Covariant derivative Definition and 174 Threads

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...

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...

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...

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...

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...

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.

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...

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...

Hi all, I'm trying to figure out the link between the connection coefficients (Christoffel symbols), the propagator, and the coordinate description of the covariant derivative with the connection coefficients. As in...

Homework Statement My teacher solved this in class but I'm not understanding some parts of tis solution. Show that \nabla_i V^i is scalar. Homework Equations \nabla_i V^i = \frac{\partial V^{i}}{\partial q^{i}} + \Gamma^{i}_{ik} V^{k} The Attempt at a Solution To start this...

Homework Statement In Wald's text on General Relativity he makes an assertion that I'm not sure why it is allowed mathematically. Here's the basic setup: Let \omega_{b} be a dual vector, \nabla_{b} and \tilde{\nabla}_{b} be two covariant derivatives and f\in\mathscr{F}. Then we may let...

Hi there! I saw this exercise that we have to calculate the covariant derivative of a vector field (in polar coordinates). Most of them equals zero, but two of them are non-zero, sugesting that this vector field is not constant. What i want to understand is the physical meaning of this values...

I am trying to calculate the covariant derivative of the Ricci Tensor the way Einstein did it, but I keep coming up with \nabla_{μ}R_{αβ}=\frac{∂}{∂x^{μ}}R_{αβ}-2\Gamma^{α}_{μ\gamma}R_{αβ} or \nabla_{μ}R_{αβ}=\frac{∂}{∂x^{μ}}R_{αβ}-\Gamma^{α}_{μ\gamma}R_{αβ}-\Gamma^{β}_{μ\gamma}R_{αβ}...

I am trying to calculate the covariant derivative of the Ricci Tensor the way Einstein did it, but I keep coming up with \nabla_{μ}R_{αβ}=\frac{∂}{∂x^{μ}}R_{αβ}-2\Gamma^{α}_{μ\gamma}R_{αβ} or \nabla_{μ}R_{αβ}=\frac{∂}{∂x^{μ}}R_{αβ}-\Gamma^{α}_{μ\gamma}R_{αβ}-\Gamma^{β}_{μ\gamma}R_{αβ}...

The connection \nabla is defined in terms of its action on tensor fields. For example, acting on a vector field Y with respect to another vector field X we get \nabla_X Y = X^\mu ({Y^\alpha}_{,\mu} + Y^\nu {\Gamma^\alpha}_{\mu\nu})e_\alpha = X^\mu {Y^\alpha}_{;\mu}e_\alpha and we call...

Homework Statement Using the definition of divergence d(i_{X}dV) = (div X)dV where X:M\rightarrow TM is a vector field, dV is a volume element and i_X is a contraction operator e.g. i_{X}T = X^{k}T^{i_{1}...i_{r}}_{kj_{2}...j_{s}}, prove that if we use Levi-Civita connection then the...

My book defines the covariant derivative of a tangent vector field as the directional derivative of each component, and then we subtract out the normal component to the surface. I am a little confused about proving some properties. One of them states: If x(u, v) is an orthogonal patch, x_u...

Hi, I am familiar with the covariant derivative of the tangent vector to a path, \nabla_{\alpha}u^{\beta} and some interesting ways to use it. I am wondering about \nabla_{\alpha}x^{\beta}=\frac{\partial x^\beta}{\partial...

Hi there, I was doing some calculations with tensors and ran into a result which seems a bit odd to me. I hope someone can validate this or tell me where my mistake is. So I have a normal orthonormal frame field \{E_i\} in the neighbourhood of a point p in a Riemannian manifold (M,g), i.e...

On a 2 dimensional Riemannian manifold how does one derive the covariant derivative from the connection 1 form on the tangent unit circle bundle?

The covariant derivative is different in form for different tensors, depending on their rank. What about other mathematical entities? The electromagnetic field A is a vector, but it has complex values. Is the covariant derivative different for complex valued vectors? And what about...

hi, I understand that Tab,b=0 because the change in density equals the negative divergence, but why do the christoffel symbols vanish for Tab;b=0?

what would Rabcd;e look like in terms of it's christoffels? or Rab;c

How can the derivative of a basis vector at a point be the linear combination of tangent vectors at that point? For example, if you take a sphere, then the derivative of the polar basis vector with respect to the polar coordinate is in the radial direction. How can something in the radial...

As you may guess from the title this question is about the covariant derivatives, more precisely about the difference between the usual covariant derivative, the one used in General Relativity defined by:\nabla_{e_{\mu}}=\left(\frac{\partial v^{\beta}} {\partial...

As you may guess from the title this question is about the covariant derivatives, more precisely about the difference between the usual covariant derivative, the one used in General Relativity defined by:\nabla_{e_{\mu}}=\left(\frac{\partial v^{\beta}} {\partial...

My apologies about lack of precision in nomenclature. So I wanted to know how to express a certain idea about choice of basis on a manifold... Let's suppose I am solving a reaction-diffusion equation with finite elements. If I consider a surface that is constrained to lie in a flat plane or...

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...

What exactly is the physical meaning of the fact that the covariant derivative of the metric tensor vanishes?

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...

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...

Edit: Solved Don't know how to delete thread though!

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.

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...

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].

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...

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...

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...

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...

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?

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...

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...

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...

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...

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...

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...

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}=...

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...

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...

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} +...