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  "body": "\n## Definition\n\nLet $S \\subseteq \\mathbb{R}^3$ be an [[oriented surface|0EKZW08FQ5]] with unit normal field $n$, and let $F$ be a continuous [[vector field|0N67NBKQ5T]] on $S$.\nThe flux of $F$ through $S$ is the [[surface integral|0DQ8ERJZNS]]\n$$\\iint_S F \\cdot n \\, dS.$$\n\n## Remarks\n\nIf $r \\colon D \\to \\mathbb{R}^3$ is a regular [[parametrisation|05788J51WZ]] of $S$ with $r_u \\times r_v$ pointing in the direction of $n$, then the flux is $\\iint_D F\\big( r(u, v) \\big) \\cdot \\big( r_u \\times r_v \\big) \\, du \\, dv$.\nReversing the orientation of $S$ changes the sign of the flux. The notation $\\iint_S F \\cdot d\\mathbf{S}$, with $d\\mathbf{S} = n \\, dS$, is common.\nExample: for $F(x, y, z) = (x, y, z)$ and the unit sphere with the outward normal $n(p) = p$, we get $F \\cdot n = 1$, so the flux is the area $4\\pi$.\nIf $F$ is the velocity field of a fluid, the flux is the volume of fluid that crosses $S$ per unit time. Flow in the direction of $n$ counts as positive.\n\n## Explanations\n\n### L1 human\n\nFlux measures how much of a field goes through a surface.\nOnly the part of the field perpendicular to the surface counts. Flow along the surface does not cross it.\n",
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