Closed non-exact $2$-form on $T = S^1 \times S^1$

Question

I would like to find a closed non-exact $2$-form on $T = S^1 \times S^1$.

Here are my thoughts:

Since $d\theta$ and $d\varphi$ are closed non-exact $1$-forms an obvious candidate is $d\theta \wedge d \varphi$. This is closed since

$$ d(d\theta \wedge d \varphi) = d\theta \wedge d \varphi \wedge d \theta + d\theta \wedge d \varphi \wedge d \varphi = 0$$

But how can I show that it is not exact?

(I'm sure it is indeed not exact)

I had the idea that by contradiction assume that it is exact. Then by Stokes' theorem

$$ \int_T d\theta \wedge d \varphi = \int_T d \psi = \int_{\partial T} \psi = \int_\varnothing \psi = 0$$

But I don't see why this should be a contradiction...

Answer 1

To achieve a contradiction, show by some other means that the integral is nonzero. Since $d\theta \wedge d\phi$ is in fact the standard volume form on $T$, this shouldn't be too difficult.