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  "body": "\n## Definition\n\nLet $S \\subseteq \\R^3$ be a [[regular surface|6080X3FAD4]], let $x \\colon U \\to S$ be a local parametrisation, and let $N = x_u \\times x_v / \\norm{x_u \\times x_v}$ be the unit [[normal vector|0BJDQ2HBH1]].\nThe coefficients of the *second fundamental form* of $S$ with respect to $x$ are\n$$e = \\ip{x_{uu}}{N}, \\qquad f = \\ip{x_{uv}}{N}, \\qquad g = \\ip{x_{vv}}{N}.$$\nAt $p = x(q)$, the second fundamental form is the quadratic form $I\\!I_p$ on the tangent plane $T_p S$ with $I\\!I_p(a x_u + b x_v) = e a^2 + 2 f ab + g b^2$.\n\n## Remarks\n\nOne writes $I\\!I = e \\, du^2 + 2 f \\, du \\, dv + g \\, dv^2$.\nEquivalently, $I\\!I_p(w) = -\\ip{dN_p(w)}{w}$, where $dN_p$ is the differential of the unit normal field; this shows that $I\\!I_p$ depends only on $S$ and the choice of $N$.\nReplacing $N$ by $-N$ changes the sign of $I\\!I$.\nExample: for the cylinder $x(\\theta, z) = (r \\cos \\theta, r \\sin \\theta, z)$ with the inner normal $N = -(\\cos \\theta, \\sin \\theta, 0)$, one gets $e = r$, $f = 0$ and $g = 0$.\nFor the plane, $x_{uu}$, $x_{uv}$ and $x_{vv}$ vanish for a linear parametrisation, so $I\\!I = 0$.\nNon-example: on a Möbius strip there is no continuous unit normal on the whole surface, so the second fundamental form is defined only locally, up to sign.\nFor $w \\in T_p S$ with $I_p(w) = 1$, the number $I\\!I_p(w)$ is the normal curvature of $S$ in the direction $w$, where $I_p$ is the [[first fundamental form|MAJ3DH7MZS]].\n",
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