 Solution

For a steady flow the local time derivative in the Euler equations vanishes. The vector identityreduces to a gradient when the vorticity $\nabla \times u$ is zero. With constant density and gravity $-gk$, the momentum equation becomesThus on each connected flow region,This is Bernoulli's theorem for a steady irrotational flow: irrotationality makes the same constant apply throughout a connected region, rather than only along an individual streamline.