Let $N=\mathbb R^2$ with coordinates $x$ and $y$. $z=x+iy$ is complex-valued function on $N$. $dz=dx+idy$ is complex-valued 1-form on $N$. Consider $a_n=Re(z^ndz)$ and $b_n=Im(z^ndz)$. I need to find $da_n$ and $db_n$.
So, can I do it someway without computation? I know the answer ,they both equal to $0$ . Because this is not so easy to compute it.
Consider $\omega = z^n\,dz$. Then $d\omega = 0$ (e.g., because $z^n$ is holomorphic). But $d\omega = d(a_n + ib_n) = da_n + idb_n$, and the real and imaginary parts of the $2$-form $d\omega = 0$ are both $0$.