Poincaré Index of a Vector Field

Concept Overview

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.

Mathematical Definition

Let V(x, y) = (u(x, y), v(x, y)) be a continuous vector field, and let C be a simple closed curve that does not pass through any critical points (where V = 0). The index of V along C is given by the contour integral of the angle change divided by 2π.

I_C(V) = (1 / 2π) ∮_C dθ = (1 / 2π) ∮_C (u dv - v du) / (u² + v²)

The resulting index is always an integer. It depends only on the behavior of the field along the boundary of the loop. If the loop encloses no critical points (and the field is smooth), the index is 0. If it encloses multiple critical points, the index is the sum of their individual indices.

Worked Example

Consider the saddle field V(x, y) = (x, -y) and the unit circle C parameterized by (cos(t), sin(t)) for t from 0 to 2π.

Evaluate the field along the contour: u = cos(t) and v = -sin(t).
Compute the differential changes: du = -sin(t) dt and dv = -cos(t) dt.
Calculate the integrand: u dv - v du = cos(t)(-cos(t)) - (-sin(t))(-sin(t)) = -cos²(t) - sin²(t) = -1.
Integrate over the full turn: ∮_C dθ = ∫_0^2π (-1 / 1) dt = -2π.
Divide by 2π: The index is -2π / 2π = -1.

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 Saddle and ensure Center x is 0 and Radius is 1.
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 2 and Radius to 0.5. 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.

Limitations

This numerical integration computes the winding index strictly along the declared finite domain:

Boundary Criticality: If the contour passes exactly through a critical point (e.g., Center x = 1, Radius = 1 for the origin), the field magnitude vanishes. The index is undefined in such states.
Collapsed Contours: Setting the radius to 0 yields a collapsed point contour, where measuring a vector direction turn is impossible.
Approximation Constraints: The numerical calculation steps along the boundary with finite resolution. While theoretically exact integers, minor floating-point residuals are absorbed by tolerance checks.

References

Detailed theory on indices of planar vector fields and the Poincaré-Hopf theorem can be explored in this resource: Introduction to the Poincaré-Hopf Theorem (Michigan).

Poincaré Index of a Vector Field