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  "body": "\n## Definition\n\nLet $S \\subseteq \\R^3$ be a [[regular surface|6080X3FAD4]], let $W \\subseteq S$ be open in $S$, and let $N \\colon W \\to \\R^3$ be a smooth unit [[normal vector|0BJDQ2HBH1]] field.\nFor $p \\in W$, the differential $dN_p$ maps the tangent plane $T_p S$ into itself, because $\\ip{N}{N} = 1$.\nThe *shape operator* of $S$ at $p$ is the linear map\n$$S_p = -dN_p \\colon T_p S \\to T_p S.$$\n\n## Remarks\n\nThe map $S_p$ is [[self-adjoint|0MB8HQW495]] for the inner product of $T_p S$, and $\\ip{S_p v}{w} = I\\!I_p(v, w)$, where $I\\!I_p$ is the polar form of the [[second fundamental form|9KHQR16P1B]].\nIn a local parametrisation, the matrix of $S_p$ in the basis $x_u, x_v$ is\n$$\\begin{pmatrix} E \u0026 F \\\\ F \u0026 G \\end{pmatrix}^{-1} \\begin{pmatrix} e \u0026 f \\\\ f \u0026 g \\end{pmatrix},$$\nwith $E, F, G$ from the [[first fundamental form|MAJ3DH7MZS]] and $e, f, g$ from the second.\nExample: on the sphere of radius $r$ centred at $0$ with the outer normal $N(p) = p / r$, one has $dN_p = \\frac{1}{r} \\id$, so $S_p = -\\frac{1}{r} \\id$.\nWith the inner normal, $S_p = \\frac{1}{r} \\id$.\nNon-example: on a Möbius strip there is no global continuous unit normal, so the shape operator is defined only locally, up to sign.\n",
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