Why is $\alpha = x^2dx^1 - x^1dx^2$ not a differential of a function? Why is $\alpha = x^2dx^1 - x^1dx^2$ not a differential of a function?
I know that a $df$ can be expressed:
$$ df= \frac{\partial f}{\partial x^i}dx^i$$
And I assumed that $\alpha$ is the differential of a function and get a necessary condition $f=0$ 
($f_{x^1}=x^2$ etc).
Does this show that $\alpha$ is not a differential of a function? If so can someone please try to tell me why? Thanks

Hint
We have
$$
\frac{\partial f}{\partial x^1}=x^2
$$
and
$$
\frac{\partial f}{\partial x^2}=-x^1
$$
so:
$$
\frac{\partial^2 f}{\partial x^1\partial x^2}\ne \frac{\partial^2 f}{\partial x^2\partial x^1}
$$
