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. 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...
  2. 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...
  3. 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...
  4. 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 +...
  5. 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...
  6. 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...
  7. S

    I Why are contravariant and covariant vectors important in general relativity?

    1) I read different texts on Contravariant , Covariant vectors. 2) Contravariant - they say is like vector . Covariant is like gradient From what I see they have those vector spaces because it eventually helps get scalar out of it if we multiply contravariant by covariant Also Contravariant...
  8. 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...
  9. C

    Calculating Covariant Riemann Tensor with Diag Metric gab

    Using Ray D'Inverno's Introducing Einstein's Relativity. Ex 6.31 Pg 90. I am trying to calculate the purely covariant Riemann Tensor, Rabcd, for the metric gab=diag(ev,-eλ,-r2,-r2sin2θ) where v=v(t,r) and λ=λ(t,r). I have calculated the Christoffel Symbols and I am now attempting the...
  10. D

    What is the trace of a second rank covariant tensor?

    What is the trace of a second rank tensor covariant in both indices? For a tensor covariant in one index and contravariant in another ##T^i_j##, the trace is ##T^k_k## but what is the trace for ##T_{ij}## because ##T_{kk}## is not even a tensor?
  11. M

    Covariant and contravariant basis vectors /Euclidean space

    I want ask another basic question related to this paper - http://www.tandfonline.com/doi/pdf/10.1080/16742834.2011.11446922 If I have basis vectors for a curvilinear coordinate system(Euclidean space) that are completely orthogonal to each other(basis vectors will change from point to point)...
  12. E

    Covariant Characterization of Causality in Continuum: T^{ik}v_k

    Hi! Let ##T^{ik}## be the stress-energy-tensor, and ##v_k## some future-pointing, time-like four vector. How can I see that the object ##T^{ik}v_k## is future-pointing and not space-like? Thank you for your help!
  13. S

    Contravariant and covariant vectors

    I know if the number of coordinates are same in both the old and new frame then A.B=A`.B` . But if the number of coordinates are not same in both old and new frame then A.B=0 means that both the vectors A and B are perpendicular. Why is it so that if the number of coordinates of both the frames...
  14. joneall

    Units if conversion between covariant/contravariant tensors

    I am still at the stage of trying to assimilate contravariant and covariant tensors, so my question probably has a simpler answer than I realize. A covariant tensor is like a gradient, as its units increase when the coordinate units do. A contravariant tensor's components decrease when the...
  15. 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...
  16. 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...
  17. G

    Covariant fields and new physics

    I assume that all the fundamental physics known - as of today - can be reduced to quantum general covariant fields (including spacetime itself to be seen as a field of those...). Now, sorry if my question is quite abstract and based on tomorrow's hypothetical new physics, but would it be against...
  18. Urs Schreiber

    Insights Higher Prequantum Geometry IV: The Covariant Phase Space - Transgressively - Comments

    Urs Schreiber submitted a new PF Insights post Higher Prequantum Geometry IV: The Covariant Phase Space - Transgressively Continue reading the Original PF Insights Post.
  19. P

    Double covariant derivative of function of scalar

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

    Covariant & Contravariant Components

    Homework Statement This is really 3 questions in one but I figure it can be grouped together: 1. The vector A = i xy + j (2y-z2) + k xz. is in rectangular coordinates (bold i,j,k denote unit vectors). Transform the vector to spherical coordinates in the unit vector basis. 2. Transform the...
  21. 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...
  22. I

    Commutator of two covariant derivatives

    Hello all, I'm trying to calculate a commutator of two covariant derivatives, as it was done in Caroll, on page 122. The problem is, I don't get the terms he does :-/ If ##\nabla_{\mu}, \nabla_{\nu}## denote two covariant derivatives and ##V^{\rho}## is a vector field, i need to compute...
  23. putongren

    Change of Basis, Covariant Vectors, and Contravariant Vector

    I'm having trouble understanding those concepts in the title. Can someone explain those concepts in an easy to understand manner? Please don't refer me to a wikipedia page. I know some linear algebra and multi-variable calculus. Thank you.
  24. 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?
  25. G

    A question about covariant representation of a vector

    Homework Statement Hi I am reviewing the following document on tensor: https://www.grc.nasa.gov/www/k-12/Numbers/Math/documents/Tensors_TM2002211716.pdf Homework Equations In the middle of page 27, the author says: Now, using the covariant representation, the expression $$\vec V=\vec V^*$$...
  26. C

    Covariant derivatives commutator - field strength tensor

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  27. S

    Covariant form of Newton's law of gravity

    Hii! Newton's law of gravity is ∇.(∇Φ) = 4πGρ. A book on GR gives a suggestion to make it Lorentz covariant by using de' Alembertian operator on 'Φ' in the LHS of above equation instead of Laplacian. Then it explains that this won't work because we have to include in 'ρ' all the energy...
  28. N

    Tensors with both covariant and contravariant components

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  29. Telemachus

    Differentiation with respect to covariant component of a vector

    I want to prove that differentiation with respec to covariant component gives a contravariant vector operator. I'm following Jackson's Classical Electrodynamics. In the first place he shows that differentiation with respecto to a contravariant component of the coordinate vector transforms as the...
  30. D

    Help! EM Exam: Covariant LW Fields Derivation Questions

    Hi I am doing some exam revision for an EM class and I'm trying to understand a few things about this derivation. Specifically in equation 18.23 why do we not consider the derivative of the four velocity i.e ##\partial^{\alpha}U^\beta## Then going from 18.23 to 18.25 why is...
  31. 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...
  32. T

    Covariant and partial derivative of metric determinant

    Homework Statement is this statement is true : ##\nabla_\mu \nabla_\nu \sqrt{g} \phi = \partial_\mu \sqrt{g} \partial_\nu \phi## Homework EquationsThe Attempt at a Solution well we know ##\nabla_\mu \sqrt{g} =0## so it moves back : ## \nabla_\mu \sqrt{g} \nabla_\nu \phi =\sqrt{g} \nabla_\mu...
  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. N

    Covariant equation intuition confusing me

    Hey all, I've just started tensor analysis but do not understand why in contravarient uses 1 and covarient uses 2, could someone please explain these? Perhaps my understanding of the definitions is causing me to misunderstand why its written like this. Any help appreciated.
  37. binbagsss

    Meaning of tensor invariant, covariant differentiation

    E.g - considering co variant differentiation, The issue with the normal differentiation is it varies with coordinate system change. Covariant differentiation fixes this as it is in tensor form and so is invariant under coordinate transformations.'If a tensor is zero in one coordinate system...
  38. L

    Difference Between Covariant & Contravariant Vectors Explained

    Can someone explain to me what is the difference between covariant and contravariant vectors ? Thank You
  39. T

    Why Do Covariant and Contravariant Vectors Use Opposite Indexing?

    I am reading a notes about tensor when I came across this which the notes did not elaborate more on it. As a result I don't quite understand why. Here it is : " Note that we mark the covariant basis vectors with an upper index and the contravariant basis vectors with a lower index. This may...
  40. putongren

    Covariant and Contravariant Tensors

    Not sure where to post this thread. That being said, can someone explain to me simply what covariant and contravariant tensors are and how covariant and contravariant transformation works? My understanding of it from googling these two mathematical concepts is that when you change the basis of...
  41. B

    Lorentz Transform on Covariant Vector (Lahiri QFT 1.5)

    Homework Statement Given that ##x_\mu x^\mu = y_\mu y^\mu## under a Lorentz transform (##x^\mu \rightarrow y^\mu##, ##x_\mu \rightarrow y_\mu##), and that ##x^\mu \rightarrow y^\mu = \Lambda^\mu{}_\nu x^\nu##, show that ##x_\mu \rightarrow y_\mu = \Lambda_\mu{}^\nu x_\nu##. Homework Equations...
  42. N

    GR vs SR: Reconciling Contravariant & Covariant Vector Components

    I am trying to reconcile the definition of contravariant and covariant components of a vector between Special Relativity and General Relativity. In GR I understand the difference is defined by the way that the vector components transform under a change in coordinate systems. In SR it seems...
  43. 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...
  44. arpon

    How to define covariant basis in curved space 'intrinsicly'?

    In Euclidean space, we may define covariant basis by the partial derivative of position vector with respect to each coordinates, i.e. ##∂R/(∂z^i )=z_i## But in curved space (such as, the two dimensional space on a sphere) how can we define covariant basis 'intrinsicly'?(as we have no position...
  45. 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...
  46. Dale

    Covariant macroscopic electromagnetism

    I wondered if anyone had a good online reference on the covariant formulation of Maxwell's macroscopic equations and the other equations of classical electromagnetism? The wikipedia article talks about constituitive equations in vacuum, which doesn't make a lot of sense to me since M and P...
  47. 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...
  48. 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...
  49. Mr-R

    Covariant differentiation twice

    Doing some problems in D'INVERNO GR textbook and I am stuck on taking the covariant derivation of a tensor twice. Please see the attached picture and please do inform me if something is not clear :smile:
  50. F

    Covariant derivate problem (christoffel symbols)

    Homework Statement I need to calculate \square A_\mu + R_{\mu \nu} A^\nu if \square = \nabla_\alpha \nabla^\alpha , and is the covariant derivate SEE THIS PDF arXiv:0807.2528v1 i want to get the equation (5) from (3) Homework Equations A^{i}_{{;}{\alpha}} =...
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