Covariant Derivative: Definition & Meaning

In summary, a covariant derivative is a mathematical tool used to measure the change of a geometric object in a curved space, taking into account the curvature of the space. It differs from a regular derivative, which measures change in a flat space, and is important in physics for understanding objects in the universe. The covariant derivative is calculated by taking a regular derivative and adding correction terms based on the metric tensor, and has applications in general relativity, differential geometry, and gauge theories.
  • #1
salparadise
23
0
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 the index via the metric of ordinary covariant derivative.

I tried to search in different textbooks for this, but couldn't find anything.

Thanks in advance for your help.
 
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  • #2
[tex]A^{ab;c} = g^{cd} \nabla_d A^{ab} = \nabla_d (g^{cd} A^{ab})[/tex]

Note that [itex]\nabla_c g_{ab} = 0[/itex].
 

What is a covariant derivative?

A covariant derivative is a mathematical tool used in differential geometry to measure the change of a geometric object along a curve or surface in a curved space. It takes into account the curvature of the space in which the object is being measured.

What is the difference between a covariant derivative and a regular derivative?

A regular derivative measures the change of a function in a flat space, while a covariant derivative measures the change of a geometric object in a curved space. The covariant derivative takes into account the curvature of the space, while a regular derivative does not.

Why is the covariant derivative important in physics?

The covariant derivative is important in physics because it allows us to define the concept of a derivative in a curved space, which is necessary for understanding the behavior of objects in the universe. It is used in various theories such as general relativity and gauge theories.

How is the covariant derivative calculated?

The covariant derivative is calculated by taking the regular derivative of a geometric object and then adding correction terms that account for the curvature of the space. These correction terms are determined by the metric tensor, which describes the geometry of the space.

What are some applications of the covariant derivative?

The covariant derivative has many applications in physics and mathematics. It is used in general relativity to describe the behavior of objects in curved space-time, in differential geometry to study the geometry of curved spaces, and in gauge theories to describe the behavior of quantum fields on curved spaces.

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