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.

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  1. 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}=...
  2. 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...
  3. E

    Understanding Covariant Derivatives Along a Curve

    I'm teaching myself about connections and came across something I am not completely sure about. The text I am using defines a connection as taking in two vector fields and outputting a vector field. However, later when discussing covariant derivatives along a curve I see this equation...
  4. 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...
  5. marcus

    Covariant loop quantum gravity and its low-energy limit (throwing the eagle)

    Rovelli has given his team and himself just seven months to advance LQG to a new stage. Why do I call this "throwing the eagle"? Because a Roman general would on occasion hurl his legion's eagle standard into the opposing army's midst, confident his side could rout their foes the recover it...
  6. 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} +...
  7. J

    What's the relation between SUSY Generators and Covariant Derivatives?

    Hello once again. I'm trying to understand the relation between the superspace representation of the SUSY generators Q_\alpha,\overline Q_{\dot\beta} and the covariant derivatives on superspaces D_\alpha, \overline D_{\dot\beta}: Q_\alpha = \frac{\partial}{\partial\theta^\alpha} -...
  8. N

    Covariant derivative repect to connection?

    Could anyone tell me about the covariant derivative with respect to the connection?
  9. 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)
  10. Peeter

    Covariant form of the Lagrangian for Lorentz force.

    If I use the following Lagrangian: \mathcal{L} = \frac{1}{2} m v^2 + e A \cdot v/c = \frac{1}{2} m \dot{x}_\mu \dot{x}^\mu + e A_\nu \dot{x}^\nu /c I can arrive at the Lorentz force equation in tensor form: m \ddot{x}_\mu &= (q/c) F_{\mu\beta} \dot{x}^\beta details offline...
  11. T

    Stokes theorem under covariant derivaties?

    in my GR book, it claims that integral of a covariant divergence reduces to a surface term. I'm not sure if I see this... So, is it true that: \int_{\Sigma}\sqrt{-g}\nabla_{\mu} V^{\mu} d^nx= \int_{\partial\Sigma}\sqrt{-g} V^{\mu} d^{n-1}x if so, how do I make sense of the d^{n-1}x term? would...
  12. snoopies622

    Covariant vs. contravariant time component

    ...of the four-momentum vector. Why is the energy of a particle identified with p0 instead of p0? Is there a theoretical basis for this, or was it simply observed that p0 is conserved in a larger set of circumstances?
  13. 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...
  14. N

    Change of basis- contravariant, covariant components of a vector.

    Homework Statement Let e_{i} with i=1,2 be an orthonormal basis in two-dimensional Euclidean space ie. the metric is g_{ij} = \delta _{ij}. In the this basis the vector v has contravariant components v^{i} = (1,2). Consider the new basis e_{1}^{'} = 5e_{1} - 2e_{2} e_{2}^{'} = 3e_{1} - e_{2}...
  15. F

    Covariant vectors vs reciprocal vectors

    If there is a contravariant vector v=aa+bb+cc with a reciprocal vector system where [abc]v=xb×c+ya×c+za×b would the vector expressed in the reciprocal vector system be a covariant vector? Is there any connection between the reciprocal vector system of a covariant vector and a...
  16. M

    The Exterior Covariant Derivative: Understanding Connections and Fibre Bundles

    Can you give me the definition of exterior covariant derivative or any reference web page ? Wiki does not involve enough info.I am not able to do calculation with respect to given definition there. Thanks in advance
  17. marcus

    Ashtekar: deriving the covariant entropy bound from LQC

    Here are some papers on the covariant entropy bound conjectured by Raphael Bousso http://arxiv.org/abs/hep-th/9905177 http://arxiv.org/abs/hep-th/9908070 http://arxiv.org/abs/hep-th/0305149 It would be a significant development if the conjectured bound could be proven to hold in LQC...
  18. M

    Intuition for Covariant derivative of a Tensor Field

    Hi. I am attempting to gain some intuition for what the covariant derivative of a tensor field is. I have a good intuition about the covariant derivative of vector fields (measuring how the vector changes as you move in a particular direction), and I understand how to extend the covariant...
  19. A

    Use of covariant derivative in general relativity.

    When we derive equation of motion by variation of the action, we use rules of ordinary differentiation and integration. So only ordinary derivatives can appear in the equation. Now in general relativity we are supposed to replace all those ordinary derivatives by covariant derivatives. Is that...
  20. F

    What is a covector, covariant tensor, etc?

    Sorry, but some of you may think this is a stupid question. (I'm only 16 years old.) I have just now gotten into the field of tensors and topology, after studying vector calculus and differential equations and I have two questions: a) What exactly is a covector? b) What is the...
  21. D

    Covariant derivative of metric tensor

    Hi, I'm trying to verify that the covariant derivative of the metric tensor is D(g) = 0. But I have a few questions: 1) This is a scalar 0 or a tensorial 0? Because it is suposed that the covariant derivative of a (m,n) tensor is a (m,n+1) tensor, and g is a (0,2) tensor so I think this 0...
  22. B

    Proving Covariant Derivative Transforms as Tensor

    Homework Statement Help! I wish to prove the following important statements: (1) The presence of Christoffel symbols in the covariant derivative of a tensor assures that this covariant derivative can transform like a tensor. (2) The reason for this is because, under transformation, the...
  23. E

    Prove Covariant Four Vector: \frac{\partial\phi}{\partial x^{\mu}}

    Homework Statement Show that \frac{\partial\phi}{\partial x^{\mu}} is a covariant four vector . Homework Equations All covariant four vector transformations . The Attempt at a Solution I really didn't understand what question implies . How can this vector be showed as being a...
  24. E

    Covariant Derivative: Proving Rank-2 Tensor Components

    Homework Statement I am trying to show that the components of the covariant derivative [tex] \del_b v^a are the mixed components of a rank-2 tensor. If I scan in my calculations, will someone have a look at them? Homework Equations The Attempt at a Solution
  25. C

    Identities for covariant derivative

    Hi. I'm considering the covariant derivative \nabla_\mu V^\nu = \partial_\mu V^\nu + \Gamma_{\mu\nu}^\lambda V^\lambda in spherical coordinates in flat 3D space (x = r cos sin, y = r sin sin, z = r cos; usual stuff). Now I wrote down the gradient of a scalar function f, for which I got...
  26. E

    Covariant and contravariant analysis

    Homework Statement Can someone explain/help me prove the formulas \vec{e_a}' = \frac{ \partial{x^b}}{\partial x'^a} \vec{e_b} \vec{e^a}' = \frac{ \partial{x'^a}}{\partial x^b} \vec{e^b} I do not understand why the partial derivative flip? Homework Equations The Attempt at a Solution
  27. S

    Covariant Derivative: What Is $\nabla^0 A_{\alpha}$?

    just a quick query, I know that, \nabla_0 A_{\alpha}= \partial_0 A_{\alpha} - \Gamma^{\beta}_{0 \alpha} A_{\beta} But what does \nabla^0 A_{\alpha} equal?
  28. Jim Kata

    Covariant quantum field theories

    I am not as well read as most of the people in here so I thought I would ask you guys first. What work has been done in the way of developing a covariant field theory? I'm going to ramble for just a little bit so try to follow. It seems to me that QFT is built on two principles Poincare...
  29. B

    Invariant and covariant in special relativity

    In anglo-american literature -a physical quantity is invariant if it has the same magnitude in all inertial reference frame, -an expression relating more physical quantities is covariant if it has the same algebraic structure in all inertial reference frames (rr-cctt) ? Thanks in advance
  30. W

    Gauge covariant derivative in curvilinear coordinates

    If we work in cartesian coordinates, we say for instance, that D_x \phi = \left( \frac{\partial}{\partial x} + i g \sum_a T_a A^a_x \right) \phi where g is the gauge coupling, and \{T^a\} are the generators of the gauge group, and \{A^a_\mu\} is the gauge vector field. But what happens when...
  31. C

    Covariant derivative and general relativity

    I'm not really sure where to put this, so I thought it post it here! I'm reading through my GR lecture notes, and have come across a comment that has confused me. I quote Now, I don't really see how this is true. For example, consider a scalar field f. The covariant derivative of this is...
  32. T

    Uncovering the Mystery of Covariant Derivatives: Sean Carroll's Perspective

    I've heard of something called a covariant derivative. what motivates it and what is it?
  33. S

    Lie vs Covariant Derivative: Intuitive Understanding

    Loosely speaking or Intuitively how should one understand the difference between Lie Derivative and Covariant derivative? Definitions for both sounds awfully similar...
  34. W

    Lie vs. covariant derivative

    Is there any relationship between the Lie (\pounds) and covariant derivative (\nabla)? Say I have 2 vector fields V, W and a metric g, the Lie and covariant derivative of W along V are: \pounds_{V}W = [V,W] V^\alpha \nabla_\alpha W^\mu = V^\alpha \partial_\alpha W^\mu + V^\alpha...
  35. T

    Covariant and Contravariant Rank-2 Tensors

    Dear Fellows, Do anyone have an idea of whether there must be a system tensor in order to be able to transform from the covariant form of a certain tensor to its contravariant one? This is a bit important to get rigid basics about tensors. Schwartz Vandslire...
  36. C

    Covariant vs. Contravariant: Understanding the Physical and Historical Basis

    Does anyone know the physical (or historical) basis for the terms covariant and contravariant? I'm guessing a particular class of mapping always tranforms components (of ..?) in exactly two different ways, so I'm wondering what the mappings are (Change of coordinate charts? Lorentz...
  37. A

    Partial and Covariant derivatives in invarint actions

    It's physics based but actually a maths question so I'm asking it here rather than the physics forums. I = \int \mathcal{L}\; d^{4}x I is invariant under some transformation \delta_{\epsilon} if \delta_{\epsilon}\mathcal{L} = \partial_{\mu}X^{\mu} for some function/tensor/field thingy X^{\mu}...
  38. Demystifier

    Covariant Loop Quantum Gravity: Experts' Views

    I find the covariant version of loop quantum gravity http://arxiv.org/abs/gr-qc?papernum=0608135 more appealing than the usual LQG approach. What the experts think?
  39. L

    Why should the covariant derivative of the metric tensor be 0 ?

    That's a crucial point of GR ! And I have always problems with that. Back to the basics, with your help. Thanks Michel
  40. H

    Why we have to definte covariant derivative?

    Why we have to definte covariant derivative?
  41. H

    What different between covariant metric tensor and contravariant metric tensor

    I read some books and see that the definition of covariant tensor and contravariant tensor. Covariant tensor(rank 2) A'_ab=(&x_u/&x'_a)(&x_v/&x'_b)A_uv Where A_uv=(&x_u/&x_p)(&x_u/&x_p) Where p is a scalar Contravariant tensor(rank 2) A'^uv=(&x'^u/&x^a)(&x'^v/&x^b)A^ab Where A^ab=dx_a...
  42. E

    Covariant derivative of the gradient

    If we define the Gradient of a function: \uparrow u= Gra(f) wich is a vector then what would be the covariant derivative: \nabla _{u}u where the vector u has been defined above...i know the covariant derivative is a vector but i don,t know well how to calculate it...thank you.
  43. J

    Covariant Derivative: A^μₛᵦ Definition & Use

    The covariant derivative is A^\mu_{\sigma} = \frac{\partial A^\mu}{\partial x_{\sigma}} + \Gamma^\mu_{\sigma \alpha}A^\alpha ... why?
  44. J

    Covariant Derivative: Deriving the Equation

    How is the covariant derivative derived?
  45. Y

    What Are Contravariant and Covariant Tensors in Relativity?

    Can anyone explain to me what is contravariant and covariant? I just know that they are tensors with specific transformation properties (from website of MathWorld), i also know that the relation between two is the -ve sign. Then dose it mean that: given a 4-velocity of a particle is the...
  46. S

    Vector Field Commutator Identity in Covariant Derivative

    I am trying to solve an exercise from MTW Gravitation and the following issue has come up: Let D denote uppercase delta (covariant derivative operator) [ _ , _ ] denotes the commutator f is a scalar field, and A and B are vector fields Question: Is it true that [D_A,D_B]f = D_[A,B]f ?
  47. C

    Is Feynman's time-ordering prescription covariant?

    We all know that time-ordering depends on the choice of Lorentz frame. So my question is somewhat obvious... Please give me a hint on where to look up that problem, eg. why the S-matrix theory is covariant. I guess the time-ordering prescription is implicitly defined for each single...
  48. N

    Solving Laplace's Equation with Covariant Derivative

    Hello! I am trying t solution Navier-Stokes equation and I cannot find something about Laplacian. I would like to solution Laplace’a equation for each component.I am trying to transform cylindrical coordinate. I would like to search equation for covariant derivative. For divergence of a...
  49. S

    Rank 2 covariant tensors and dimesionality

    I've already handed in my (I can only assume) incorrect solution, but I just felt like posting, though I'm not sure if anyone will be able to help. I have a rank-2 covariant tensor, T sub i,j. This can be written in the form of t sub i,j + alpha*metric tensor*T super k, sub k (I hope my...
  50. T

    The covariant derivative of a contravariant vector

    Since there are some equations in my question. I write my question in the following attachment. It is about the covariant derivative of a contravariant vector. Thank you so much!
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