How to transform material permittivity tensor from Cartesian coordinates to another orthogonal coordinate system? I have a material specified by a permittivity tensor written in Cartesian coordiantes:
$$\begin{pmatrix}
\epsilon_{xx} &  \epsilon_{xy} & \epsilon_{xz}\\ 
\epsilon_{yx} &\epsilon_{yy}  & \epsilon_{yz}\\ 
\epsilon_{zx} & \epsilon_{zy} &\epsilon_{zz} 
\end{pmatrix}$$
I want to write all my equations in cylindrical coordinates  and so I have to write the permittivity tensor in cylindrical coordinates  too. How can I do that?
 A: In cylindrical coordinates  the tensor components would look like
$ T_{r\theta z}= \left( \begin{array}{ccc}
\epsilon_{rr} & \epsilon_{r\theta} &\epsilon_{rz} \\
\epsilon_{\theta r} & \epsilon_{\theta \theta} & \epsilon_{\theta z} \\
\epsilon_{zr} & \epsilon_{z\theta} & \epsilon_{zz} \end{array} \right)\ $
Since both coordinate systems are orthogonal, the transformation of your Cartesian tensor $T_{xyz}$ to a tensor $T_{r\theta z}$ would be given by
$ T_{r\theta z}= {Q^T}T_{xyz} {Q} $
with the transformation matrix $Q$ is the same matrix you would use to transform a vector from Cartesian to cylindrical, i.e. 
$ Q= \left( \begin{array}{ccc}
\cos\theta & \sin\theta & 0 \\
-\sin\theta & \cos\theta & 0 \\
0 & 0 & 1 \end{array} \right)\ $
and ${Q^T}$ is the transpose of $Q$.
Note that this discussion assumes that the goal is to solve the non-covariant Maxwell's equations in cylindrical coordinates, neglecting special relativity. A more general treatment of the permittivity tensor and its transformation would be necessary if solving the covariant form of the Maxwell's equations
