In Fluid Mechanics, you can define a velocity potential as a closed line integral in a vector field, which equals the sum of curl (vorticity) inside the line over which you are integrating (by Stokes' Theorem).
But if there's a singularity (or it's a non-simply connected domain), you get that the velocity potential is (can be?) multi-valued.
Is there a way to understand how this happens without Complex Analysis? How does the 'singularity' inside the domain make the line integral multi-valued? Is there something in Stokes' theorem that I am skimming over?