this is my first question here and i hope you could help me with this one:
Let $ F: \Bbb{R}^d\setminus\{0\} \rightarrow \Bbb{R}^d $ be a continous vector field with \begin{equation} F(x)=\varphi(\|x\|^2)x \end{equation} for a continous $ \varphi:(0,\infty) \rightarrow \Bbb{R}$. Show that the one-form $\omega^F(p)(v):=\langle F(p),v\rangle$, got an antiderivative.
I have no idea. Do i have to show that $\Bbb{R}^d\setminus\{0\} $ is star-like and that $F$ is closed?
HINT: Well, first of all $\phi$ is only continuous so you won't be able to differentiate $\omega$ (and once you pull out the origin, the space is no longer starlike). So you really want to try to construct a function $f$ directly with $df(p)(v) = \langle F(p),v\rangle$. That is, you want $$df = \varphi(\|x\|)^2\sum\limits_{i=1}^d x_i\,dx_i.$$ (By the way, $d$ is an awkward letter to use for the dimension. :)) So look for $f(x) = g(\|x\|^2)$ for some appropriate differentiable function $g$. (It might help to look for $g(u)$, where ultimately $u=\|x\|^2$.)