I've been learning about line integrals in vector fields and am a little confused about singularities.
From my understanding, a closed line integral in a conservative field is always $0$ unless it goes around a singularity. In addition, going around a singularity accounts for a certain change in the value of the line integral.
My question is: What does the value added by integrating around a singularity represent? I am assuming it represents something because going around the singularity with any path results in the same value. Is there a physical meaning for what the value represents?