# 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. ### How Do Perturbation Equations Affect FRW Cosmology Metrics?

The perturbed line element: ##g = a(\tau)^2[-d\tau^2 + (\delta_{ij} + h_{ij})dx^i dx^j]## Expanding the covariant derivative with ##\nu = 0##, you get a few pieces. Here on keeping only terms linear in the perturbations, ##\partial_{\mu} T^{\mu 0} = a^{-2} \bar{\rho} \left[ \delta' -...
2. ### I Newton Galilean spacetime as fiber bundle

Hi, Penrose in his book "The Road to Reality" claims that Newton/Galilean spacetime has actually a structure of fiber bundle. The base is one-dimensional Euclidean space (time) and each fiber is a copy of ##\mathbb E^3##. The projection on the base space is the "universal time mapping" that...

15. ### I Is Covariant Derivative Notation Misleading in Vector Calculus?

[Moderator's note: Thread spun off from previous thread due to topic change.] This thread brings a pet peeve I have with the notation for covariant derivatives. When people write ##\nabla_\mu V^\nu## what it looks like is the result of operating on the component ##V^\nu##. But the components...
16. E

### B A few questions about the covariant derivative

Hey everyone, I was trying to learn in an unrigorous way a bit about making derivatives in the general manifold, but I'm getting confused by a few things. Take a vector field ##V \in \mathfrak{X}(M): M \rightarrow TM##, then in some arbitrary basis ##\{ e_{\mu} \}## of ##\mathfrak{X}(M)## we...

20. ### B Standard version of covariant derivative properties

[Throughout we're considering the intrinsic version of the covariant derivative. The extrinsic version isn't of any concern.] I'm having trouble reconciling different versions of the properties to be satisfied by the covariant derivative. Essentially ##\nabla## sends ##(p,q)##-tensors to...
21. ### I Parallel transport vs Lie dragging along a Killing vector field

Hi, I would like to ask for a clarification about the difference between parallel transport vs Lie dragging in the following scenario. Take a vector field ##V## defined on spacetime manifold and a curve ##C## on it. The manifold is endowed with the metric connection (I'm aware of it does exist...
22. ### I Dirac Lagrangian and Covariant derivative

This is from Griffiths particle physics, page 360. We have the full Dirac Lagrangian: $$\mathcal L = [i\hbar c \bar \psi \gamma^{\mu} \partial_{\mu} \psi - mc^2 \bar \psi \psi] - [\frac 1 {16\pi} F^{\mu \nu}F_{\mu \nu}] - (q\bar \psi \gamma^{\mu} \psi)A_{\mu}$$ This is invariant under the joint...
23. ### I About Covariant Derivative as a tensor

Hi, I've been watching lectures from XylyXylyX on YouTube. I believe they are really great ! One doubt about the introduction of Covariant Derivative. At minute 54:00 he explains why covariant derivative is a (1,1) tensor: basically he takes the limit of a fraction in which the numerator is a...
24. ### I Covariant Derivative: Limits on Making a Tensor?

Can you take any non invariant quantity like components and take the covariant derivative of them and arrive at an invariant tensor quantity? Or are there limits on what you can make a tensor?
25. ### I Ricci Tensor: Covariant Derivative & Its Significance

I read recently that Einstein initially tried the Ricci tensor alone as the left hand side his field equation but the covariant derivative wasn't zero as the energy tensor was. What is the covariant derivative of the Ricci tensor if not zero?
26. ### I Gauge Transformations and the Covariant Derivative

This is from QFT for Gifted Amateur, chapter 14. We have a Lagrangian density: $$\mathcal{L} = (D^{\mu}\psi)^*(D_{\mu}\psi)$$ Where $$D_{\mu} = \partial_{\mu} + iq A_{\mu}(x)$$ is the covariant derivative. And a global gauge transformation$$\psi(x) \rightarrow \psi(x)e^{i\alpha(x)}$$ We are...

30. ### I Metric compatibility and covariant derivative

Sean Carroll says that if we have metric compatibility then we may lower the index on a vector in a covariant derivative. As far as I know, metric compatibility means ##\nabla_\rho g_{\mu\nu}=\nabla_\rho g^{\mu\nu}=0##, so in that case ##\nabla_\lambda p^\mu=\nabla_\lambda p_\mu##. I can't see...
31. ### I Covariant Derivative: 2nd Diff - My Question

My question is shown in Summary section. Please see the attached file.
32. ### I Christoffel symbols and covariant derivative intuition

So I'm trying to get sort of an intuitive, geometrical grip on the covariant derivative, and am seeking any input that someone with more experience might have. When I see ##\frac {\partial v^{\alpha}}{\partial x^{\beta}} + v^{\gamma}\Gamma^{\alpha}{}_{\gamma \beta}##, I pretty easily see a...
33. ### I The vanishing of the covariant derivative of the metric tensor

I brought up this subject here about a decade ago so this time I'll try to be more specific to avoid redundancy. In chapter five of Bernard F. Schutz's A First Course In General Relativity, he arrives at the conclusion that in flat space the covariant derivative of the metric tensor is zero...
34. ### A Semicolon notation in component of covariant derivative

Can someone clarify the use of semicolon in I know that semicolon can mean covariant derivative, here is it being used in the same way (is expandable?) Or is a compact notation solely for the components of?
35. ### I Covariant derivative of the contracted energy-momentum tensor of a particle

The energy-momentum tensor of a free particle with mass ##m## moving along its worldline ##x^\mu (\tau )## is $$T^{\mu\nu}(y^\sigma)=m\int d \tau \frac{\delta^{(4) }(y^\sigma-x^\sigma(\tau ))}{\sqrt{-g}}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau}.$$ Let contract...
36. ### A Question about covariant derivatives

I am reading I am reading Spacetime and Geometry : An Introduction to General Relativity -- by Sean M Carroll and have arrived at chapter 3 where he introduces the covariant derivative ##{\mathrm{\nabla }}_{\mu }##. He makes demands on this which are \begin{align} \mathrm{1.\...
37. ### I Connecting Geodesic Curves and the Covariant Derivative

In Carrol's gr notes the covariant derivative of a vector is given as ∇μAϑ=∂μAϑ+ΓϑμλAλ...(1) For a geodesic in 2-D cartesian coordinates the tangent vector is V=##a\hat x+b\hat y##(a and b are constt.)where the tangent vector direction along the curve is ##\hat n=\frac{a\hat x+b\hat...
38. ### I Covariant derivative of tangent vector for geodesic

For the simple case of a 2-D curve in polar coordinated (r,θ) parametrised by λ (length along the curve). At any λ the tangent vector components are V1=dr(λ)/dλ along ##\hat r## and V2=dθ(λ)/dλ along ##\hat θ##. The non-zero christoffel symbol are Γ122 and Γ212. From covariant derivative...
39. ### Expressing Covariant Derivative in Matrix Form

Homework Statement We are given a Lorentz four-vector in "isospin space" with three components ##\vec v^{\mu} = (v^{\mu}_1, v^{\mu}_2, v^{\mu}_3)## and want to express the covariant derivative $$D^{\mu} = {\partial}^{\mu} - ig\frac {\vec \tau} {2}\cdot \vec v^{\mu}$$ explicitly in ##2\times 2##...
40. ### Covariant derivative summation convention help

Homework Statement Assume that you want to the derivative of a vector V with respect to a component Zk, the derivative is then ∂ViZi/∂Zk=Zi∂Vi/∂Zk+Vi∂Zi/∂Zk = Zi∂Vi/∂Zk+ViΓmikZm Now why is it that I can change m to i and i to j in ViΓmikZm?
41. ### I Covariant Derivative Equivalence: Exploring an Intriguing Result

If we are representing the basis vectors as partial derivatives, then ##\frac{\partial}{\partial x^\nu + \Delta x^\nu} = \frac{\partial}{\partial x^\nu} + \Gamma^\sigma{}_{\mu \nu} \Delta x^\mu \frac{\partial}{\partial x^\sigma}## to first order in ##\Delta x##, correct? But in the same manner...
42. ### I EoM via varying action - covariant derivative when integrate

##\int d^4 x \sqrt {g} ... ## if I am given an action like this , were the ##\sqrt{\pm g} ## , sign depending on the signature , is to keep the integral factor invariant, when finding an eom via variation of calculus, often one needs to integrate by parts. When you integrate by parts, with...
43. ### Tensor Covariant Derivative Expressions Algebra (Fermi- Walk

Homework Statement Hi I am looking at part a). Homework Equations below The Attempt at a Solution I can follow the solution once I agree that ## A^u U_u =0 ##. However I don't understand this. So in terms of the notation ( ) brackets denote the symmetrized summation and the [ ] the...
44. ### A Compute Commutator of Covariant Derivative & D/ds on Vector Fields

Hi, let ##\gamma (\lambda, s)## be a family of geodesics, where ##s## is the parameter and ##\lambda## distinguishes between geodesics. Let furthermore ##Z^\nu = \partial_\lambda \gamma^\nu ## be a vector field and ##\nabla_\alpha Z^\mu := \partial_\alpha Z^\mu + \Gamma^\mu_{\:\: \nu \gamma}...
45. ### I What is the covariant derivative of the position vector?

What is the covariant derivative of the position vector $\vec R$ in a general coordinate system? In which cases it is the same as the partial derivative ?
46. ### A Interpretation of covariant derivative of a vector field

On Riemannian manifolds ##\mathcal{M}## the covariant derivative can be used for parallel transport by using the Levi-Civita connection. That is Let ##\gamma(s)## be a smooth curve, and ##l_0 \in T_p\mathcal{M}## the tangent vector at ##\gamma(s_0)=p##. Then we can parallel transport ##l_0##...
47. ### Covariant derivative and 'see-saw rule'

Homework Statement Apologies if this is a stupid question but just thinking about the see-saw rule applied to something like: ## w_v \nabla_u V^v = w^v \nabla_u V_v ## It is not obvious that the two are equivalent to me since one comes with a minus sign for the connection and one with a plus...
48. ### I Confirming Covariant Derivative in Hartle's Gravity

In Hartle's Gravity we have the covariant derivative (first in an LIF) which is: ##\nabla_{\beta} v^{\alpha} = \frac{\partial v^{\alpha}}{\partial x^{\beta}}## As the components of the tensor ##\bf{ t = \nabla v}##. But, it's not clear which components they are! My guess is that...
49. ### I Covariant derivative of Ricci scalar causing me grief

Hi all I'm having trouble understanding what I'm missing here. Basically, if I write the Ricci scalar as the contracted Ricci tensor, then take the covariant derivative, I get something that disagrees with the Bianchi identity: \begin{align*} R&=g^{\mu\nu}R_{\mu\nu}\\ \Rightarrow \nabla...
50. ### A Covariant derivative in Standard Model

The covariant derivative in standard model is given byDμ = ∂μ + igs Gaμ La + ig Wbμ Tb + ig'BμYwhere Gaμ are the eight gluon fields, Wbμ the three weak interaction bosons and Bμ the single hypercharge boson. The La's are SU(3)C generators (the 3×3 Gell-Mann matrices ½ λa for triplets, 0 for...