Pullback of a volume form on $S^1$ to $\mathbb{R}$

Question

Let $F:\mathbb{R} \longrightarrow S^{1}$, $F(t)=(cost,sint)$ and $\omega$ a volume form on $S^{1}$. I want to know if $F^{*}\omega$ (pullback of $\omega$ by $F$) is a volume form too. Thanks for the help.

Answer 1

$F$ is a local diffeomorphism (here it is equivalent to saying that its differential does not vanish since $S^1$ and $\mathbb{R}$ are $1$-dimensional). This implies that the pullback of a volume form by $F$ is a volume form.