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  "body": "\n## Definition\n\nLet $D \\subseteq \\mathbb{R}^2$ be a set that is the closure of an open set.\nA parametrised surface is a map $r \\colon D \\to \\mathbb{R}^3$, $r(u, v) = \\big( x(u, v), y(u, v), z(u, v) \\big)$, of [[class C^1|0TCBZMEYGN]] on an open set that contains $D$.\nIts image $S = r(D)$ is the surface that $r$ parametrises.\nThe parametrisation is regular at $(u, v)$ if the [[partial derivatives|05AXG3QRKR]] $r_u = \\partial r / \\partial u$ and $r_v = \\partial r / \\partial v$ are linearly independent, that is, $r_u \\times r_v \\ne 0$.\n\n## Remarks\n\nThe graph of a function $f \\colon D \\to \\mathbb{R}$ is parametrised by $r(x, y) = (x, y, f(x, y))$. Here $r_x \\times r_y = (-f_x, -f_y, 1) \\ne 0$, so this parametrisation is regular.\nThe unit sphere is parametrised by $r(\\varphi, \\theta) = (\\sin \\varphi \\cos \\theta, \\sin \\varphi \\sin \\theta, \\cos \\varphi)$ on $[0, \\pi] \\times [0, 2\\pi]$. It is not regular at the poles $\\varphi = 0$ and $\\varphi = \\pi$.\nThe cylinder of radius $R \u003e 0$ about the $z$-axis is parametrised by $r(\\theta, z) = (R \\cos \\theta, R \\sin \\theta, z)$.\nOne surface has many parametrisations. Areas and surface integrals do not depend on the choice.\n\n## Explanations\n\n### L1 human\n\nA parametrised surface is to a surface what a [[parametrised curve|0JF1XXGWB4]] is to a curve, with two parameters instead of one.\nThink of $(u, v)$ as coordinates on a flat sheet, and of $r$ as a map that bends the sheet into space.\n",
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