I am trying to show that a potential flow must satisfy the incompressible Euler's equation: \begin{align} \rho \partial_t v + \rho (v \cdot \nabla ) v = - \nabla p \end{align}
As the flow is steady we have the following: \begin{align} \rho\partial_t\nu +\rho(\nu\cdot \nabla)\nu= \rho\cdot 0+\rho\nu\cdot \nabla\nu \end{align}
I believe that $\nabla\nu$ is the covariant derivative of $\nu$? Assuming that $p$ is the dynamic pressure and is hence equal to $- \frac{\rho}{2} |v|^2$, then I must show that $\rho\nu\cdot \nabla\nu = -\nabla(- \frac{\rho}{2} |v|^2)$.