Convert tensor from cartesian to cylindrical coordinate

[ngkamsengpeter]

Homework Statement

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

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
<br /> F^{\mu \nu }F_{\mu \nu }≠2(B^2-\frac{E^2}{c^2})<br />

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?