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  "body": "\n## Definition\n\nLet $S \\subseteq \\R^3$ be a [[regular surface|6080X3FAD4]], let $p \\in S$, and choose a smooth unit normal field $N$ near $p$.\nLet $S_p$ be the [[shape operator|DQCD11X1VM]] at $p$ and let $k_1, k_2$ be the [[principal curvatures|FA8QE1SGP4]] at $p$.\nThe *mean curvature* of $S$ at $p$ is\n$$H(p) = \\frac{k_1 + k_2}{2} = \\frac{1}{2} \\tr S_p.$$\n\n## Remarks\n\nIn a local parametrisation, with $E, F, G$ the coefficients of the [[first fundamental form|MAJ3DH7MZS]] and $e, f, g$ those of the [[second fundamental form|9KHQR16P1B]],\n$$H = \\frac{e G - 2 f F + g E}{2 (E G - F^2)}.$$\nExample: the cylinder of radius $r$ with the inner normal has $k_1 = 1/r$ and $k_2 = 0$, so $H = 1/(2r)$.\nThe saddle $z = xy$ has $k_1 = 1$ and $k_2 = -1$ at the origin, so $H = 0$ there.\nNon-example: replacing $N$ by $-N$ changes $H$ into $-H$, so on a surface without a global unit normal, such as a Möbius strip, only $\\abs{H}$ is defined globally.\nSome authors define $H$ as $k_1 + k_2$, without the factor $\\frac{1}{2}$.\n",
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