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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 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 *Gaussian curvature* of $S$ at $p$ is\n$$K(p) = \\det S_p = k_1 k_2.$$\nIt does not depend on the choice of the unit normal.\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$$K = \\frac{e g - f^2}{E G - F^2}.$$\nExample: on the sphere of radius $r$, $S_p = \\pm \\frac{1}{r} \\id$, so $K = 1/r^2$.\nOn a cylinder $k_1 = 1/r$ and $k_2 = 0$, so $K = 0$; on the saddle $z = xy$ at the origin $k_1 = 1$ and $k_2 = -1$, so $K = -1$.\nNon-example: on that saddle, the normal curvatures in the orthogonal directions $(1, 0, 0)$ and $(0, 1, 0)$ are both $0$.\nTheir product $0$ is not $K = -1$, because these directions are not principal directions.\nFor the graph $z = h(x, y)$ of a smooth function, $K = (h_{xx} h_{yy} - h_{xy}^2) / (1 + h_x^2 + h_y^2)^2$.\n",
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