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  "body": "\n## Definition\n\nLet $S \\subseteq \\R^3$ be a subset.\nThe set $S$ is a *regular surface* if each $p \\in S$ has the following property.\nThere are an [[open set|01ZQQDCERP]] $V \\subseteq \\R^3$ containing $p$, an open set $U \\subseteq \\R^2$, and a map $x \\colon U \\to V \\cap S$ such that:\n\n1. The map $x$ is [[smooth|0BE3QDH73A]] as a map from $U$ to $\\R^3$.\n2. The map $x$ is a [[homeomorphism|05GZ11PWRY]] from $U$ onto $V \\cap S$, with the subspace topology on $V \\cap S$.\n3. For each $q \\in U$, the [[Jacobian matrix|0YF0VXKXGB]] of $x$ at $q$ has rank $2$, that is, $x_u(q)$ and $x_v(q)$ are linearly independent.\n\nSuch a map $x$ is a *local parametrisation* of $S$ at $p$.\n\n## Remarks\n\nExample: the unit sphere is a regular surface.\nAt points with $z \u003e 0$, the map $x(u, v) = (u, v, \\sqrt{1 - u^2 - v^2})$ on the open unit disc satisfies the three clauses, and five similar maps cover the rest.\nMore generally, if $f \\colon \\R^3 \\to \\R$ is smooth and $\\nabla f \\ne 0$ on $f^{-1}(c)$, then the level set $f^{-1}(c)$, when nonempty, is a regular surface.\nNon-example: the double cone $x^2 + y^2 = z^2$ fails clause 2 at the origin.\nIndeed, removing the origin from a small neighbourhood of it in the cone leaves two pieces, while a punctured disc is connected.\nEach local parametrisation is a regular [[parametrised surface|05788J51WZ]] in the sense of a map, but a regular surface is a set, covered by such maps.\n",
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