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  "body": "\n## Definition\n\nThe polar coordinate map is $P \\colon [0, \\infty) \\times \\mathbb{R} \\to \\mathbb{R}^2$, $P(r, \\theta) = (r \\cos \\theta, r \\sin \\theta)$.\nA pair $(r, \\theta)$ with $P(r, \\theta) = (x, y)$ is a pair of polar coordinates of the point $(x, y)$. Here $r = \\sqrt{x^2 + y^2}$ is the radial coordinate and $\\theta$ is the angular coordinate.\n\n## Remarks\n\nEach point other than the origin has exactly one pair of polar coordinates with $r \u003e 0$ and $\\theta \\in [0, 2\\pi)$. The origin has $r = 0$ and any $\\theta$.\nFor $x \u003e 0$, $\\theta = \\arctan(y / x)$ is one valid angle. In general one uses the two-argument function $\\operatorname{atan2}(y, x)$, with values in $(-\\pi, \\pi]$.\nThe restriction of $P$ to $(0, \\infty) \\times (0, 2\\pi)$ is a $C^\\infty$ [[diffeomorphism|0Z1Y7JHGBR]] onto the plane without the half-line $\\{(x, 0) : x \\ge 0\\}$. Its Jacobian determinant is $r$.\nThe circle of radius $a$ about the origin is $r = a$, and the ray at angle $\\alpha$ is $\\theta = \\alpha$.\n\n## Explanations\n\n### L1 human\n\nInstead of moving $x$ units right and $y$ units up, you turn by the angle $\\theta$ and then walk the distance $r$.\nBoth descriptions name the same point.\n",
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