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

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A scalar potential exists if the curl of a vector point function is zero, as demonstrated by the closed curve integral being zero for any closed curve. This implies that the one-way integral between two points is path-independent, leading to a well-defined potential. By establishing a starting point, p_0, the integral values from this point to any other points can be calculated, confirming the existence of a scalar potential. The process relies on the properties of vector fields and their integrals over surfaces. Thus, the conditions for a scalar potential are satisfied when the curl of the vector field is zero.
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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
 
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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
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immortalsameer13 said:
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.
 
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