What is Covariant derivative: Definition and 171 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.
Can you give me the definition of exterior covariant derivative or any reference web page ?
Wiki does not involve enough info.I am not able to do calculation with respect to given definition there.
Thanks in advance
Hi. I am attempting to gain some intuition for what the covariant derivative of a tensor field is.
I have a good intuition about the covariant derivative of vector fields (measuring how the vector changes as you move in a particular direction), and I understand how to extend the covariant...
When we derive equation of motion by variation of the action, we use rules of ordinary differentiation and integration. So only ordinary derivatives can appear in the equation. Now in general relativity we are supposed to replace all those ordinary derivatives by covariant derivatives. Is that...
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?
If we work in cartesian coordinates, we say for instance, that
D_x \phi = \left( \frac{\partial}{\partial x} + i g \sum_a T_a A^a_x \right) \phi
where g is the gauge coupling, and \{T^a\} are the generators of the gauge group, and \{A^a_\mu\} is the gauge vector field.
But what happens when...
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...
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...
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.
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!