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Poincaré–Hopf theorem
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Poincaré–Hopf theorem (also known as the Poincaré–Hopf index formula, Poincaré–Hopf index theorem, or Hopf index theorem) is an important theorem that is used in <a href="/facts/Differential_topology/Y21eGQe0">differential topology</a>. It is named after <a href="/facts/Henri_Poincar%C3%A9/AazL2j4X">Henri Poincaré</a> and <a href="/facts/Heinz_Hopf/PMks10NV">Heinz Hopf</a>. </p><p>The Poincaré–Hopf theorem is often illustrated by the special case of the <a href="/facts/Hairy_ball_theorem/pSX6zDBz">hairy ball theorem</a>, which simply states that there is no smooth <a href="/facts/Vector_field/ccFNuPPp">vector field</a> on an even-dimensional <a href="/facts/N-sphere/bFu2S7E2">n-sphere</a> having no sources or sinks. </p>