Are there differential forms on $\Bbb{R}^n$ which are neither exact nor closed

Question

There are exact differential forms on $\Bbb{R}^n$ (all of which are closed). There are closed forms which are not exact. I'd like an example of a differential form on $\Bbb{R}^n$ which is neither exact nor closed.

Answer 1

Most forms on $\mathbb R^n$ are not closed. For zero forms, any nonconstant function is not closed. For $1\leq k<n$, let $I$ be a subset of $[n]=\{1,\ldots,n\}$ with $|I|=k$. define $$\omega=\left(\prod_{i\in[n]\setminus I}x^i\right)dx^I$$ Then $$d\omega=\sum_{j\in[n]\setminus I}\left(\prod_{i\in[n]\setminus(I\cup\{j\})}x^i\right)dx^jdx^I\neq 0$$ Indeed, for any smooth function $f:\mathbb R^n\to\mathbb R$ with some nonzero partial derivative, you can define a differential form which is not closed.