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  "body": "\n## Statement\n\nLet $R = (\\alpha, \\beta) \\times (\\gamma, \\delta)$ be an open rectangle in $\\mathbb{R}^2$.\nLet $M, N : R \\to \\mathbb{R}$ be continuous functions such that the partial derivatives $\\partial M / \\partial y$ and $\\partial N / \\partial x$ exist and are continuous on $R$.\nThen the equation $M\\, dx + N\\, dy = 0$ is [[exact|0EC7AGJ0PX]] on $R$ if and only if\n$$\\frac{\\partial M}{\\partial y} = \\frac{\\partial N}{\\partial x} \\quad \\text{on } R.$$\n\n## Remarks\n\nThe condition is necessary on each open set: if $\\psi_x = M$ and $\\psi_y = N$, the mixed partial derivatives of $\\psi$ are equal, so $M_y = N_x$.\nFor the converse, fix $(x_0, y_0) \\in R$ and set $\\psi(x, y) = \\int_{x_0}^{x} M(s, y_0)\\, ds + \\int_{y_0}^{y} N(x, t)\\, dt$.\nThe converse holds on each simply connected open set. It can fail on other sets: $(-y\\, dx + x\\, dy)/(x^2 + y^2) = 0$ satisfies the condition on $\\mathbb{R}^2 \\setminus \\{0\\}$, but it is not exact there.\n\n## Explanations\n\n### L1 human\n\nIf $\\psi$ exists, then $M$ and $N$ are its two partial derivatives.\nTheir cross derivatives must agree, because the order of differentiation does not matter for nice functions.\nOn a rectangle, this easy test is also enough.\n",
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