\(- \epsilon_{nqi} \frac{\partial}{\partial x_q} \left( \frac{1}{\rho} \frac{\partial p}{\partial x_i} \right) = 0 + \underbrace{\frac{1}{\rho^2} \epsilon_{nqi} \frac{\partial \rho}{\partial x_q} \frac{\partial p}{\partial x_i}}_{\text{the curl of the pressure gradient is zero}}\)
\(- \nu \epsilon_{nqi} \epsilon_{ijk} \frac{\partial^2 u_k}{\partial x_q \partial x_j} = - \nu \underbrace{(\delta_{nj}\delta_{qk} - \delta_{nk}\delta_{qj})}_{\epsilon_{nqi} \epsilon_{ijk}} \frac{\partial^2 u_k}{\partial x_q \partial x_j} = - \nu \underbrace{\frac{\partial^2 u_q}{\partial x_q \partial x_n}}_{=0} + \nu \frac{\partial^2 u_n}{\partial x_j^2} = \nu \frac{\partial^2 \omega_n}{\partial x_j^2}\)
\(\therefore \frac{\partial \omega_n}{\partial t} = (\omega_j + 2\Omega_j) \frac{\partial u_n}{\partial x_j} - u_j \frac{\partial \omega_n}{\partial x_j} + \frac{1}{\rho^2} \epsilon_{nqi} \frac{\partial \rho}{\partial x_q} \frac{\partial p}{\partial x_i} + \nu \frac{\partial^2 \omega_n}{\partial x_j^2}\)
\(\because \frac{D F}{D t} \equiv \frac{\partial F}{\partial t} + \mathbf{u} \cdot \nabla F \text{ or } \frac{D F}{D t} \equiv \frac{\partial F}{\partial t} + u_i \frac{\partial F}{\partial x_i}\)
\(\therefore \boxed{ \frac{D \omega_n}{D t} = (\omega_j + 2\Omega_j) \frac{\partial u_n}{\partial x_j} + \frac{\epsilon_{nqi}}{\rho^2} \frac{\partial \rho}{\partial x_q} \frac{\partial p}{\partial x_i} + \nu \frac{\partial^2 \omega_n}{\partial x_j^2} \text{ or } \frac{D \boldsymbol{\omega}}{D t} = (\boldsymbol{\omega} + 2\boldsymbol{\Omega}) \cdot \nabla \mathbf{u} + \frac{1}{\rho^2} \nabla \rho \times \nabla p + \nu \nabla^2 \boldsymbol{\omega}}\)