Poincaré Index of a Vector Field
Track vector directions around a loop and classify enclosed critical points.
Free theory articleTrack vector directions around a loop and classify enclosed critical points.
Concept & interpretation
The Poincaré index measures the total number of counterclockwise rotations a vector field makes as one traverses a simple closed curve in the counterclockwise direction. For a loop enclosing an isolated critical point, this topological invariant characterizes the nature of the point—distinguishing saddles, sources, sinks, and centers.
Experiment
You can observe the winding behavior of different vector fields interactively in the Poincaré Index of a Vector Field module:
- Set the Field to
Saddleand ensure Center x is0and Radius is1. - Observe the unwrapped Angle Trace graph. As the spatial angle traverses one positive turn, the field angle makes exactly one negative turn, yielding an index of -1.
- Change the Field to
Complex square. Notice the trace now climbs up to two full turns, giving an index of 2. - Change Center x to
2and Radius to0.5. Because the critical point at the origin is no longer enclosed, the trace returns to its starting angle with no net rotation, and the index is 0.