What is Covariant: Definition and 361 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.

View More On Wikipedia.org
Recent contents Top users
  1. Hans de Vries

    A New Covariant QED representation of the E.M. field

    90 years have gone by since P.A.M. Dirac published his equation in 1928. Some of its most basic consequences however are only discovered just now. (At least I have never encountered this before). We present the Covariant QED representation of the Electromagnetic field. 1 - Definition of the...
  2. LesterTU

    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##...
  3. George Keeling

    Covariant coordinates don't co-vary

    Homework Statement I am studying co- and contra- variant vectors and I found the video at youtube.com/watch?v=8vBfTyBPu-4 very useful. It discusses the slanted coordinate system above where the X, Y axes are at an angle of α. One can get the components of v either by dropping perpendiculars...
  4. K

    I Covariant Derivatives: Doubt on Jolt & Proving Zj Γjk Vi = 0

    I've just learned about the covariant derivatives (##\nabla_i## and ##\delta/\delta t##) and I have a doubt. We should be able to say that $$ J^i = \frac{\delta A^i}{\delta t} = \frac{\delta^2 V^i}{\delta^2 t} = \frac{\delta^3 Z^i}{\delta^3 t} $$ where ##J## is the jolt. This...
  5. D

    I Transforming Contra & Covariant Vectors

    Hi. The book I am using gives the following equations for the the Lorentz transformations of contravariant and covariant vectors x/μ = Λμν xν ( 1 ) xμ/ = Λμν xv ( 2 ) where the 2 Lorentz transformation matrices are the inverses of each other. I am trying to get equation 2...
  6. Gene Naden

    I Do Isometries Preserve Covariant Derivatives?

    O'Neill's Elementary Differential Geometry, in problem 3.4.5, asks the student to prove that isometries preserve covariant derivatives. Before solving the problem in general, I decided to work through the case where the isometry is a simple inversion: ##F(p)=-p##, using a couple of simple vector...
  7. M

    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?
  8. K

    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...
  9. binbagsss

    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...
  10. K

    I Comparing Lie & Covariant Derivatives of Vector Fields

    I subtracted the ##\mu##-th component of the Lie Derivative of a Vector ##U## along a vector ##V## from the ##\mu##-th component of the Covariant derivative of the same vector ##U## along the same vector ##V## and I got ##(\nabla_V U)^\mu - (\mathcal{L}_V U)^\mu = U^\nu \partial_\nu V^\mu -...
  11. bananabandana

    I Deriving Covariant Form of $E_{1}E_{2}|\vec{v}|$

    Given a two particle scattering problem with (initial) relative velocity $|\vec{v}|$, apparently the product $E_{1}$E_{2}|\mathb{v}|$ can be expressed in the covariant form: $$ E_{1}E_{2}|\vec{v}| = \sqrt{ (p_{1}\cdot p_{2} - m_{1}^{2}m_{2}^{2}} $$ My textbook gives no further explanation -...
  12. binbagsss

    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...
  13. P

    B How is the gradient covariant?

    Hi, basic cartesian coordinates and we want to know the gradient of a scalar function of x,y, and z. So we can use the most basic basis there is of three orthogonal unit vectors and come up with the gradient of the scalar function. Now without rescaling the coordinate system or altering it in...
  14. P

    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}...
  15. binbagsss

    A Prove EL Geodesic and Covariant Geodesic Defs are Same via Riemmanian Geometry

    ... via plugging in the Fundamental theorem of Riemmanian Geometry : ##\Gamma^u_{ab}=\frac{1}{2}g^{uc}(\partial_ag_{bc}+\partial_bg_{ca}-\partial_cg_{ab})## Expanding out the covariant definition gives the geodesic equation as: (1) ##\ddot{x^u}+\Gamma^u_{ab} x^a x^b =0 ## (2) Lagrangian is...
  16. JuanC97

    I Does a covariant version of Euler-Lagrange exist?

    Hello everyone. I've seen the usual Euler-Lagrange equation for lagrangians that depend on a vector field and its first derivatives. In curved space the equation looks the same, you just replace the lagrangian density for {-g}½ times the lagrangian density. I noticed that you can replace...
  17. Alain De Vos

    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 ?
  18. Gene Naden

    I Contravariant first index, covariant on second, Vice versa?

    I am working through a derivation of the Dirac matrix transformation properties. I have a tensor for the Lorentz transformation that is covariant on the first index and contravariant on the second index. For the derivation, I need vice versa, i.e. covariant on the second index and contravariant...
  19. P

    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##...
  20. binbagsss

    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...
  21. PeroK

    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...
  22. F

    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...
  23. S

    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...
  24. T

    I Finding Covariant and Contravariant components of Cylin Coor

    I'm going through Introduction_to_Tensor_Calculus by Wiskundige_Ingenieurstechnieken. I want to find the covariant and contravariant components of the cylindrical cooridnates. Ingerniurstechnieken sets tangent vectors as basis (E1, E2, E3). He also sets the normal vectors as another basis (E1...
  25. 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? ?
  26. 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( {...
  27. JTC

    A Understanding the Dual Basis and Its Directions

    Please help. I do understand the representation of a vector as: vi∂xi I also understand the representation of a vector as: vidxi So far, so good. I do understand that when the basis transforms covariantly, the coordinates transform contravariantly, and v.v., etc. Then, I study this thing...
  28. davidge

    I Solving Covariant Derivatives: Minkowskian Metric

    How does one solve a problem like this? Suppose we have $$(e_\theta + f(\theta)e_\varphi) (e_\theta + f(\theta)e_\varphi)$$ What is the result of the above operation? As I remember it from the theory of covariant derivatives, the above relation would look like this $$e_\theta[e_\theta] +...
  29. saadhusayn

    I Transformation of covariant vector components

    Riley Hobson and Bence define covariant and contravariant bases in the following fashion for a position vector $$\textbf{r}(u_1, u_2, u_3)$$: $$\textbf{e}_i = \frac{\partial \textbf{r}}{\partial u^{i}} $$ And $$ \textbf{e}^i = \nabla u^{i} $$ In the primed...
  30. binbagsss

    General relativity, geodesic, KVF, chain rule covariant derivatives

    Homework Statement To show that ##K=V^uK_u## is conserved along an affinely parameterised geodesic with ##V^u## the tangent vector to some affinely parameterised geodesic and ##K_u## a killing vector field satisfying ##\nabla_a K_b+\nabla_b K_a=0## Homework Equations see above The Attempt at...
  31. 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}) =...
  32. 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...
  33. P

    I Differential forms as a basis for covariant antisym. tensors

    In a text I am reading (that I unfortunately can't find online) it says: "[...] differential forms should be thought of as the basis of the vector space of totally antisymmetric covariant tensors. Changing the usual basis dx^{\mu_1} \otimes ... \otimes dx^{\mu_n} with dx^{\mu_1} \wedge ...
  34. Dyatlov

    I Worked example on a covariant vector transformation

    Hello. I would like to check my understanding of how you transform the covariant coordinates of a vector between two bases. I worked a simple example in the attached word document. Let me know what you think.
  35. 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...
  36. O

    A What Are the Different Types of Derivatives in Calculus?

    Derivatives in first year calculus Gateaux Derivatives Frechet Derivatives Covariant Derivatives Lie Derivatives Exterior Derivatives Material Derivatives So, I learn about Gateaux and Frechet when studying calculus of variations I learn about Covariant, Lie and Exterior when studying calculus...
  37. J

    Books on Covariant formulation of Electrodynamics

    Hello! I am an undergraduate currently enrolled in a course on theoretical physics. One big part of the course is on the classical field theory of electromagnetism(on its covariant formulation using Lagrangians mostly). So, I would like to ask which are some good books on the subject. Thanks in...
  38. M

    B Covariant Quantum Fields

    I just read Carlo Rovelli new book "Reality is Not What It Seems: The Journey to Quantum Gravity" in one sitting. I'd like to know about the following: "Fields that live on themselves, without the need of a spacetime to serve as a substratum, as a support, and which are capable by themselves of...
  39. ShayanJ

    Different versions of covariant Maxwell's equations

    The standard way of writing Maxwell's equations is by assuming a vector potential ## A^\mu ## and then defining ## F_{\mu \nu}=\partial_\mu A_\nu-\partial_\nu A_\mu ##. Then by considering the action ## \displaystyle \mathcal S=-\int d^4 x \left[ \frac 1 {16 \pi} F_{\mu \nu}F^{\mu\nu}+\frac 1 c...
  40. 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...
  41. S

    A What is the true definition of the covariant gamma matrix ##\gamma_{5}##?

    Covariant gamma matrices are defined by $$\gamma_{\mu}=\eta_{\mu\nu}\gamma^{\nu}=\{\gamma^{0},-\gamma^{1},-\gamma^{2},-\gamma^{3}\}.$$ -----------------------------------------------------------------------------------------------------------------------------------------------------------...
  42. G

    Different formulations of the covariant EM Lagrangian

    Homework Statement I'm reading through A. Zee's "Quantum Field Theory in a nutshell" for personal learning and am a bit confused about a passage he goes through when discussing field theory for the electromagnetic field. I am well versed in non relativistic quantum mechanics but have no...
  43. 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...
  44. naima

    I Commutator of covariant derivatives

    Hi there I came across this paper. the author defines a covariant derivative in (1.3) ##D_\mu = \partial_\mu - ig A_\mu## He defines in (1.6) ##F_{jk} = i/g [D_j,D_k]## Why is it equal to ##\partial_j A_k - \partial_k A_j - ig [A_j, A_k]##? I suppose that it comes from a property of Lie...
  45. 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...
  46. 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...
  47. S

    Manifestly covariant Maxwell's equations

    Consider the following Maxwell's equation in tensor notation: ##\partial_{k}F_{ij}=0## ##-\partial_{k}\epsilon_{ijm}B_{m}=0## ##\partial_{k}\epsilon_{ijm}B_{m}=0## ##\partial_{k}B_{k}=0## I wonder how you go from the third line to the fourth line.
  48. mertcan

    I Covariant and contravariat components

    hi, Initially I would like to ask a little and basic question: I know that $$v=\sum_{i=0} e_i v^i$$ where $$v^i=e^i v$$ But sometimes I think we can write the first equation like $$v=\sum_{i=0} e_i e^i v$$, and I am aware that $$e_i e^i=1$$ , then our equation becomes $$v=\sum_{i=0} v$$, ın...
  49. 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...
  50. 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...
Back
Top