I was given this general $2$-form on $\mathbb{R}^3$
$$\omega= A(x^1,x^2,x^3) \ dx^1 \wedge dx^2 +B(x^1,x^2,x^3) \ dx^1 \wedge dx^3+C(x^1,x^2,x^3) \ dx^2 \wedge dx^3$$
and was asked to write down explicit conditions on $A$, $B$, and $C$ which are equivalent to:
i. $d\omega=0$
ii. $\omega = d \alpha$, where $\alpha$ is a $1$-form and $\alpha=F\ dx^1+G\ dx^2+ H\ dx^3$.
iii. Show that the conditions from (ii.) imply those of (i.).
Here is what I got so far:
i. The condition that makes $d\omega=0$ is $\frac{\partial A}{\partial x^3}-\frac{\partial B}{\partial x^2}+\frac{\partial C}{\partial x^1}=0$.
ii. The conditions that make $\omega=d\alpha$ are
$A=\frac{\partial G}{\partial x^1}-\frac{\partial F}{\partial x^2}$
$B=\frac{\partial H}{\partial x^1}-\frac{\partial F}{\partial x^3} $
$C=\frac{\partial H}{\partial x^2}-\frac{\partial G}{\partial x^3}$
iii. $\frac{\partial A}{\partial x^3}-\frac{\partial B}{\partial x^2}+\frac{\partial C}{\partial x^1}=\big( \frac{\partial^2 G}{\partial x^1 \partial x^3}-\frac{\partial^2 G}{\partial x^3 \partial x^1}\big) +\big( \frac{\partial^2 F}{\partial x^3 \partial x^2}-\frac{\partial^2 F}{\partial x^2 \partial x^3} \big)+\big(\frac{\partial^2 H}{\partial x^2 \partial x^1} -\frac{\partial^2 H}{\partial x^1 \partial x^2}\big)$
The thing I am not sure about is weather this expression in (iii) equals $0$ or not. Does it equal $0$? Thank you!