This problem is from V.I Arnold's book Mathematics of Classical Mechanics. Q) Show that every differential 1-form on line is differential of some function
- Relevant equations The differential of any function is $$df_{x}(\psi): TM_{x} \rightarrow R$$
- The attempt at a solution
The tangent to line is line itself. The differential 1-form is $dy-dx=0$. Here I am struct. I don't know how to find out the differential. Can anyone help?
$dy-dx$ is not a differential form on $\mathbb{R}$, since $\mathbb{R}$ has only one coordinate for every chart you choose. It is instead a differential form on $\mathbb{R}^2$, where is the differential of $f(x,y)=y-x$.
About Arnold's exercise, I would insted prove it in this way: Let $\omega(x)dx $ be the general differential form on $\mathbb{R}$, where $\omega(x)$ is $C^{\infty}[\mathbb{R}]$. Then, let $f:=\int_0^x\omega(t)dt$. By definition of differential and by the fundamental theorem of calculus, we obtain that: $df=\partial_{x}( \int_0^x \omega(t)dt)dx=\omega(x)dx$ QED