The question is $$ Let\quad f : \Bbb R \to \Bbb R$$ $$\omega = f(||\mathbf x||)(\sum_{i=1}^n x_{i}dx_{i}) \in \mathcal A^1(\Bbb R^n) $$
$ (a) $ Assuming f is differentiable, prove that $d\omega = 0$ on $\Bbb R^n - \mathbf {0}$.
$(b) $ Assuming f is continuous, prove that $\omega$ is exact.
I'm not too sure if this is right, I'm mostly confused by the space $\mathcal A^1(\Bbb R^n)$ and how the partial derivatives will evaluate. Also once I show $d\omega = 0$, how do I show $\omega$ is exact ?
Edit: I'd like to understand the space $\mathcal A^1(\Bbb R^n)$ so I can show that it is simply connected.
Notice with your definition of $g$: $$ \partial_i g_j = x_j \partial_i f( ||x| |) = x_j f'( ||x|| ) \partial_i ||x||= \frac{ x_j x_i }{ ||x||} f'(||x||)$$ (remember $f:\mathbb{R} \to \mathbb{R}$)Thus $$ \partial_i g_j - \partial_j g_i = 0$$ thats the only step I'd say you missed to justify in a). For b), define $$ \eta = \int_{x_0}^x \omega $$ where the integral is any path from $x_0 \in \mathbb{R}^n$ to $x \in \mathbb{R}^n$. What is $d \eta$?