Covariant Derivative: 2nd Diff - My Question

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Thread starter Kisok
Start date Sep 22, 2019
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Covariant Covariant derivative Derivative Differentiation

In summary, a covariant derivative is a mathematical tool used to measure changes in a vector field along a given direction, taking into account the curvature of the underlying space. It differs from an ordinary derivative by considering the curvature of the space, making it more suitable for describing vectors in curved spaces. The purpose of a covariant derivative is to establish a consistent and meaningful calculus in curved spaces. It is calculated using a connection, which relates tangent spaces at different points, and is used to define the covariant derivative operator applied to the vector field. The covariant derivative is used in various fields of science, including mathematics, physics, and engineering, and is particularly significant in the study of general relativity.

Sep 22, 2019

Kisok

TL;DR Summary: On p158 of General Relativity written by Hobson et al., there is second differentiation of the covariant derivative. However hard I tried to understand why the first equality for the second differentiation holds, I could not. I would be very appreciated if someone help me why it should be.

My question is shown in Summary section. Please see the attached file.

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Sep 22, 2019

martinbn

Science Advisor

Well, what is this ##\nabla_cT_{ba}## equal to?

1. What is a covariant derivative?

A covariant derivative is a mathematical concept used in differential geometry to describe how a vector field changes as it moves along a curved surface or manifold. It takes into account the curvature of the surface and ensures that the vector field is "parallel" to the surface as it moves.

2. Why is the covariant derivative important?

The covariant derivative is important because it allows us to define a notion of "parallel transport" on a curved surface. This is essential in many areas of physics, including general relativity, where the curvature of spacetime must be taken into account.

3. How is the covariant derivative different from the ordinary derivative?

The covariant derivative takes into account the curvature of the surface, while the ordinary derivative does not. This means that the covariant derivative of a vector field will be different depending on the direction in which it is being transported, whereas the ordinary derivative will always be the same.

4. What is the formula for calculating the covariant derivative?

The formula for the covariant derivative depends on the specific metric and curvature of the surface. In general, it involves taking the ordinary derivative of the vector field and subtracting terms that account for the curvature of the surface.

5. How is the covariant derivative used in practical applications?

The covariant derivative is used in many areas of physics and engineering, including general relativity, fluid dynamics, and electromagnetism. It is also used in computer graphics to simulate the behavior of light on curved surfaces. Additionally, it has applications in optimization and control theory.