I cannot figure out how to derive a formula in a scientific paper (LINDBORG, 2007, DOI: 10.1175/JAS3864.1). I will list all the information needed below:
The starting point of the derivation is this equation (eq 4 in original paper, with Einstein summation convention): $$ Q_{zz} = -\epsilon_{3ik} \epsilon_{3jl} \frac{\partial R_{kl}} {\partial r_i \partial r_j} $$ Here the variable $r$ is in a cartesian coordinate, we need to change it into cylindrical coordinate ($\rho, \phi, z$) to get the final results. The tensor $\mathbf R$ in cylindrical coordinate is written as (eq 5 in original paper): $$ \mathbf R = \mathbf e_{z} \mathbf e_{z} R_{zz} + \mathbf e_{\rho} \mathbf e_{\rho} R_{\rho\rho} + \mathbf e_{\phi} \mathbf e_{\phi} R_{\phi\phi} \\ + \mathbf e_{z} \mathbf e_{\rho} R_{z\rho} + \mathbf e_{\rho} \mathbf e_{z} R_{\rho z} + \mathbf e_{z} \mathbf e_{\phi} R_{z\phi} \\ + \mathbf e_{\phi} \mathbf e_{z} R_{\phi z} + \mathbf e_{\rho} \mathbf e_{\phi} R_{\rho\phi} + \mathbf e_{\phi} \mathbf e_{\rho} R_{\phi\rho} $$ And we have following relations (eq 6-8 in original paper): $$ \frac{\partial}{\partial r_i} = e_{\rho_i} \frac{\partial}{\partial \rho} + \frac{e_{\phi_i}}{\rho} \frac{\partial}{\partial \phi} \\ \frac{\partial}{\partial \rho} e_{\rho_i} = \frac{\partial}{\partial \rho} e_{\phi_i} = 0 \\ \frac{\partial}{\partial \phi} e_{\rho_i} = e_{\phi_i} \\ \frac{\partial}{\partial \phi} e_{\phi_i} = -e_{\rho_i} \\ \epsilon_{3ik} = e_{\rho_i} e_{\phi_k} - e_{\phi_i} e_{\rho_k} $$ Then the final expression of $Q_{zz}$ will be (eq 9 in original paper): $$ Q_{zz} = \frac{1}{\rho} \frac{\partial R_{\rho\rho}}{\partial \rho} - \frac{1}{\rho^2} \frac{\partial}{\partial \rho} \left( \rho^2 \frac {\partial R_{\phi\phi}}{\partial \rho} \right) - \frac{1}{\rho^2} \frac {\partial^2 R_{\rho\rho}}{\partial \phi^2} + \frac{1}{\rho} \frac {\partial^2}{\partial \rho \partial \phi} (R_{\rho\phi} + R_{\phi\rho}) $$
My quesion is how the final equation be derived.
- It seems that the product between vector components of unit vector (ie. $e_{\rho_i} e_{\rho_j}$) don't have a definite value. If so, how are they eliminated in the final expression?
- If we use the definition of Levi-Cvita symbol mentioned above, then we will find that a unit vector in cartesian coordinate system have four vector components (ie. for $\mathbf e_{\rho}$, we have $e_{\rho_i}, e_{\rho_j}, e_{\rho_k}, e_{\rho_l}$). How to understand this?