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  "body": "\n## Statement\n\nLet $M$ be a compact [[smooth manifold|F45VQ47GPV]] of dimension $n$ without boundary.\nLet $v$ be a continuous [[vector field|0299VCW1TH]] on $M$ with only isolated zeros.\nFor a zero $x$ of $v$, choose a chart around $x$ and a small sphere $S$ around $x$ in it that encloses no other zero.\nThe *index* $\\operatorname{ind}_x(v)$ is the [[degree|TMFG2P912R]] of the map $S \\to S^{n-1}$, $u \\mapsto v(u) / \\norm{v(u)}$.\nThen $v$ has finitely many zeros, and\n$$\\sum_{v(x) = 0} \\operatorname{ind}_x(v) = \\chi(M),$$\nthe [[Euler characteristic|RNDDWDAFJ5]] of $M$.\n\n## Remarks\n\nThe index does not depend on the chart.\nOne proof first shows that the sum of indices does not depend on $v$.\nFor this it compares $v$ with a field on a tubular neighbourhood of $M$ in $\\R^N$ and uses the degree of the Gauss map.\nThen one computes the sum for the gradient of a Morse function, whose zeros are the critical points.\nThe index at a critical point of Morse index $\\lambda$ is $(-1)^\\lambda$, and the sum of these signs is $\\chi(M)$ by Morse theory.\nFor a manifold with boundary the same formula holds if $v$ points outward along the boundary.\n\nExample: on the sphere $S^2$, the gradient of the height function vanishes only at the two poles, a source and a sink, each of index $+1$, and $1 + 1 = 2 = \\chi(S^2)$.\nOn the torus, the field tangent to one family of circles has no zero, and $\\chi(T^2) = 0$.\n\nCompactness is needed: the constant field on $\\R^n$ has no zero, but $\\chi(\\R^n) = 1$.\n\n## Explanations\n\n### L1 human\n\nCount the points where a flow on a closed surface stops, giving each one a sign or multiplicity according to how the flow turns around it.\nThe total does not depend on the flow; it is the Euler characteristic, a number that only depends on the shape of the surface.\nOn a sphere the total is $2$, so every flow on a sphere must stop somewhere.\n\n### L2 human\n\nThe theorem links analysis, through vector fields, with topology, through the Euler characteristic.\nIt gives the [[hairy ball theorem|Z49RAKVCWG]] at once.\nIt shows that a closed manifold has a nowhere vanishing vector field only if $\\chi(M) = 0$, and Hopf proved the converse for connected closed manifolds.\nIt can be derived from the Lefschetz fixed-point theorem applied to the flow of $v$.\nIt is closely related to the Gauss–Bonnet theorem and to the Euler class of the tangent bundle.\n",
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