Covariant derivative and vector functions

Thread starter tom.young84
Start date Nov 21, 2010
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Covariant Covariant derivative Derivative Functions Vector

In summary, a covariant derivative is a mathematical operation used to differentiate vector functions in curved spaces, taking into account the curvature of the space. It differs from a regular derivative by considering multiple variables in a curved space. Covariant derivatives are important in physics for calculating physical quantities in curved spaces, and an example of its use is in calculating the acceleration of an object moving along a curved path. They are closely related to tensors, which are generalized vectors that also take into account the curvature of the space.

Nov 21, 2010

tom.young84

So given this identity:

[V,W] = [tex]\nabla[/tex]_VW-[tex]\nabla[/tex]_WV

^^I got the above identity from O'Neil 5.1 #9.

From this I'm not sure how to make the jump with vector functions, or if it is even possible to apply that definition to a vector function [x_u,x_v].

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Nov 21, 2010

arkajad

You question is not clear. Can you be more precise?

Related to Covariant derivative and vector functions

1. What is a covariant derivative?

A covariant derivative is a mathematical operation that allows for the differentiation of vector functions in curved spaces. It takes into account the curvature of the space and adjusts the derivative accordingly.

2. How is a covariant derivative different from a regular derivative?

A regular derivative only takes into account the change in a function with respect to one variable, while a covariant derivative takes into account the change in a function with respect to multiple variables in a curved space.

3. What is the importance of covariant derivatives in physics?

Covariant derivatives are important in physics because they allow for the calculation of physical quantities in curved spaces, which is necessary for understanding phenomena such as gravity and the behavior of particles in general relativity.

4. Can you give an example of a covariant derivative in action?

One example of a covariant derivative in action is the calculation of the acceleration of an object moving along a curved path. The covariant derivative takes into account the curvature of the path and adjusts the acceleration accordingly.

5. What is the relationship between covariant derivatives and tensors?

Covariant derivatives are closely related to tensors, as they are used to define the transformation properties of tensors in curved spaces. Tensors can be thought of as generalized vectors that also take into account the curvature of the space.