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Since I am not going ot upgrade yet, I will make these notes public.  The steady-state Euler equation reads  \begin{equation}  \left(\mathbf{v} \cdot \nabla \right)\mathbf{v} + \frac{1}{\rho}\nabla P + \nabla \Phi_G -\nu\left[\nabla^2\mathbf{v} + \frac{1}{3}\left(\nabla \cdot \mathbf{v} \right) \right] = 0  \end{equation}  where it is understoond that gravitational potential $\Phi_G$ and the coeficcient of kinematic viscosity $\nu$ are time independent.  Integrate the momentum equation for a steady, nonviscous, flow along a streamline  \begin{equation}  \int{\mathbf{ds}\cdot (\rm{momentum equation})}