Christoffel Symbols of f(u,v) Surface

[Lee33]

Homework Statement

Find the Christoffel symbols of a surface in the form $g= f(u,v).$

Homework Equations

$f_{u_1u_1} = \Gamma^1_{11} f_{u_1} + \Gamma^2_{11}f_{u_2} + A \vec{N}$
$f_{u_1u_2} = f_{u_2u_1} = \Gamma^1_{12} f_{u_1} + \Gamma^2_{12}f_{u_2} + B \vec{N}$
$f_{u_2u_2} = \Gamma^1_{22} f_{u_1} + \Gamma^2_{22}f_{u_2} + C\vec{N}$

The Attempt at a Solution

I know how to calculate the Christoffel symbols if it explicitly shows a parameterized surface but how can I calculate it of the form $g= f(u,v)?$

 

[mighty2000]

To calculate the Christoffel symbols for a surface in the form $g=f(u,v)$, we can use the first and second fundamental forms. The first fundamental form, given by $ds^2 = E du^2 + 2F dudv + G dv^2$, can be calculated from the metric tensor $g$ as follows:

$E = g_{11} = f_{u} \cdot f_{u}$
$F = g_{12} = g_{21} = f_{u} \cdot f_{v}$
$G = g_{22} = f_{v} \cdot f_{v}$

The second fundamental form, given by $d\vec{N} = L du^2 + 2M dudv + N dv^2$, can be calculated from the unit normal vector $\vec{N}$ as follows:

$L = \vec{N} \cdot f_{uu}$
$M = \vec{N} \cdot f_{uv}$
$N = \vec{N} \cdot f_{vv}$

Once we have calculated the first and second fundamental forms, we can use the formulas given in the homework equations to calculate the Christoffel symbols. For example, to calculate $\Gamma^1_{11}$, we can use the formula $\Gamma^1_{11} = \frac{1}{2} \frac{\partial E}{\partial u}$. Similarly, we can calculate the other Christoffel symbols using the appropriate formulas.

In summary, to calculate the Christoffel symbols for a surface in the form $g=f(u,v)$, we need to first calculate the first and second fundamental forms using the metric tensor and the unit normal vector, and then use the formulas given in the homework equations to calculate the Christoffel symbols.