Consider the equation of the dielectric tensor $\bar{\epsilon}$ of a nematic liquid crystal in flat space in cartesian coordinates,
\begin{equation} \bar{\epsilon}=\epsilon_{iso}\delta_{ij}+\epsilon_a\left(n_in_j-\frac{\delta_{ij}}{3} \right), \end{equation}
where $\epsilon_{iso}=\frac{\epsilon_\|+2\epsilon_\bot}{3}$, $\epsilon_a=\epsilon_\|-\epsilon_\bot$, $\epsilon_\|$ and $\epsilon_\bot$ are the molecular dielectric constants, respectively, parallel and perpendicular to the major axis of a prolate ellipsoid liquid crystalline molecule, and $n_i$ are the components of the versor $\hat{n}$ (AKA director) that gives the average orientation of the major axis of these liquid crystalline molecules.
How can I modify this equation for cylindrical coordinates in flat space? Is it sufficient to consider $\delta_{ij}$ as the metric tensor and to write it in cylindrical coordinates? Or may I multiply $n_in_j$ by $\delta_{ij}$ (imagining a scalar product)? With these suggestions, for example, I can't obtain the correct dielectric tensor for the director $\hat{n}=\hat{\theta}$. An equivalent formulation of this question: how can I generalize this equation for any coordinates?
Again, please forgive me for this basic question.
Yes, $\delta_{ij}$ should be interpreted as the metric tensor in Cartesian coordinates.
People on the more pure mathematics side of things tend to write things like this in a basis independent manner. For any vectors $a, b$,
$$\bar \epsilon(a,b) = \epsilon_\text{iso} (a \cdot b) + \epsilon_a ([\hat n \cdot a][\hat n \cdot b] - \frac{1}{3} a \cdot b)$$
Use whatever basis you like, as long as the inner products (denoted by $\cdot$) transform accordingly, using the corresponding metric.