Is $\bf{u} \times\bf{(\nabla \times u)}$ $=0$ in the contex for deriving Bernoulli’s theorem

Question

Is $\bf{u} \times\bf{(\nabla \times u)}$ $=0$

I am trying to derive the to Bernoulli’s theorem for a a steady, inviscid, homogeneous, incompressible fluid.

Using mass conservation:

Using Euler's equation:

Now somehow we get that $\nabla H$ is zero, i don't understand how?

Answer 1

What is true is that $ u \cdot \nabla H = 0$. This is all that is required, as Bernoulli’s principle says that the quantity $H$ is conserved along streamlines.