Question
Give an example of a 1-form $\alpha \in \Omega ^1(\mathbb{R}^3)$ with the property that $\alpha \wedge d\alpha = dx\wedge dy\wedge dz$.
Attempt
If we take $\alpha=fdx+gdy+hdz$,
then, $d\alpha=(\frac{\partial h}{\partial y}-\frac{\partial g}{\partial z})dy\wedge dz+(\frac{\partial g}{\partial x}-\frac{\partial f}{\partial y})dx\wedge dy+(\frac{\partial f}{\partial z}-\frac{\partial h}{\partial x})dz\wedge dx$.
This means,
$\alpha \wedge d\alpha = [f(\frac{\partial h}{\partial y}-\frac{\partial g}{\partial z})+g(\frac{\partial f}{\partial z}-\frac{\partial h}{\partial x})+h(\frac{\partial g}{\partial x}-\frac{\partial f}{\partial y})]dx\wedge dy\wedge dz$.
However, I don’t seem to know what $f,g$ and $h$ should be so that the sum of terms in the square brackets would give me 1.
Your help would be appreciated.