Is Every Differential 1-Form on a Line the Differential of Some Function?

[Abhishek11235]

Homework Statement

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

Homework 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?

 

[Orodruin]

The tangent to line is line itself.

No.

The differential 1-form is dy−dx=0dy−dx=0dy-dx=0.

No. A general differential 1-form is of the form $\omega = g(x) dx$. You have to show that there exists a function $G(x)$ such that $\omega = dG$.

 

[Abhishek11235]

No.No. A general differential 1-form is of the form $\omega = g(x) dx$. You have to show that there exists a function $G(x)$ such that $\omega = dG$.

That means I have to integrate. Can you provide me solution?

 

[Orodruin]

Can you provide me solution?

That would violate the forum rules, which you would realize if you had bothered reading them.