Convert tensor from cartesian to cylindrical coordinate

[ngkamsengpeter]

Homework Statement

Given the tensor
$$F_{\mu \nu }= <br /> \left[ \begin{array}{cccc} 0 & -E_{x} & -E_{y} & -E_{z} \\ E_{x} & 0 & B_{z} &-B_{y} \\E_{y} & -B_{z} & 0 & B_{x} \\E_{z} & B_{y} & -B_{x} & 0 \end{array} \right]$$
$$F^{\mu \nu }F_{\mu \nu }=2(B^2-\frac{E^2}{c^2})$$
and metric tensor
$$n_{\mu \nu }= <br /> \left[ \begin{array}{cccc}c^2& 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -r^2 & 0 \\ 0 & 0 & 0 & -1 \end{array} \right]$$

How to convert it into cylindrical coordinates, that is in terms of E_θ,E_z,E_r
More info of this tensor can be viewed at http://en.wikipedia.org/wiki/Electromagnetic_tensor

Homework Equations

The Attempt at a Solution

I try to convert it using the transformation matrix and tensor transformation rule but it turns out that
$$F^{\mu \nu }F_{\mu \nu }≠2(B^2-\frac{E^2}{c^2})$$

Can anyone give me some idea how to solve this?
Thanks.

Homework Statement

Homework Equations

The Attempt at a Solution

 

[ngkamsengpeter]

No one can help?