I do not know how to demonstrate that
\begin{equation}
\omega(\psi) = \frac{d H(\psi)}{d\psi},
\end{equation}
where $H$ stands for the Bernoulli's constant $H=\frac{u^2 +v^2}{2} + p/\rho$, $\psi$ for the stream function and $\omega$ for the vorticity.
Thank you in advance :)
On
Like Bob Sacamento, I had not seen anything like this either. But in the 2D case, I can make sense of your equation.
So what I think your equation means is this: differentiate along any path that is perpendicular to the stream line (i.e. the path that maximizes the rate of change of $\psi$), and along that path we have $dH/d\psi = w$, and that also happens to be the direction in which $H$ maximally changes.
Mathematically, this becomes: $$ w \nabla \psi = \nabla H .$$ But $\nabla \psi = (-u,v)$ (that's the definition of the stream function), and so $(w \nabla \psi,0) = (u,v,0) \times \text{curl}(u,v,0)$ (note $\text{curl}(u,v,0) = (0,0,w)$. And so the desired equation is simply equation (19) of http://www.maths.bris.ac.uk/~majge/week4.pdf.
Ummm, actually, I don't think that equation is quite right. Take a look at the first few lines of this.