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  "body": "\n## Statement\n\nLet $S(\\rho, \\varphi, \\theta) = (\\rho \\sin \\varphi \\cos \\theta, \\ \\rho \\sin \\varphi \\sin \\theta, \\ \\rho \\cos \\varphi)$ be the [[spherical coordinate map|0RW8KPMXF6]].\nIts [[Jacobian determinant|03W6Q1MFR5]] is\n$$\\det DS(\\rho, \\varphi, \\theta) = \\rho^2 \\sin \\varphi.$$\nHence, for a continuous function $f$ on a region $E \\subseteq \\mathbb{R}^3$ described by a region $E'$ of $(\\rho, \\varphi, \\theta)$ with $\\rho \\ge 0$, $0 \\le \\varphi \\le \\pi$, $0 \\le \\theta \\le 2\\pi$,\n$$\\iiint_E f \\, dV = \\iiint_{E'} f(S(\\rho, \\varphi, \\theta)) \\, \\rho^2 \\sin \\varphi \\, d\\rho \\, d\\varphi \\, d\\theta.$$\n\n## Remarks\n\nThe Jacobian matrix has rows $(\\sin \\varphi \\cos \\theta, \\ \\rho \\cos \\varphi \\cos \\theta, \\ -\\rho \\sin \\varphi \\sin \\theta)$, $(\\sin \\varphi \\sin \\theta, \\ \\rho \\cos \\varphi \\sin \\theta, \\ \\rho \\sin \\varphi \\cos \\theta)$ and $(\\cos \\varphi, \\ -\\rho \\sin \\varphi, \\ 0)$. Expansion along the last row gives $\\rho^2 \\sin \\varphi$.\nThe integral formula follows from the [[change of variables theorem|00N5FHP1CV]]. The set where $S$ is not injective, or where the determinant vanishes, has volume zero.\nSince $0 \\le \\varphi \\le \\pi$, the factor $\\rho^2 \\sin \\varphi$ is non-negative, so no absolute value is needed.\nExample: the volume of the ball of radius $a$ is $\\int_0^{2\\pi} \\int_0^\\pi \\int_0^a \\rho^2 \\sin \\varphi \\, d\\rho \\, d\\varphi \\, d\\theta = \\tfrac{4}{3} \\pi a^3$.\n\n## Explanations\n\n### L1 human\n\nA small spherical box has sides $d\\rho$, $\\rho \\, d\\varphi$ and $\\rho \\sin \\varphi \\, d\\theta$.\nIts volume is about $\\rho^2 \\sin \\varphi \\, d\\rho \\, d\\varphi \\, d\\theta$, and this factor appears in integrals in spherical coordinates.\n",
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        "url": "https://en.wikipedia.org/wiki/Multiple_integral"
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        "locator": "Integration and differentiation in spherical coordinates",
        "url": "https://en.wikipedia.org/wiki/Spherical_coordinate_system"
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