Poincaré Index Theorem: Vector Fields on a Disk (2D)

Field Distortion

Twist A 0.00

Arrow density 18

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Singularities inside disk

Max field speed

1.00

Sum of interior indices

Boundary winding number

Index sum ≟ winding number (Poincaré index theorem)

MATCH ✓

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How it works

On a sphere the Hairy Ball theorem forces at least one cowlick because the sphere has no boundary. On a flat disk the boundary changes the rule: a tangent field can be completely zero-free everywhere inside — but only if it isn't allowed to point any particular way on the rim. The precise statement is the Poincaré index theorem (a 2D case of Poincaré–Hopf with boundary):

Σ index(zeros inside C) = winding number of V along C

The field here is the gradient of a height function, V(x,y) = ∇H(x,y), with a cubic "twist" term added to a plain bowl:

H(x,y) = ½(x² + y²) + (A/3)(x³ − 3xy²)
V = ∇H = ( x + A(x² − y²),  y − 2Axy )

At A = 0 the bowl has one zero — the origin, a sink with index +1 — and the field winds exactly once counter-clockwise around any circle enclosing it, so both sides of the theorem read +1. This H has three more critical points at radius 1/|A| for any A ≠ 0 (a saddle at angle 180° and two more at ±60°, all found in closed form by solving ∇H = 0): as Twist A grows, 1/|A| shrinks and those three points sweep in from outside the fixed boundary circle to inside it.

Each critical point's type is read directly from its 2×2 Hessian, exactly as on the sphere:

index = +1  if det(Hessian) > 0   (local max or min)
index = −1  if det(Hessian) < 0   (a saddle)

All three incoming points are saddles (index −1), so the moment they cross the boundary the interior sum jumps from +1 to +1−1−1−1 = −2. That is not a violation — the boundary winding number, measured completely independently by walking the field's direction all the way around the fixed circle and summing the wrapped angle change, jumps by exactly the same amount at exactly the same instant. Nothing is fudged: the two numbers are computed by two different numerical methods (grid search + Newton's method on the interior vs. a direct angle integral on the boundary) and are asserted to agree live.

Field (arrows), boundary circle Index +1 (source/sink) Index −1 (saddle)