The Hodge decomposition theorem tells us that any $r$-form $\omega$ on a Riemannian manifold $M$ (without boundary and compact) may be uniquely decomposed as $$ \omega = d \gamma_1 + d^\dagger \gamma_2 + \gamma_3 $$ where $\gamma_1\in\Omega^{r-1}(M)$, $\gamma_2\in\Omega^{r+1}(M)$ and $\gamma_3\in \text{Harm}^r(M)$ and where $d^\dagger$ is the co-differential ($d^\dagger\propto \star d \star$).
My question is: given $\omega$, how do I compute $\gamma_3$?
In the proof of the theorem one typically uses some projection operator $P:\Omega^r(M)\to\text{Harm}^r(M)$, but I don't know how to explicitly compute $P(\omega)$($=\gamma_3$ by definition) if I know explicitly $\omega$.
More specifically, I want to compute $P(\omega)$ if $\omega=f F$ where $f\in\Omega^0(M)$, $F\in\Omega^5(M)$ with dim($M$)=10 and $F=d C$ for some $C\in\Omega^4(M)$.
Many thanks for your help!