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. B

    Why is grad(f) a covariant vector

    Take R2. Take a function f(x,y) defined on R2 which maps every point to a real number. The gradient of this at any point mean a vector which points in the direction of steepest incline. The magnitude of the vector is the value of the derivative of the function in that direction. Both of these...
  2. S

    Question about covariant derivatives

    Why is it that the covariant derivative of a covariant tensor does not seem to follow the product rule like contravariant tensors do when taking the covariant derivatives of those? Here is a visual of what I mean: This is the covariant derivative of a contravariant vector. As you can see, it...
  3. C

    Virtues of principal/associated bundle formulation of covariant deriv.

    For someone who does not already know Lie group and bundle theory, the formulation of covariant derivatives through parallel transport in the principal, and associated vector bundles, might seem unnecessarily complicated. In that light, I wondered what the virtues of the principal/associated...
  4. Q

    Covariant vs Canonical Formalism

    Why is a covariant formalism preferred over a canonical formalism in loop quantum gravity, in simple layman terms
  5. N

    Derivatives of contravariant and covariant vectors

    Can someone explain why the derivative with respect to a contravariant coordinate transforms as a covariant 4-vector and the derivative with respect to a covariant coordinate transforms as a contravariant 4-vector.
  6. A

    QFT (derivative the covariant and contravariant fields)

    Hi, please help me .. How can I derivative covariant and contravariant fields? as in the attached picture Thanks.. http://www.gulfup.com/?tNXcaN w.r.t alpha
  7. S

    Contravariant and covariant indices

    When we write contravariant and covariant indices, for example for the Lorentz transformation, does it matter if we write \Lambda^\mu\,_\nu or \Lambda^\mu_\nu ? i.e. if the \nu index is to the right of the \mu or they are at the same place with respect to left-right?
  8. WannabeNewton

    Centrifugal and Coriolis forces covariant formulation

    Hi guys. I was reading a paper in which a calculation was done to show that in Schwarzschild space-time, if we consider a time-like circular orbit 4-velocity ##u^{\mu} = \gamma(\xi^{\mu} + \omega \eta^{\mu})## where ##\xi^{\mu}## is the time-like Killing field, ##\eta^{\mu}## is the axial...
  9. S

    MHB Intrinsic Derivative and Covariant Derivative

    In the context of tensor calculus, what is the difference between intrinsic derivative and covariant derivative?
  10. K

    Contravariant and Covariant Vectors

    I remember I have read somewhere that contravariant/covariant vectors correspond to polar/axial vectors in physics, respectively. Examples for polar/axial vectors are position, velocity,... and angular momentum, torque,..., respectively. Is this right? Can I prove that, say, any axial...
  11. J

    For what covariant derivative?

    I will take the differential form of position vector r: ##\vec{r}=r\hat{r}## ##d\vec{r}=dr\hat{r}+rd\hat{r}## So, now I need find ##d\hat{r}## ##d\hat{r}=\frac{d\hat{r}}{dr}dr+\frac{d\hat{r}}{d\theta}d\theta## ##\frac{d\hat{r}}{dr}=\Gamma ^{r}_{rr}\hat{r}+\Gamma...
  12. J

    Covariant and Contravariant Coordinate

    Hellow everybody! A simples question: is it correct the graphic representation for covariant (x₀, y₀) and contravariant (x⁰, y⁰) coordinates of black vector?
  13. Q

    Is Gravitomagnetism Covariant and Are There Any Alternative Approaches?

    Gravitomagnetism is this thing https://en.wikipedia.org/wiki/Gravitomagnetism Like electromagnetism equations, but adapted for gravity The equations just like that are manifestly problematic. If we use four-momentum as the source( appropriate units), does the equation become covariant...
  14. C

    Relation between covariant differential and covariant derivative

    In Theodore Frankel's book, "The Geometry of Physics", he observes at page 248 that the covariant derivative of a vector field can be written as $$\nabla_X v = e_iX^j (v^i_{,j} + \omega^i_{jk} v^k)= e_i(dv^i(X) + \omega^i_k(X) v^k) = e_i (dv^i + \omega^i_k v^k)(X)$$ where ##\omega^i_k =...
  15. C

    Jacobi identity for covariant derivatives proof.

    Suppose we have a torsion free connection. Does anyone here know of a slick way to prove that covariant derivatives satisfy the Jacobi identity? I.e. that $$([\nabla_X,[\nabla_Y,\nabla_Z]] + [\nabla_Z,[\nabla_X,\nabla_Y]] +[\nabla_Y,[\nabla_Z,\nabla_X]])V = 0$$ without going into...
  16. C

    Covariant derivative of a commutator (deriving Bianchi identity)

    Hi. I'm trying to understand a derivation of the Bianchi idenity which starts from the torsion tensor in a torsion free space; $$ 0 = T(X,Y) = \nabla_X Y - \nabla_Y X - [X,Y]$$ according to the author, covariant differentiation of this identity with respect to a vector Z yields $$$ 0 =...
  17. E

    Covariant and contravariant

    If the notions of covariant and contravariant tensors were not introduced,what would happen?E.g. what form will the Einstein E.q. Guv=8πTuv be changed into ?
  18. marcus

    Systematic covariant QG, Engle style

    This thread is in part belated appreciation of J.S.Engle's seemingly orderly systematic way of doing research, and good writing style. I'm not kidding: a clear widely understandable writing style helps a lot. If you want a well-written review (from basics to latest) of spinfoams he has one here...
  19. O

    Commutator between covariant derivative, field strength

    Hello, i try to prove that ∂μFμ\nu + ig[Aμ, Fμ\nu] = [Dμ,Fμ\nu] with the Dμ = ∂μ + igAμ but i have a problem with the term Fμ\nu∂μ ... i try to demonstrate that is nil, but i don't know if it's right... Fμ\nu∂μ \Psi = \int (∂\nuFμ\nu) (∂μ\Psi) + \int Fμ\nu∂μ∂\nu \Psi = (∂\nuFμ\nu) [\Psi ]∞∞...
  20. Philosophaie

    Finding the 4x4 Cofactor of a Covariant Metric Tensor g_{ik}

    If I have a 4x4 Covarient Metric Tensor g_{ik}. I can find the determinant: G = det(g_{ik}) How do I find the 4x4 Cofactor of g_ik? G^{ik} then g^{ik}=G^{ik}/G
  21. I

    Physical intrepretation of contra-variant and covariant vectors?

    Hey all, I starting to study QED along with a slew with other materials. (I read in the QED book and when I don't understand a reference I go to Jackson's E&M and work some problems out, it has been beneficial thus far!) Most of the topics are not too far fetched but I am struggling to...
  22. F

    Covariant differentiation on 2-sphere

    Consider a 2-sphere S2 with coordinates xμ=(θ,\phi) and metric ds2=dθ2+sin2θ d\phi2 and a vector \vec{V} with components Vμ=(0,1). Calculate the following quantities. ∇θ∇\phiVθ ∇\phi∇θVθ
  23. P

    Covariant Versus Contravariant Vectors

    I'm confused about the difference between a contravariant and covariant vector. Some books and articles seem to say that there really is no difference, that a vector is a vector, and can be written in terms of contravariant components associated with a particular basis, or can be written in...
  24. L

    How do I derive the Covariant Derivative for Covectors? (Lower index)

    Hello everyone! I'm trying to learn the derivation the covariant derivative for a covector, but I can't seem to find it. I am trying to derive this: \nabla_{α} V_{μ} = \partial_{α} V_{μ} - \Gamma^{β}_{αμ} V_{β} If this is a definition, I want to know why it works with the definition...
  25. atyy

    Minimal Coupling Needed for Covariant Energy Conservation?

    Thanks. I started a new thread, because I've seen what seemed to me a contradictory claim in Carroll's GR notes (Eq 5.38) - he says diff invariance is enough to get covariant energy conservation. I've never understood whether Carroll's claims and the ones in these papers are really...
  26. L

    Covariant and Contravariant Components of a Vector

    I think this may be a simple yes or no question. I am currently reading a book Vector and Tensor Analysis by Borisenko. In it he introduces a reciprocal basis \vec{e_{i}} (where i=1,2,3) for a basis \vec{e^{i}} (where i is an index, not an exponent) that may or may not be orthogonal...
  27. bcrowell

    Newtonian force as a covariant or contravariant quantity

    I recently came across a very cool book called Div, Grad, and Curl are Dead by Burke. This is apparently a bit of a cult classic among mathematicians, not to be confused with Div, Grad, Curl, and All That. Burke was killed in a car accident before he could put the book in final, publishable...
  28. J

    Graphing Covariant Spherical Coordinates

    I am studying Riemannian Geometry and General Relativity and feel like I don't have enough practice with covariant vectors. I can convert vector components and basis vectors between contravariant and covariant but I can't do anything else with them in the covariant form. I thought converting the...
  29. M

    Prove Maxwell Eqs. Covariant: Wave Eqn & 4th-Vector Pot.

    Is it enough to see the covariance of the wave equation the fourth-vector potential (\phi, \bar{A}) satisfy? I mean, is this enough to prove the covariance of Maxwell equations? The equation would be ∂_{\mu}∂^{\mu}A^{\nu} =\frac{4\pi}{c} J^{\nu}
  30. Z

    Lie derivative of covariant vector

    Homework Statement Derive L_v(u_a)=v^b \partial_b u_a + u_b \partial_a v^b Homework Equations L_v(w^a)=v^b \partial_b w^a - w^b \partial_b v^a L_v(f)=v^a \partial_a f where f is a scalar. The Attempt at a Solution In the end I get stuck with something like this, L_v(u_a)w^a=v^b...
  31. T

    General relativity - Covariant Derivative Of F(R)

    In f(R) gravity as http://en.wikipedia.org/wiki/F%28R%29_gravity , i have problem with the term [ g_ab □ - ∇_a ∇_b ] F(R) , well actually is [ ∇_b ∇_a - ∇_a ∇_b ] F(R) , but F is a function of Ricci Factor and Ricci Factor is expressed as a(t) ( scale factor ) . for the a = b = 0 i say this...
  32. Z

    Covariant derivative - is this a typo ?

    hello, please see the attached snapshot (taken from 'Problem book in relativity and gravitation'). In the last equation I think there would be no semicolon. Here is why I believe (S is scalar by the way): S;α[βγ] = 1/2 * ( S;αβγ - S;αγβ ) Now from the equation which precedes it, we have ...
  33. S

    Covariant and contravariant vector

    Will anyone help me to under stand the covariant and contravariant vector ? And can anyone show me the derivation of dr=(dr/dx)dx + (dr/dy)dy + (dr/dz)dz
  34. S

    Covariant derivative of a 1-form

    Let $$f:U \to \mathbb{R}^3$$ be a surface with local coordinates $$f_i=\frac{\partial f}{\partial u^i}$$. Let $\omega$ be a one-form. I want to express $$\nabla \omega$$ in terms of local coordinates and Christoffel symboles. Where $$\nabla$$ is the Levi-Civita connection (thus it coincides with...
  35. P

    Bitensor covariant derivative commutation

    Hi everyone, I'm trying to prove a relation in which I need do commute covariant derivatives of a bitensor. The equation is quite long but I need to write something like this: Given a bitensor G^{\alpha}_{\beta'}(x,x'), where the unprimed indexes (\alpha,\beta, etc) are assigned to the...
  36. P

    Covariant Derivative Commutation

    Hello, Can anyone tell me the general formula for commuting covariant derivatives, I mean, given a (r,s)-tensor field what is the formula to commute covariant derivatives? I found a formula http://pt.scribd.com/doc/25834757/21/Commuting-covariant-derivatives page 25, Eq.6.18 but it doesn't...
  37. G

    Finding znew for Covariant Conservation

    suppose we have unit vector z=(0,0,0,1), we can use it to form a tensor z^\mu z^\nu , it is easy to check that ∂_\μ( z^\mu z^\nu=0 in Minkowski spacetime, now I want to generalize this equation to general curved spacetime, so that ∇μ ( znew^\mu znew^\nu)=0. But I am not sure how to...
  38. T

    Covariant and Contravariant components in Oblique System

    Homework Statement In the oblique coordinate system K' defined in class the position vector r′ can be written as: r'=a\hat{e'}_{1}+b\hat{e'}_{2} Are a and b the covariant (perpendicular) or contravariant (parallel) components of r′? Why? Give an explanation based on vectors’ properties...
  39. T

    Requesting Clear Description of Contravariant vs Covariant vectors

    Ok, so here's my problem. I just graduated with a mathematics degree and am going full force into a physics graduate program. I'm taking a course called mathematical methods for physicists, in which the first subject is tensors. Everyone else seems to be comfortable with the material, but me...
  40. T

    Covariant Derivative derivation.

    Homework Statement Using the Leibniz rule and: \nabla_{c}X^{a}=\partial_{c}X^a+\Gamma_{bc}^{a}X^b \nabla_{a}\Phi=\partial\Phi Show that \nabla_c X_a = \partial_c X_a - \Gamma^{b}_{ac}X_{b} . The question is from Ray's Introducing Einsteins relativity, My attempt...
  41. shahbaznihal

    Covariant commutation relation in Mandl and Shaw

    Hi, I am trying to study Quantum Field Theory by myself from Mandl and Shaw second edition and I am having trouble understanding the section on covariant commutation relations. I understand the idea that field at equal times at two different points commute because they cannot "communicate"...
  42. A

    MHB Covariant and Contravariant Vector

    I have been given the following problem: The covariant vector field is: \(v_{i}\) = \begin{matrix} x+y\\ x-y\end{matrix}What are the components for this vector field at (4,1)? \(v_{i}\) = \begin{matrix} 5\\ 3\end{matrix} Now I can use this information to solve the...
  43. A

    Gravitomagnetism: Why isn't mass-current covariant?

    "Gravitomagnetism is a widely used term referring specifically to the kinetic effects of gravity, in analogy to the magnetic effects of moving electric charge". http://en.wikipedia.org/wiki/Gravitoelectromagnetism Essentially the idea is to take Maxwell's equations and make a substitution...
  44. H

    Covariant Notation: Understanding Its Basics and Applications

    I posted this question in homeworks section but no one can help, or seems unable to answer my question. I am hoping if I post it here, someone might be able to clear it up for me: https://www.physicsforums.com/showthread.php?t=621238 Thanks in advance.
  45. C

    Covariant versus Contravariant

    Hi everyone, I am having a little trouble with the difference between a covariant vector and contravariant vector. The examples that I come across say that an example of a contravariant vector is velocity and that a contravariant must contra-vary with a change of basis to compensate. So...
  46. G

    Weird version of covariant derivative on wikipedia

    http://en.wikipedia.org/wiki/Four-force At the bottom of that page, the author provides the generalization of four force in general relativity, where the partial derivative is replaced with the covariant derivative. However if you notice on the second term in the third equality, there is a...
  47. G

    Covariant Derivative and metric tensor

    Hi all, I am wondering if it is possible to derive the definition of a Christoffel symbols using the Covariant Derivative of the Metric Tensor. If yes, can I get a step-by-step solution? Thanks! Joe W.
  48. G

    Contravariant and covariant vectors transformations

    Hi all, I am new to General Relativity and I started with General Relativity Course on Youtube posted by Stanford (Leonard Susskind's lectures on GR). So first thing to understand is transformation of covariant and contravariant vectors. Before I can understand a transformation, I would...
  49. P

    Covariant derivative of a vector field

    Homework Statement Show that \nabla_a(\sqrt{-det\;h}S^a)=\partial_a(\sqrt{-det\;h}S^a) where h is the metric and S^a a vector. Homework Equations \nabla_a V^b = \partial_a V^b+\Gamma^b_{ac}V^c \Gamma^a_{ab} = \frac{1}{2det\;h}\partial_b\sqrt{det\;h} \nabla_a\sqrt{-det\;h} (is that...
  50. N

    Covariant derivative along a horizontal lift in an associated vector bundle

    I am trying to familiarize myself with the use of fibre bundles and associated bundles but am having some problems actually making calculations. I would like to show that the covariant derivative along the horizontal lift of a curve in the base space vanishes (which should be just a matter of...
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