POINCARÉ–HOPF the zeros of any vector field on a surface must add up to its Euler characteristic
Put a vector field on a closed surface — wind on a globe, hair on a ball. It will have zeros (calm points, cowlicks), and each zero has an integer index (how the field spins around it: a source or sink is +1, a saddle −1). The Poincaré–Hopf theorem says the sum of all the indices equals the Euler characteristicχ — no matter which field you chose. On a sphere χ=2, so the indices must total 2; you cannot make them total zero, which is why you cannot comb a hairy ball flat — some cowlick (a zero) is forced. Change the field, move the zeros, split or merge them; the total is locked to the topology.
THE TECHNIQUE sum of vector-field indices = χ; sphere = 2 (hairy ball)
A vector field on the sphere with two sources (index +1 each). The demo sums the indices and recovers χ=2: live demo
HISTORY & CREDIT Poincaré 1885 · Hopf 1926
“Just comb the hair flat and there's no cowlick.” — on a sphere the indices must sum to 2, so a zero is unavoidable: at least one cowlick, always. The wind on Earth always has a calm point. cited
1885 · Henri Poincaré — the index sum for fields on surfaces. 1926 · Heinz Hopf — the general theorem in all dimensions: total index = χ. now · the hairy-ball theorem, Morse theory, the Euler class — topology forcing zeros to exist.
Whatever field you comb onto the surface, its zeros must bookkeep to χ. The topology does not care how you arrange the cowlicks, only that they sum correctly. Poincaré-Hopf
RECOMMEND FOR I-13 index sum = χ, computed
On the canonical compiler, two sources (index +1 each) sum to 2 = χ of the sphere — the hairy-ball total:
$ i13 run inv_poincarehopf.i13 # sum of field indices
total_index = 2
chi = 2 -- the indices are forced to total chi
Recommend: Poincaré–Hopf is invariance as a conservation law for zeros, and i13 enacts its bookkeeping directly: sum the integer indices, get χ. The “transformation” is any change of the vector field — move a cowlick, split a source into two, cancel a source against a saddle — and the total index does not budge from 2. It is the same integer-invariance as the degree (dart 283) and genus (dart 282), now phrased as a rule the singularities must obey. i13 adds the indices; topology guarantees the sum. A foreign shape — the Euler class — computed as a one-line total.