Christoffel symbols expansion for second derivatives

In summary: I thought I had given you the mathematical demonstration. All that remains for you is to solve a system of linear equations. If you need more details, why don't you first try and at least write the equations, so that if anyone is willing to add details they can just reply and use what you've written.
  • #1
mertcan
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upload_2017-9-5_16-51-3.png


Hi, I really wonder how these second derivatives can be written in terms of christofflel symbols. I have made so many search but could not find on internet What is the derivation of equations related to second derivatives in attachment?
 
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  • #2
This is the definition of the Christoffel symbols.
 
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  • #3
martinbn said:
This is the definition of the Christoffel symbols.
I am sorry I do not understand could you be more explicit using mathematical demonstration? ?

(As far as I have searched, equations in attachment should be derived from somewhere, those equations are not the definition of christoffel symbol, that symbol is different expansion )
 
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  • #4
mertcan said:
View attachment 210409

Hi, I really wonder how these second derivatives can be written in terms of christofflel symbols. I have made so many search but could not find on internet What is the derivation of equations related to second derivatives in attachment?
What attachment are you referring to?
 
  • #5
WWGD said:
What attachment are you referring to?
Can't you see a picture in my post? ?? I have copied and paste a picture which includes my relevant equations. You can look at it, if there is any problem ( you can not see my picture but I see) please let me know and I will repeat my question. I hope you will help me I really wonder...
 
  • #6
The three vectors ##\{x_u, x_v, n\}## form a basis, that means that any vector is a linear combination of those three. So, you differentiate ##x_u## along ##u## and obtain a new vector ##x_{uu}##. It is a linear combination of the basis vectors so ##x_{uu}=Ax_u+Bx_v+Cn##, similarly for the other two derivatives. Well, the definition is that these coefficients are called the Christoffel's symbols, and are denoted in a certain way.
 
  • #7
May be what you are actually trying to ask is how to express the Christoffel's symbols in terms of the first fundamental form. In any case if you want to show that these coefficients (in the equations of the attachment) are the Christoffel's symbols, you need to tell us what definition of the symbols you use. As I said, usually these equations are used to define them.
 
  • #8
martinbn said:
May be what you are actually trying to ask is how to express the Christoffel's symbols in terms of the first fundamental form. In any case if you want to show that these coefficients (in the equations of the attachment) are the Christoffel's symbols, you need to tell us what definition of the symbols you use. As I said, usually these equations are used to define them.
upload_2017-9-6_9-53-21.png

Ok, please let me change my question, how the coefficients of basis vectors ( A, B ) are written in terms of christoffel symbols defined in my last picture in this post. How this definition of christoffel symbol definition is used as coefficients of basis??
 
  • #9
Take the dot product of both sides of the equations in your first post with ##x_u## and ##x_v##. Keep in mind that ##n## is orthogonal to them so those terms will be zero. Then solve for the Gamma's.
 
  • #10
martinbn said:
Take the dot product of both sides of the equations in your first post with ##x_u## and ##x_v##. Keep in mind that ##n## is orthogonal to them so those terms will be zero. Then solve for the Gamma's.
martinbn said:
Take the dot product of both sides of the equations in your first post with ##x_u## and ##x_v##. Keep in mind that ##n## is orthogonal to them so those terms will be zero. Then solve for the Gamma's.
If you don't mind could you provide me with mathematical demonstration, because I am very confused now, I easily mixed up the things now, I try to focus on multiple things at the same time, I need explicit and visual things in order to make myself not confused.
 
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  • #11
mertcan said:
If you don't mind could you provide me with mathematical demonstration, because I am very confused now, I easily mixed up the things now, I try to focus on multiple things at the same time, I need explicit and visual things in order to make myself not confused.
I thought I had given you the mathematical demonstration. All that remains for you is to solve a system of linear equations. If you need more details, why don't you first try and at least write the equations, so that if anyone is willing to add details they can just reply and use what you've written.
 

What are Christoffel symbols and how are they related to second derivatives?

Christoffel symbols are a set of mathematical objects used to describe the curvature of a manifold, which is a higher-dimensional space. They are related to second derivatives because they represent the connection between the first and second derivatives of a function on a manifold.

How do you expand Christoffel symbols to calculate second derivatives?

To expand Christoffel symbols for second derivatives, you first need to calculate the first derivatives of the function on the manifold. Then, you can use the Christoffel symbols to compute the second derivatives by taking the derivative of the first derivatives with respect to each coordinate.

Why is the Christoffel symbols expansion important in differential geometry?

The Christoffel symbols expansion is important in differential geometry because it allows us to calculate the curvature of a manifold, which is a fundamental concept in this field. It also helps us to understand the relationship between the first and second derivatives of a function on a manifold.

Can Christoffel symbols be used to calculate second derivatives on any type of manifold?

Yes, Christoffel symbols can be used to calculate second derivatives on any type of manifold, as long as it is a smooth and differentiable space. This includes Euclidean spaces, curved spaces, and even abstract mathematical spaces.

Are there any limitations to using Christoffel symbols for second derivatives?

One limitation of using Christoffel symbols for second derivatives is that they can become very complex and difficult to calculate on higher-dimensional manifolds. In these cases, alternative methods such as numerical approximations may be used.

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