Mechanical Model of Simple Phase Singularity
Imagine a plane. Pick an origin O. Use polar coordinates, (r,theta) for arbitrary moving point P.
Pick a point 0' with fixed coordinates (a, chi).
Draw a circle of radius b < a centered at O' with coordinates (b,phi)
Let point P move around this circle whose center O' is displaced from origin O.
Obviously when a =/= 0 the total theta angle integral of the 1-form dtheta
swept out in one complete circuit round the circle is ZERO. Basically theta oscillates.
Note that the angle theta depends on the angles chi and phi.
Half of the movement is clockwise and then counter-clockwise for dtheta on successive semi-circles as P moves around the circumference of the displaced circle. This is most easily seen intuitively all at once when O' is vertical compared to O (on y-axis ordinate).
Note what happens when you move the circle to different locations on the plane.
Draw tangents from O to the circle in different locations.
In contrast, when a = 0, or alternatively, b > a the total angle integral of dtheta is 2pi.
Use trigonometry to make an algebraic proof.