{\mu}\phi## is invariant under the simultaneous phase rotation ##\phi \rightarrow e^{i\Lambda}\phi## of the field ##\phi## and the gauge transformation ##A_{\mu}\rightarrow...
  13. M

    I Trying to understand covariant tensor

    I am taking a course on GR and trying to understand Tensor calculus. I think I understand contravariant tensor (transformation of objects such as a vector from one frame to another) but I am having a hard time with covariant tensors. I looked into the Wikipedia page...
  14. redtree

    A Help with covariant differentiation

    I'm having trouble evaluating the following expression (LATEX): ##\nabla_{i}\nabla_{j}T^{k}= \nabla_{i} \frac{\delta T^{k}}{\delta z^{j}} + \Gamma^{k}_{i m} \frac{\delta T^{m}}{\delta z^{i}} + \Gamma^{k}_{i m} \Gamma^{m}_{i l} T^{l}## What are the next steps to complete the covariant...
  15. mertcan

    A Expansion of covariant derivative

    (V(s)_{||})^\mu = V(s)^\mu + s \Gamma^\mu_{\nu \lambda} \frac{dx^\nu}{ds} V(s)^\lambda + higher-order terms (Here we have parallel transported vector from point "s" to a very close point)Hi, I tried to make some calculations to reach the high-order terms for parallel transporting of vector...
  16. mertcan

    A Riemann tensor and covariant derivative

    hi, I tried to take the covariant derivative of riemann tensor using christoffel symbols, but it is such a long equation that I have always been mixing up something. So, Could you share the entire solution, pdf file, or links with me? ((( I know this is the long way to derive the einstein...
  17. BiGyElLoWhAt

    I Covariant derivative of a contravariant vector

    This is (should be) a simple question, but I'm lost on a negative sign. So you have ##D_m V_n = \partial_m V_n - \Gamma_{mn}^t V_t## with D_m the covariant derivative. When trying to deduce the rule for a contravariant vector, however, apparently you end up with a plus sign on the gamma, and I'm...
  18. G

    I Raising index on covariant derivative operator?

    In Carroll, the author states: \nabla^{\mu}R_{\rho\mu}=\frac{1}{2} \nabla_{\rho}R and he says "notice that, unlike the partial derivative, it makes sense to raise an index on the covariant derivative, due to metric compatibility." I'm not seeing this very clearly :s What's the reasoning...
  19. W

    Covariant derivative of vector fields on the sphere

    Homework Statement Given two vector fields ##W_ρ## and ##U^ρ## on the sphere (with ρ = θ, φ), calculate ##D_v W_ρ## and ##D_v U^ρ##. As a small check, show that ##(D_v W_ρ)U^ρ + W_ρ(D_v U^ρ) = ∂_v(W_ρU^ρ)## Homework Equations ##D_vW_ρ = ∂_vW_ρ - \Gamma_{vρ}^σ W_σ## ##D_vU^ρ = ∂_vU^ρ +...
  20. J

    A Covariant derivative definition in Wald

    I'm working through Wald's "General Relativity" right now. My questions are actually about the math, but I figure that a few of you that frequent this part of the forums may have read this book and so will be in a good position to answer my questions. I have two questions: 1) Wald first defines...
  21. D

    Covariant derivative of Killing vector and Riemann Tensor

    I need to prove that $$D_\mu D_\nu \xi^\alpha = - R^\alpha_{\mu\nu\beta} \xi^\beta$$ where D is covariant derivative and R is Riemann tensor. ##\xi## is a Killing vector. I have proved that $$D_\mu D_\nu \xi_\alpha = R_{\alpha\nu\mu\beta} \xi^\beta$$ I can't figure out a way to get the required...
  22. D

    A Confusion on notion of connection & covariant derivative

    I have been reading Nakahara's book "Geometry, Topology & Physics" with the aim of teaching myself some differential geometry. Unfortunately I've gotten a little stuck on the notion of a connection and how it relates to the covariant derivative. As I understand it a connection ##\nabla...
  23. LarryS

    Complex numbers sometimes *Required* in Classical Physics?

    In general, one thinks of complex numbers as being absolutely required in Quantum Physics but as being optional in Classical Physics. But what about modern classical electromagnetic field theory (gauge theory) in which the electromagnetic field is coupled to the field of charged particles by...
  24. N

    Relation between affine connection and covariant derivative

    I now study general relativity and have a few questions regarding the mathematical formulation: 1) What ist the relation between an connection and a covariant derivative? Can you explain the exact difference? 2) One a lorentzian manifold, what ist the relation between the...
  25. T

    Metric variation of the covariant derivative

    Homework Statement Hi all, I currently have a modified Einstein-Hilbert action, with extra terms coming from some vector field A_\mu = (A_0(t),0,0,0), given by \mathcal{L}_A = -\frac{1}{2} \nabla _\mu A_\nu \nabla ^\mu A ^\nu +\frac{1}{2} R_{\mu \nu} A^\mu A^\nu . The resulting field...
  26. P

    Double covariant derivative of function of scalar

    If R is Ricci scalar ∇i∇j F(R) = ? , where ∇i is covariant derivative.
  27. D

    Ricci rotation coefficients and non-coordinate bases

    I'm currently working through chapter 7 on Riemannian geometry in Nakahara's book "Geometry, topology & physics" and I'm having a bit of trouble reproducing his calculation for the metric compatibility in a non-coordinate basis, using the Ricci rotation coefficients...
  28. S

    Partial Derivative of x^2 on Manifold (M,g)

    How can I figure out ##\partial_\mu x^2## on the manifold ##(M,g)##? I thought that it should be ##2x_\mu##, but I think I'm wrong and the answer is ##2x_\mu+x^\nu x^\lambda \partial_\mu g_{\nu\lambda}##, right?! In particular, it seems to me, we can't write...
  29. Ravendark

    Is the Sign in the Covariant Derivative Important for Local Gauge Invariance?

    Homework Statement Consider the fermionic part of the QCD Lagrangian: $$\mathcal{L} = \bar\psi (\mathrm{i} {\not{\!\partial}} - m) \psi \; ,$$ where I used a matrix notation to supress all the colour indices (i.e., ##\psi## is understood to be a three-component vector in colour space whilst...
  30. P

    Checking derivation of the curvature tensor

    Homework Statement I am trying to derive the curvature tensor by finding the commutator of two covariant derivatives. I think I've got it, but my head is spinning with Nablas and indices. Would anyone be willing to check my work? Thanks Homework Equations I am trying to derive the curvature...
  31. dhalilsim

    D'Alembert operator is commute covariant derivative?

    For example: [itex] D_α D_β D^β F_ab= D_β D^β D_α F_ab is true or not? Are there any books sources?
  32. U

    Covariant Derivative - where does the minus sign come from?

    I was reading through hobson and my notes where the covariant acts on contravariant and covariant tensors as \nabla_\alpha V^\mu = \partial_\alpha V^\mu + \Gamma^\mu_{\alpha \gamma} V^\gamma \nabla_\alpha V_\mu = \partial_\alpha V_\mu - \Gamma^\gamma_{\alpha \mu} V_\gamma Why is there a minus...
  33. D

    Parallel Transport & Covariant Derivative: Overview

    I have been reading section 3.1 of Wald's GR book in which he introduces the notion of a covariant derivative. As I understand, this is introduced as the (partial) derivative operators \partial_{a} are dependent on the coordinate system one chooses and thus not naturally associated with the...
  34. F

    Understanding Covariant Derivative & Parallel Transport

    Hello, I try to apprehend the notion of covariant derivative. In order to undertsand better, here is a figure on which we are searching for express the difference \vec{V} = \vec{V}(M') - \vec{V}(M) : In order to evaluate this difference, we do a parallel transport of \vec{V}(M') at point...
  35. binbagsss

    Covariant Derivative: Understanding Motivation Quickly

    As we can not meaningfully compare a vector at 2 points acted upon by this operator , because it does not take into account the change due to the coordinate system constantly changing, I conclude that the elementary differential operator must describe a change with respect to space-time, How do...
  36. K

    Covariant derivative for four velocity

    Homework Statement Show U^a \nabla_a U^b = 0 Homework Equations U^a refers to 4-velocity so U^0 =\gamma and U^{1 - 3} = \gamma v^{1 - 3} The Attempt at a Solution I get as far as this: U^a \nabla_a U^b = U^a ( \partial_a U^b + \Gamma^b_{c a} U^c) And I think that the...
  37. P

    Covariant derivative of covector

    I was trying to see what is the covariant derivative of a covector. I started with $$ \nabla_\mu (U_\nu V^\nu) = \partial_\mu (U_\nu V^\nu) = (\partial_\mu U\nu) V^\nu + U_\nu (\partial_\mu V^\nu) $$ since the covariant derivative of a scalar is the partial derivative of the latter. Then I...
  38. C

    Covariant Derivative Wrt Superscript Sign: Explained

    Dear all, I was reading this https://sites.google.com/site/generalrelativity101/appendix-c-the-covariant-derivative-of-the-ricci-tensor, and it said that if you take the covariant derivative of a tensor with respect to a superscript, then the partial derivative term has a MINUS sign. How? The...
  39. Greg Bernhardt

    What is a covariant derivative

    Definition/Summary Covariant derivative, D, is a coordinate-dependent adjustment to ordinary derivative which makes each partial derivative of each coordinate unit vector zero: D\hat{\mathbf{e}}_i/\partial x_j\ =\ 0 The adjustment is made by a linear operator known both as the connection...
  40. S

    MHB Intrinsic Derivative and Covariant Derivative

    In the context of tensor calculus, what is the difference between intrinsic derivative and covariant derivative?
  41. J

    For what covariant derivative?

    I will take the differential form of position vector r: ##\vec{r}=r\hat{r}## ##d\vec{r}=dr\hat{r}+rd\hat{r}## So, now I need find ##d\hat{r}## ##d\hat{r}=\frac{d\hat{r}}{dr}dr+\frac{d\hat{r}}{d\theta}d\theta## ##\frac{d\hat{r}}{dr}=\Gamma ^{r}_{rr}\hat{r}+\Gamma...
  42. C

    Relation between covariant differential and covariant derivative

    In Theodore Frankel's book, "The Geometry of Physics", he observes at page 248 that the covariant derivative of a vector field can be written as $$\nabla_X v = e_iX^j (v^i_{,j} + \omega^i_{jk} v^k)= e_i(dv^i(X) + \omega^i_k(X) v^k) = e_i (dv^i + \omega^i_k v^k)(X)$$ where ##\omega^i_k =...
  43. C

    Covariant derivative of a commutator (deriving Bianchi identity)

    Hi. I'm trying to understand a derivation of the Bianchi idenity which starts from the torsion tensor in a torsion free space; $$ 0 = T(X,Y) = \nabla_X Y - \nabla_Y X - [X,Y]$$ according to the author, covariant differentiation of this identity with respect to a vector Z yields $$$ 0 =...
  44. O

    Commutator between covariant derivative, field strength

    Hello, i try to prove that ∂μFμ\nu + ig[Aμ, Fμ\nu] = [Dμ,Fμ\nu] with the Dμ = ∂μ + igAμ but i have a problem with the term Fμ\nu∂μ ... i try to demonstrate that is nil, but i don't know if it's right... Fμ\nu∂μ \Psi = \int (∂\nuFμ\nu) (∂μ\Psi) + \int Fμ\nu∂μ∂\nu \Psi = (∂\nuFμ\nu) [\Psi ]∞∞...
  45. L

    How do I derive the Covariant Derivative for Covectors? (Lower index)

    Hello everyone! I'm trying to learn the derivation the covariant derivative for a covector, but I can't seem to find it. I am trying to derive this: \nabla_{α} V_{μ} = \partial_{α} V_{μ} - \Gamma^{β}_{αμ} V_{β} If this is a definition, I want to know why it works with the definition...
  46. T

    General relativity - Covariant Derivative Of F(R)

    In f(R) gravity as http://en.wikipedia.org/wiki/F%28R%29_gravity , i have problem with the term [ g_ab □ - ∇_a ∇_b ] F(R) , well actually is [ ∇_b ∇_a - ∇_a ∇_b ] F(R) , but F is a function of Ricci Factor and Ricci Factor is expressed as a(t) ( scale factor ) . for the a = b = 0 i say this...
  47. Z

    Covariant derivative - is this a typo ?

    hello, please see the attached snapshot (taken from 'Problem book in relativity and gravitation'). In the last equation I think there would be no semicolon. Here is why I believe (S is scalar by the way): S;α[βγ] = 1/2 * ( S;αβγ - S;αγβ ) Now from the equation which precedes it, we have ...
  48. S

    Covariant derivative of a 1-form

    Let $$f:U \to \mathbb{R}^3$$ be a surface with local coordinates $$f_i=\frac{\partial f}{\partial u^i}$$. Let $\omega$ be a one-form. I want to express $$\nabla \omega$$ in terms of local coordinates and Christoffel symboles. Where $$\nabla$$ is the Levi-Civita connection (thus it coincides with...
  49. P

    Bitensor covariant derivative commutation

    Hi everyone, I'm trying to prove a relation in which I need do commute covariant derivatives of a bitensor. The equation is quite long but I need to write something like this: Given a bitensor G^{\alpha}_{\beta'}(x,x'), where the unprimed indexes (\alpha,\beta, etc) are assigned to the...
  50. P

    Covariant Derivative Commutation

    Hello, Can anyone tell me the general formula for commuting covariant derivatives, I mean, given a (r,s)-tensor field what is the formula to commute covariant derivatives? I found a formula http://pt.scribd.com/doc/25834757/21/Commuting-covariant-derivatives page 25, Eq.6.18 but it doesn't...
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