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  "body": "\n## Definition\n\nThe cylindrical coordinate map is $C \\colon [0, \\infty) \\times \\mathbb{R} \\times \\mathbb{R} \\to \\mathbb{R}^3$,\n$$C(r, \\theta, z) = (r \\cos \\theta, \\ r \\sin \\theta, \\ z).$$\nA triple $(r, \\theta, z)$ with $C(r, \\theta, z) = (x, y, z)$ is a triple of cylindrical coordinates of $(x, y, z)$.\nHere $r = \\sqrt{x^2 + y^2}$ is the distance to the $z$-axis, $\\theta$ is the azimuth, and $z$ is the height.\n\n## Remarks\n\nThe pair $(r, \\theta)$ is a pair of [[polar coordinates|0RJZV05MJA]] of $(x, y)$. So, away from the $z$-axis, the coordinates are unique if $r \u003e 0$ and $\\theta \\in [0, 2\\pi)$.\nThe Jacobian determinant of $C$ is $r$, so the volume element is $dV = r \\, dr \\, d\\theta \\, dz$.\nExample: the solid cylinder $x^2 + y^2 \\le a^2$, $0 \\le z \\le h$ has volume $\\int_0^h \\int_0^{2\\pi} \\int_0^a r \\, dr \\, d\\theta \\, dz = \\pi a^2 h$.\nSome texts write $\\rho$ or $s$ for $r$, and $\\varphi$ for $\\theta$.\n\n## Explanations\n\n### L1 human\n\nCylindrical coordinates are polar coordinates in the floor plan, together with the height.\nSurfaces $r = $ constant are cylinders around the $z$-axis, which gives the coordinates their name.\n",
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