Transformation law for Christoffel symbol of first kind

In summary, the conversation is about a problem with proving the transformation law for the Christoffel symbol of the first kind. The person asking for help has read books on the topic but is struggling to eliminate some terms. They are looking for someone to show them how to prove it. Another person offers to provide the results and calculations for the second kind symbols and encourages the first person to try and find the first kind symbols themselves. The first person also mentions that their understanding is based on an "old style Tensor" and would appreciate a derivation based on their current learning style. The conversation ends with another person offering to help with the manipulation of suffixes and derivatives to find the first kind symbols' transformation law.
  • #1
Will_C
Hi,
I have met a problem, that is how to prove transformation law for Christoffel symbol of first kind. I have read books about that, but many of them just state: cyclic permutation of the 3 indices and substitution. When I tried to work out, I could not elimate some terms...
Can anyone show me how to prove it once.
Will.
 
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  • #2
Will_C said:
Hi,
I have met a problem, that is how to prove transformation law for Christoffel symbol of first kind. I have read books about that, but many of them just state: cyclic permutation of the 3 indices and substitution. When I tried to work out, I could not elimate some terms...
Can anyone show me how to prove it once.
Will.

By prove,u mean calculate it,right??I'll give u the results and the calculations for the second kind symbols and let u strive to find the first kind symbols.
 

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  • #3
Thx dextercioby,
I think ,what I have read, is a old style Tensor. So most books, I have read, derive the second kind transformation law from the first kind. Thank you very much for your help. It would be more happy to me if I can get some derives base on the style, what I am learning.

Will.
 
  • #4
Will_C said:
Thx dextercioby,
I think ,what I have read, is a old style Tensor. So most books, I have read, derive the second kind transformation law from the first kind. Thank you very much for your help. It would be more happy to me if I can get some derives base on the style, what I am learning.

Will.

No,what u've read was a about 90% of a page from a pdf-format book on GR.That's why it used Greek suffixes.If it's old style,well,i doubt it,since he takes diff.geometry from zero,from the definiton of a topological space.
I believe u can manipulate suffices,derivatives and other tensorial quantities so that u can get the first kind symbols' transformation law directly from the expression of a covariant derivative of a covector.

Daniel.
 
  • #5
thanks .
 

1. What is the transformation law for the Christoffel symbol of first kind?

The transformation law for the Christoffel symbol of first kind is a mathematical rule that describes how the components of the Christoffel symbol change when coordinates of a curved space are transformed. It is used in the study of differential geometry and general relativity.

2. Why is the transformation law important?

The transformation law is important because it allows us to relate the Christoffel symbol in one coordinate system to another coordinate system, which is necessary for solving equations and making predictions in curved spaces. It also helps us understand the geometric properties of the space.

3. What are the key components of the transformation law?

The key components of the transformation law are the metric tensor, the inverse metric tensor, and the partial derivatives of the metric tensor. These components are used to calculate the new Christoffel symbol components in the transformed coordinate system.

4. How is the transformation law derived?

The transformation law is derived using the chain rule of differentiation, along with the properties of the metric tensor. It involves calculating the partial derivatives of the metric tensor in the new coordinate system and using them to construct the new Christoffel symbol components.

5. Can the transformation law be applied to any curved space?

Yes, the transformation law can be applied to any curved space, as long as the space has a metric tensor defined. This includes spaces with positive, negative, or zero curvature, and also spaces with different dimensions. The transformation law is a fundamental concept in the study of curved spaces and is applicable in various fields of science and mathematics.

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