How to prove that a scalar potential exists if the curl of the vector point function is zero?

[immortalsameer13]

scalar potential can be obtained by integrating the vector point function whose curl is zero but how to arrive at this result that a potential exist

 

[wrobel]

First show that if $\mathrm{rot}\, v=0$ then $\int_Cv_xdx+v_ydy+v_zdz=0$ for any closed curve $C$. To do that
consider a 2-dimensional surface $S$ such that $\partial S=C$

 

[FactChecker]

scalar potential can be obtained by integrating the vector point function whose curl is zero but how to arrive at this result that a potential exist

Because the closed curve integral is zero, the one-way integral from one point to another has only one answer no matter which path is taken. So the one-way integral gives you a well-defined definition of the potential.

ADDED: Establish a starting point, $p_0$, for the beginning of a path to any and all other points. The integral values from $p_0$ to the other points gives a well-defined potential at those points.