Covariant derivative Definition and 167 Threads

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...

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...

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

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...

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?

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...

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

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

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...

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

Why we have to definte covariant derivative?

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.

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

How is the covariant derivative derived?

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 ?

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...

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!