Solving Curvature Math Problems with Christoffel Symbols

  • Thread starter Feynman
  • Start date
In summary, the first post has symbols that look like hieroglyphics, and the second post has formulas for calculating curvurture.
  • #1
Feynman
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0
Hello
I have a small pb:
Let [tex]\displaystyle{M=\mathbb{R^{2}\{(x,y);x=0\quad ou \quad y=-1}[/tex]
Let D such that Christoffel symboles different from 0 are\\\\
[tex]\displaystyle\Gamma_{11}^{1}(x,y)=-\frac{1}{x}\\\\\\\\
\Gamma_{22}^{2}(x,y)=\frac{1}{1+y}[/tex]\\\\\\
How calculate curvurture?\\
 
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  • #2
The Riemann-curvature is given by:

[tex]R^a_{b c d} = \frac{\partial \Gamma^a_{b d}}{\partial x^c} - \frac{\partial \Gamma^a_{b c}}{\partial x^d} + \Gamma^a_{c e} \Gamma^e_{b d} - \Gamma^a_{d e} \Gamma^e_{b c}[/tex]
 
  • #3
excuse me [tex]\displaystyle\Gamma_{11}^{1}(x,y)=-\frac{1}{x}\quad and \quad Gamma_{22}^{2}(x,y)=\frac{1}{1+y}[/tex]
And
[tex]\displaystyle{M={R^{2}\{(x,y);x=0\quad or \quad y=-1}}[/tex]
So How we use this formula in this case
and how we obtain this formula?
 
  • #4
a, b, c, d and e are indices, so in this case they can take the values 1 or 2: [tex]x = x^1[/tex] and [tex]y = x^2[/tex]. Repeates indices are summed over (Einstein summation convention).

So for example:

[tex]R^1_{1 2 2} = \frac{\partial \Gamma^1_{1 2}}{\partial x^2} - \frac{\partial \Gamma^1_{1 2}}{\partial x^2} + \sum_{e=1}^2 (\Gamma^1_{2 e} \Gamma^e_{1 2} - \Gamma^1_{2 e} \Gamma^e_{1 2})[/tex]

which in this case is 0 because only [tex]\Gamma^1_{1 1}[/tex] and [tex]\Gamma^2_{2 2}[/tex] are different from 0.

If you want to see a derivation of this formula, you could have a look at (chapter 3 of) Sean M. Carroll's lecture notes: http://pancake.uchicago.edu/~carroll/notes/
 
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  • #5
But in my case
R=0?
 
  • #6
Are u looking for the tensor,or for its contractions (Ricci tensor,Ricci scalar)...?

Daniel.
 
  • #7
What are those hieroglyphic looking symbols in the first post? I clicked on it and the latex code doesn't look anything like it. :confused:
 
  • #8
U didn't close \mathbb function right after R...:tongue:

Daniel.
 
  • #9
[tex]\displaystyle{M=\mathbb{R}^{2}\{(x,y);x=0\quad ou \quad y=-1}[/tex]

Ah, there we go. I still don't know what that means, but it looks more recognizable. I was hoping those were tensor diagrams like Penrose uses. I don't get how they work, but they sure look nifty.
 

What is a "Curveture math problem"?

A "Curveture math problem" is a mathematical problem that involves calculating the curvature of a mathematical curve or surface. It is commonly used in fields such as physics and engineering to model and understand the behavior of physical objects.

What is curvature?

Curvature is a measure of how much a curve or surface deviates from being a straight line or flat plane. It is often described as the amount of bending or curving present in a particular mathematical object.

How is curvature calculated?

The calculation of curvature depends on the specific curve or surface being analyzed. In general, it involves determining the rate of change of the direction of a tangent to the curve or surface at a given point. This can be done using various mathematical techniques and formulas.

What real-world applications use curvature math problems?

Curvature math problems have many real-world applications, particularly in the fields of physics and engineering. For example, they are used to model the behavior of objects in motion, such as the trajectory of a projectile or the shape of a roller coaster. They are also used in the design of structures, such as bridges and buildings, to ensure they can withstand forces and stresses.

What skills are needed to solve curvature math problems?

Solving curvature math problems requires a strong understanding of various mathematical concepts, including calculus, geometry, and algebra. It also requires critical thinking and problem-solving skills, as well as the ability to apply mathematical concepts to real-world situations.

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