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In a paper (LINDBORG, 2007, DOI:10.1175/JAS3864.1) it is said that the Levi-Civita symbol in cylindrical coordinate can be written as(eq.8): $$ \epsilon_{3ik} = e_{\rho_i} e_{\phi_k} - e_{\phi_i} e_{\rho_k} $$ where the $\rho$ and $\phi$ are polar radius and polar angle in cylindrical coordinate.

This does not look like the normal form of Levi-Civita symbol. I can not figure out how to derive this.

And the eq.9 in that paper also seems not that "starightforward" to me, how to derive this:

$$ Q_{zz} = \frac{1}{\rho} \frac{\partial R_{\rho\rho}}{\partial rho} - \frac{1}{\rho^2} \frac{\partial}{\partial \rho} \left( \rho^2 \frac {\partial R_{\phi\phi}}{\partial \rho} \right) - \frac{1}{\rho^2} \frac {\partial^2 R_{\rho\rho}}{\partial \phi^2} + \frac{1}{\rho} \frac {\partial^2}{\partial \rho \partial \phi} (R_{\rho\phi} + R_{\phi\rho}) $$ where the $R$ and $Q$ are two-point velocity correlation function and two-point correlation functions of vertical vorticity.

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To me this looks like the normal form of the Levi-Civita symbol. In Cartesian coordinates we know that the cross product can be written as $$ (\boldsymbol{x}\times\boldsymbol {y})_i=\epsilon_{i\,j\,k}\;x_j\,y_k\,. $$ In particular, for the basis vectors we have as only non zero element $$ (\boldsymbol{e}_x\times\boldsymbol{e}_y)_3=\epsilon_{3\,j\,k}\;(\boldsymbol{e}_x)_j\,(\boldsymbol{e}_y)_k=(\boldsymbol{e}_x)_1(\boldsymbol{e}_y)_2- (\boldsymbol{e}_x)_2(\boldsymbol{e}_y)_1\,. $$ The last term is zero and the first term is one. I have just deliberately reproduced their clumsy notation.

In cylindrical coordinates things are not different as I showed here. For spherical polar coordinates see here.

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