= Solution
Using the product rule for directional <derivatives>,
$$
\partial_t(u\cdot w)
=(w\cdot\nabla)\left(\frac12|u|^2-P\right)
+(u\cdot\nabla)(-u\cdot w).
$$
Thus $f=|u|^2/2-P$ and $g=-h$. If $\nabla\cdot u=0$ and $w=\nabla\times u$, then $\nabla\cdot w=0$. Integrating the two directional <derivatives> as divergences gives only boundary fluxes, which vanish because $u\cdot n=w\cdot n=0$. This is <helicity conservation by tangent boundary conditions>, so $dH/dt=0$.
For the stated field, direct use of the cylindrical curl gives $u\cdot w=-2a\rho^4\sin^2z$. Therefore
$$
\boxed{H=-2a\int_0^{2\pi}d\phi\int_0^\pi\sin^2z\,dz
\int_0^a\rho^5d\rho
=-\frac{\pi^2a^7}{3}.}
$$
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