Does anyone know the Christoffel symbols of second kind for the elliptic coordinate system:
\begin{matrix} x = R\cosh(u)\cos(v)\\ y = R\sinh(u)\sin(v)\\ z = z \end{matrix}
the metric tensor is given by:
$$\begin{bmatrix} R^2(\sinh^2(u)+sin^2(v)) & 0 & 0\\ 0 & R^2(\sinh^2(u)+sin^2(v)) & 0\\ 0&0 &1 \end{bmatrix}$$
I calculated them but they differ from the ones published on Wolfram Mathematica website. I would appreciate any help!
This is what I obtained for the "u" element:
\begin{equation} \left( \begin{array}{ccc} \frac{\cosh (u) \sinh (u)}{\sin ^2(v)+\sinh ^2(u)} & \frac{\cos (v) \sin (v)}{\sin ^2(v)+\sinh ^2(u)} & 0 \\ \frac{\cos (v) \sin (v)}{\sin ^2(v)+\sinh ^2(u)} & -\frac{\cosh (u) \sinh (u)}{\sin ^2(v)+\sinh ^2(u)} & 0 \\ 0 & 0 & 0 \\ \end{array} \right) \end{equation}
using the package here: http://library.wolfram.com/infocenter/MathSource/8329
I don't know if it conflicts with your site.
The "v" element
\begin{equation} \left( \begin{array}{ccc} -\frac{\cos (v) \sin (v)}{\sin ^2(v)+\sinh ^2(u)} & \frac{\cosh (u) \sinh (u)}{\sin ^2(v)+\sinh ^2(u)} & 0 \\ \frac{\cosh (u) \sinh (u)}{\sin ^2(v)+\sinh ^2(u)} & \frac{\cos (v) \sin (v)}{\sin ^2(v)+\sinh ^2(u)} & 0 \\ 0 & 0 & 0 \\ \end{array} \right) \end{equation}