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. 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...
  2. 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?
  3. 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...
  4. V

    Covariant and Contravariant Vectors

    Dear friends, while reading about schwarzschild geometry, I learned that E=-p_0 and L=p_{\phi} are constant along a geodesic or are constant of motion. I further read that p^0=g^{00}p_0=m(1-2M/r)^{-1}E and p^{\phi}=g^{\phi\phi}p_{\phi}=m(1/r^2)L, which I can see depends on radius r. This made...
  5. R

    Components of a covariant vector

    Homework Statement Consider the following two basis sets (or triads) in {R}^3: \{\vec{e}_1, \vec{e}_2, \vec{e}_3\} := \{(1, 0, 0), (0,1, 0), (0, 0, 1)\} \{\widehat{\vec{e}_1}, \widehat{\vec{e}_2}, \widehat{\vec{e}_3}\} := \{(1, 0, 0), (1,1, 0), (1, 1, 1)\}. Let a covariant...
  6. 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...
  7. L

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

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  8. 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.
  9. D

    Wrong example of Covariant Vector

    I have a question about covariant and contravariant vectors. I tried making concrete examples and in one example I succeed, in another I fail. It is said that displacement vectors transform contravariantly, and gradients of a scalar transform covariantly. I can get the whole story working in...
  10. 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...
  11. D

    Two covariant derivatives of a vector field

    V_{a;b} = V_{a,b} - \Gamma^d_{ad}V_d Now take the second derivative... V_{a;b;c} = (V_{a;b})_{,c} - \Gamma^f_{ac}V_{f;b} - \Gamma^f_{bc}V_{a;f} But I have no idea how to get the parts with the Christoffel symbols. V_{a;b;c} = (V_{a;b})_{,c} - \Gamma^f_{(a;b)c}V_{af} = (V_{a;b})_{,c} -...
  12. D

    Covariant and contravariant components

    Homework Statement Consider the two-dimensional space given by ds^2 = e^y dx^2 + e^x dy^2 Calculate the covariant and contravariant components of the metric tensor for this spacetime. The Attempt at a Solution Are the covariant components just e^y and e^x with the...
  13. 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].
  14. D

    Is the Boltzmann Equation in GR Truly Coordinate Independent?

    Hi everyone, have a question about covariant, coordinate independent quantities in GR. Reading Kolb and Turner's book The Early Universe one can find the Boltzmann equation in a GR setting. Now, one of the terms in that equation is - \Gamma_{\alpha \beta}^\gamma p^\alpha p^\beta...
  15. 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...
  16. V

    Covariant derivate same as normal derivative?

    hey, I'm getting really confused with something. If i have the covariant derivative of say, constant * exp(2f), is this equivalent to 2*constant*exp(2f) * cov.deriv. of f ?
  17. 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...
  18. 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...
  19. T

    General Relativity - Contravariant and Covariant Vectors: aaargghh

    Homework Statement I know this is an easy question, I just can't seem to grasp what I am actually doing: Let M be a manifold. Let Va be contravariant, and Wa be covariant. Show that \mu=VaWa Homework Equations (couldn't get Latex to work consistently, sorry) (1) V 'a = (dx 'a /...
  20. Phrak

    What Lorentz Covariant Objects Can You Name?

    For starters, there is the covariant vector (E/c, p). Dividing by the scalar invariant, h_bar/2∏, where k is the propagation vector, there is (ω/c, k). There must be a significant number of covariant objects in electromagnetism...
  21. R

    Difference between covariant and contravariant levi-civita tensor?

    The title says it all, basically I'm trying to figure out what the difference is between the two tensors (levi-civita) that are 3rd rank. Do they expand out in matrix form differently?
  22. 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...
  23. M

    Representation of covariant and contravariant vectors on spacetime diagrams

    Hi, How can we represent covariant and contravariant vectors on curved spacetime diagrams? How can we draw these vectors on a spacetime diagram? Contravariant vectors are really vectors, therefore we can represent them on the diagram with directed line elements. Covariant vectors are...
  24. Rasalhague

    Covariant and exterior derivatives

    Reading Roger Penrose's The Road to Reality, I wondered what is the relationship/difference between these? Can one be expressed simply in terms of the other? The exterior derivative seems to be only defined for form fields. He says the covariant derivative of a scalar field (0-form field) is the...
  25. A

    Covariant and Contravariant Tensors

    we have studied in Tensor's analysis that there are two kinds of tensors that usually used in transformation. one is Contravariant & covariant. what is the difference between them and and why they are same for Rectangular coordinates?
  26. D

    Dis-ambiguate: derivative of a vector field Y on a curve is the covariant of Y

    Homework Statement This two-part problem is from O'Neill's Elementary Differential Geometry, section 2.5. Let W be a vector field defined on a region containing a regular curve a(t). Then W(a(t)) is a vector field on a(t) called the restriction of W to a(t). 1. Prove that Cov W w.r.t...
  27. P

    Covariant & contravariant tensors

    Homework Statement I have a few introductory problems dealing with proofs of tensor properties, and one about a transformation from rectangular to spherical coordinates. If someone has the time and inclination to help out this week, I can email you the specific problem set. (I'd prefer not...
  28. 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?
  29. R

    Is space covariant or contravariant?

    I often meet the question whether the (physical) space is 'covariant' or 'contravariant'. I once replied to that question with: Space is space. The COMPONENTS of a tensor are covariant/contravariant if the basis is CHOSEN TO BE contravariant/covariant. As far as I know tensors the...
  30. C

    Contravariant components to covariant

    Hi, everyone I was playing with the coordinate transformations and metric tensors to get a feeling of how it all behaves, and got stuck with some basic problem I am hoping you can help me with. So, I have defined a coordinate system (s,t), with the s axis going along the x-axis in the...
  31. E

    Electromagnetic with Covariant method

    Hi Now I'm studying Electromagnetic we started to study the EM with the Covariant method (contravariant method too) we use cgs units almost of time so I ask if someone can give a hand to with recomendation to where study the notation and the mathematical approach of EM by covariant method (EM...
  32. R

    Covariant formulation of Coulomb's Law

    I was noodling about with 4-vectors and Maxwell's Equations when I took it upon myself to figure out how to write the field of a point charge in terms of 4-vectors, since I'd never seen it before (if anyone knows a book that has this I'd be happy to see it). It's actually relatively simple...
  33. E

    Covariant Maxwell Equations in Materials

    Hi everybody, I have this simple question. ¿Where can I find the covariant maxwell equations in materials?. I've already one and proved they correctly represent the non-homogene maxwell equations, is this one \partial_{\nu}F^{\nu\mu}+\Pi^{\mu\nu}A_{\nu}=J^{\mu}_{libre} with the tensor...
  34. K

    Covariant differentiation commutes with contraction?

    Homework Statement I've been reading a textbook on tensor analysis for a while. The book uses the conclusion of "covariant differentiation commutes with contraction" directly and I searched around and found most people just use the conclusion without proof.Homework Equations For example...
  35. pellman

    EM vector potential - covariant or contravariant?

    Are potentials appearing in the Maxwell equations the components of a contravariant vector or a covariant vector? Let us be specific. metric is (+,-,-,-) . Let us write the potentials which appear in the Maxwell equations as \Phi and \vec{A}=(A_x,A_y,A_z) Is it then the case that...
  36. 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...
  37. V

    Does the Riemman tensor and the covariant derivate commute?

    A random question - Does the Riemman tensor and the covariant derivate commute? a yes/no answer would suffice, but any explanation would be welcome:) From the equations, it looks as though they do for flat metrics - but if we have other manifolds, it seems to me that the Christoffel...
  38. 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...
  39. I

    Proving Identity for Non-Zero Symmetric Covariant Tensors

    Homework Statement For ease of writing, a covariant tensor \bf G.. will be written as \bf G and a,b,c,d are vectors. Let \bf S and \bf G be two non-zero symmetric covariant tensors in a four-dimensional vector space. Furthermore, let S and G satisfy the identity: [\bf G \otimes \bf...
  40. X

    Uncertainty relations aren't Lorentz covariant

    Heisenberg's uncertainty relations are not covariant under Lorentz transformation. That means they don't have the same form in any inertial frame. So, how to modify these relations to leave them invariant in form under a Lorentz transformation ?
  41. 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...
  42. 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...
  43. T

    Covariant derivatives in Wolfram Math

    In the Wolfram Mathworld section on spherical coordinates there's given a list of nine covariant derivatives. The derivatives are given with respect to radius, azmuth, and zenith using the usual symbols r, theta and phi. The question is: what would be examples of the vectors whose derivatives...
  44. M

    Proving Tangent Vector Field X on \Re^{3} to a Cylinder in \Re^{3}

    How do I show that a Vector field X on \Re^{3} is tangent to a Cylinder in \Re^{3}?
  45. maverick280857

    Covariant divergence question from Landau and Lifshitz

    Hi everyone, I'm trying to work through section 86 of Landau and Lifgarbagez volume 2 (The Classical Theory of Fields). Basically, I am unable to get equation (86.6) from equations (86.4) and (86.5). I've detailed my working/question in the attached jpg file. I would appreciate any inputs...
  46. 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...
  47. T

    Covariant vs contravariant indices

    I'm having some trouble breaking into tensors. What is specifically bewildering me is contravariant vs. covariant indices. Could someone please explain this to me (or link me)? I barely understand what each one means in its own right, let alone the differences between the two.
  48. R

    Proving Epsilon and Covariant Derivatives with Christoffel Symbols

    Homework Statement 1) Show that \epsilon_{ijk,m}=0 and (\sqrt{g})_{,k}=0 . Where ' ,k ' , stands for covariant derivative and \epsilon is the epsilon permutation symbol. 2) where the {} is for christoffel symbol of the second kind. Homework Equations The Attempt at a...
  49. J

    Covariant Green's function for wave equation

    This comes from Jackson's Classical Electrodynamics 3rd edition, page 613. He finds the Green's function for the covariant form of the wave equation as: D(z) = -1/(2\pi)^{4}\int d^{4}k\: \frac{e^{-ik\cdot z}}{k\cdot k} Where z = x - x' the 4 vector difference, k\cdot z = k_0z_0 -...
  50. 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...
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