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  "body": "\n## Definition\n\nLet $U \\subseteq \\mathbb{R}^n$ be open, and let $\\omega$ be a [[differential k-form|0QHTX7WQEY]] on $U$ with $C^1$ coefficients, where $k \\ge 1$.\nThe form $\\omega$ is closed if $d\\omega = 0$, where $d$ is the [[exterior derivative|004FRCXH6M]].\nThe form $\\omega$ is exact if $\\omega = d\\eta$ for some $(k - 1)$-form $\\eta$ on $U$.\n\n## Remarks\n\nEach exact form $d\\eta$ with $\\eta$ of class $C^2$ is closed, because [[d d = 0|0X308R2DBE]].\nA [[one-form|0MTENFBW3Q]] $\\sum_i f_i \\, dx_i$ is closed if and only if $\\partial_j f_i = \\partial_i f_j$ for all $i, j$. It is exact if and only if $(f_1, \\dots, f_n)$ is a [[conservative|05R9NF2J49]] field. In $\\mathbb{R}^3$, closed means that the [[curl|0GDNHS9J2X]] is zero.\nThe angle form $d\\theta = \\frac{-y \\, dx + x \\, dy}{x^2 + y^2}$ on $\\mathbb{R}^2 \\setminus \\{0\\}$ is closed but not exact. Its integral around the unit circle is $2\\pi$, not $0$.\nOn a convex open set, each closed form is exact. This is the [[Poincaré lemma|05MC3HND79]].\n\n## Explanations\n\n### L1 human\n\nExact forms are derivatives of other forms, and closed forms only pass the local test for this: their derivative is zero.\nThe gap between the two notions is a measure of the holes of the domain.\n",
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