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  "body": "\n## Statement\n\nLet $D \\subseteq \\mathbb{R}^2$ be the compact region bounded by a piecewise smooth simple closed curve $\\gamma$, oriented counterclockwise.\nLet $r$ be an injective regular [[parametrised surface|05788J51WZ]] of class $C^2$ on an open set that contains $D$. Let $S = r(D)$ have the unit normal $n = (r_u \\times r_v) / \\| r_u \\times r_v \\|$, and let $\\partial S$ be the closed curve $r \\circ \\gamma$.\nIf $F$ is a vector field of class $C^1$ on an open set that contains $S$, then\n$$\\oint_{\\partial S} F \\cdot d\\mathbf{r} = \\iint_S (\\nabla \\times F) \\cdot n \\, dS.$$\n\n## Remarks\n\nThe left side is the circulation of $F$ around the boundary. The right side is the [[flux|0YPRY7T4PK]] of the [[curl|0GDNHS9J2X]] of $F$ through $S$.\nThe orientations match by the right-hand rule: with the thumb along $n$, the fingers curl in the direction of $\\partial S$.\nExample: for the upper unit hemisphere and $F = (-y, x, 0)$, $\\nabla \\times F = (0, 0, 2)$, and both sides equal $2\\pi$.\nFor a flat surface in the $xy$-plane, the theorem reduces to [[Green's theorem|0HR3N02317]]. The proof pulls the integrals back to $D$ and applies Green's theorem there.\n\n## History\n\nThe theorem first appears in a letter from William Thomson to George Stokes in 1850.\nStokes set it as a question on the Smith's Prize examination at Cambridge in 1854, and the result then got his name.\n\n## Explanations\n\n### L1 human\n\nThe total swirl of a field over a surface equals the circulation of the field around the edge of the surface.\nThe surface can be any shape: only its boundary curve matters.\n",
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