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.
I'm starting to learn differential geometetry on my own, but I'm having a little trouble figuring out the difference between covariant and contravariant vector fields. It seems that contravariant fields are just the normal vector fields they introduced in multivariable calculus, but if so, I...
title says it all. I've heard these two phrases.
Lorentz invariant: Equation (Lagrangian, or ...?) takes same form under lorentz transforms.
Lorentz covariant: Equation is in covariant form.
I'm don't think I know what I mean when I say the latter. Can someone elucidate the...
What is meant by " if an electron has size it would be difficult to be make covariant" in quantum field theory.Does this mean the electron would
behave differently in different frames of reference,or does it just mean that the electron would not be in a state that allows it to fit into the...
I am doing the excercises on Chapter 2 of Ziebach's new book A First Course in String Theory. Part (b) of Problem 2.3 asks us to show that the objects \partial/{\partial x^{\mu}} transform under a boost along the x^1 axis in the same way as the a_{\mu} do.
In other words, to show the...
A spin foam is a "mousse de spin"
On Friday Rovelli is giving a symposium talk on spin foams and he was Etera Livine's thesis director.
My uninformed guess is that Rovelli will talk about Livine's thesis and in particular chapter 8 (Covariant loop gravity) which reflects potentially important...
In the online text on differential geometry
http://people.hofstra.edu/faculty/Stefan_Waner/RealWorld/pdfs/DiffGeom.pdf
The author calls the "derivative along the curve" (aka absolute derivative) the "covariant derivative" which is wrong.
It's on box 8.2 on page 59.
Does anyone...
I have a little question. I hope someone can help me.
When we learn the theory of relativity and its formalism, we'll meet concepts : covariant and contravariant, such as covariant vector, covariant tensor...
I wonder that why we need to use the concepts ? What are advantages of them ?
I...