Calculating Christoffell Symbols of Second Kind for Cartesian Space

[zheng89120]

Homework Statement

Let a surface be define by z = x^(a=3) = f[x^(a=1,2)]

Show that the Christoffell sybols of the 2nd kind are:

[Christoffell symbol]^a_bc = { f_af_bc }/ { f^$$\alpha$$ f_sub_$$\alpha$$ }

where indices on f indicates partial derivatives

Homework Equations

(d^2 x/dt^2)^$$\alpha$$ + [Christoffell symbol]^$$\alpha$$_BC (dx/dt)^B (dx/dt)^C = 0

compare with:

Euler-Lagrangian Equation

The Attempt at a Solution

E-L equatiion: x** - m dz*/dx* = -g dz/dx

compare with the first relevant equation...

how? what is x*^B and x*^C in the first equation

 

[sgd37]

have you got an online version of the question or picture because it is hard to make sense of this thing

 

[zheng89120]

http://www.flickr.com/photos/59383047@N05/5436668502/

 

[sgd37]

you have a 2d metric that only depends on the derivatives of f . You can derive it to be

$$g = \begin{pmatrix} 1+(\partial_1 f)^2 & \partial_1 f \partial_2 f \\ \partial_1 f \partial_2 f & 1+(\partial_2 f)^2 \end{pmatrix}$$

or in component form and using your notation

$$g_{ab} = \delta_{ab} +f_a f_b$$

then you can use the standard formula for the Christoffel symbols

$$\Gamma^{a}_{bc} = \frac{1}{2} g^{ad} (\partial_c g_{bd} +\partial_b g_{cd} - \partial_d g_{bc} )$$

the inverse metric is a bit harder to write down in component form but it is possible

 

[paranormal]

?

I would first clarify the notation used in the problem. It seems that the surface is defined by a function z that is dependent on x raised to the power of a=3. The Christoffel symbols of the 2nd kind are then shown to be related to the partial derivatives of this function.

Next, I would explain the relevance of the Christoffel symbols in this context. They are used in the study of Riemannian geometry and are a way of measuring curvature on a surface. In this case, they can help us understand the curvature of the surface defined by z = x^(a=3).

To solve for the Christoffel symbols, we can use the Euler-Lagrange equation which relates the second derivatives of a function to its partial derivatives and the Christoffel symbols. By comparing this equation to the first relevant equation, we can see that the x*^B and x*^C in the first equation correspond to the dx/dt terms in the Euler-Lagrange equation. This allows us to solve for the Christoffel symbols using the partial derivatives of z.

In summary, the Christoffel symbols of the 2nd kind can be calculated using the partial derivatives of the function defining the surface and the Euler-Lagrange equation. They provide valuable information about the curvature of the surface and are important in the study of Riemannian geometry.