I am studying differential forms from Arnold's book Mathematics of Classical Mechanics and have two difficulty with two problems on page 175:
1.Show that every differential 1-form (not necessarily closed)on line is differential of some function.
2.Find differential 1-forms(not necessarily closed) on the circle and the plane which are not the differential of any function.
Can anyone help me?
A differential $1$-form on the line is always closed. If $\alpha= f(x)dx$, it is the differential of $g(x)=\int_0^xf(t)dt$.
If $S^1$ is a circle, it is the quotient of $\mathbb{R}$ by the translation $t(x)=x+1$, consider the form $dx$ of $\mathbb{R}$ it is invariant by $t$ and induces a $1$-form on $S^1$ which is not exact, since there does not exist an element $x$ such that $\alpha_x=0$. If $\alpha=df$, $f$ has a maximum $x_0$ since $S^1$ is compact and $df_{x_0}=0$.
On $\mathbb{R}^2$, the form $xdy$ is not closed since its differential is $dx\wedge dy$.