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  "body": "\n## Statement\n\nLet $C$ be a positively oriented, piecewise smooth, simple [[closed curve|0D032NSZKY]] in $\\mathbb{R}^2$, and let $D$ be the bounded region enclosed by $C$.\nLet $L$ and $M$ be real functions with continuous [[partial derivatives|05AXG3QRKR]] on an open set that contains $D \\cup C$.\nThen\n$$\\oint_C (L \\, dx + M \\, dy) = \\iint_D \\left( \\frac{\\partial M}{\\partial x} - \\frac{\\partial L}{\\partial y} \\right) dA.$$\n\n## Remarks\n\nPositive orientation means that $C$ goes counterclockwise, so $D$ is on the left.\nProof sketch: first let $D$ be a region between the graphs of two functions of $x$. Write the [[double integral|0NKBMFC18R]] of $\\partial L / \\partial y$ as an iterated integral, and use the fundamental theorem of calculus. This gives $\\oint_C L \\, dx = -\\iint_D \\partial L / \\partial y \\, dA$. The $M$ term is similar, and general regions are cut into such pieces.\nExample: with $L = -y$ and $M = x$, the right side is $2$ times the [[area of D|09CBQB6V67]].\nGreen's theorem is the planar case of [[Stokes' theorem|0AKRKVHW94]]. George Green stated a similar result in 1828, and Riemann gave a proof in 1851.\n\n## Explanations\n\n### L1 human\n\nThe theorem says that the total circulation of a plane field around a closed curve equals the sum of the small circulations inside.\nThe small circulations of neighbouring squares cancel along shared edges, so only the outer boundary is left.\n",
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