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.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
The curl of a vector field measures the tendency of the field to induce rotation around a point. If the curl of a vector field is zero, it indicates that the field is irrotational, meaning there is no local rotation at any point in the field. This is a necessary condition for the existence of a scalar potential function.
A scalar potential is a scalar function whose gradient gives rise to a vector field. In physics, scalar potentials are important because they simplify the analysis of conservative forces, such as gravitational and electrostatic forces. The existence of a scalar potential allows one to derive the properties of the vector field easily and compute work done along a path.
To determine if a scalar potential exists for a vector field with zero curl, we can use the fact that if the curl of a vector field is zero in a simply connected domain, then the vector field can be expressed as the gradient of a scalar potential. This can be shown through the application of Stokes' theorem and the fundamental theorem of line integrals.
A simply connected domain is one that is path-connected and contains no holes. The significance of this condition is that it ensures any closed loop can be continuously contracted to a point without leaving the domain. In such domains, a vector field with zero curl can be shown to have a scalar potential, as there are no obstructions that could prevent the existence of such a potential.
Yes, a vector field can have zero curl but not possess a scalar potential if the domain is not simply connected. In such cases, even though the curl is zero, there may be topological constraints (like holes) that prevent the existence of a single-valued scalar potential function throughout the entire domain.