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  "body": "\n## Definition\n\nLet $S \\subseteq \\R^3$ be a [[regular surface|6080X3FAD4]] and let $p \\in S$.\nLet $T_p S \\subseteq \\R^3$ be the [[tangent plane|0BJDQ2HBH1]] of $S$ at $p$, viewed as a $2$-dimensional linear subspace.\nThe *first fundamental form* of $S$ at $p$ is the [[quadratic form|0HZNE5QZDJ]] $I_p \\colon T_p S \\to \\R$, $I_p(w) = \\ip{w}{w}$, where $\\ip{\\cdot}{\\cdot}$ is the [[dot product|05HQYM0R4G]] of $\\R^3$.\nIn a local parametrisation $x(u, v)$ of $S$, its coefficients are\n$$E = \\ip{x_u}{x_u}, \\qquad F = \\ip{x_u}{x_v}, \\qquad G = \\ip{x_v}{x_v}.$$\nFor $w = a x_u + b x_v$, this gives $I_p(w) = E a^2 + 2F ab + G b^2$.\n\n## Remarks\n\nOne writes $I = E \\, du^2 + 2F \\, du \\, dv + G \\, dv^2$.\nThe length of a curve $t \\mapsto x(u(t), v(t))$, $t \\in [a, b]$, is $\\int_a^b \\sqrt{E u'^2 + 2F u' v' + G v'^2} \\, dt$, and the area of $x(R)$ is $\\iint_R \\sqrt{EG - F^2} \\, du \\, dv$.\nExample: for the unit sphere with $x(\\theta, \\varphi) = (\\sin \\theta \\cos \\varphi, \\sin \\theta \\sin \\varphi, \\cos \\theta)$, one gets $E = 1$, $F = 0$ and $G = \\sin^2 \\theta$.\nFor $0 \u003c \\theta \u003c \\pi$ the form is positive definite, as it must be, since $EG - F^2 = \\sin^2 \\theta \u003e 0$.\nNon-example: at $\\theta = 0$ the same formulas give $G = 0$, a degenerate form; the map is not a local parametrisation there, because $x_\\varphi = 0$.\n",
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