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  "body": "\n## Statement\n\nLet $U \\subseteq \\mathbb{R}^n$ be open and convex, and let $\\omega = \\sum_i f_i \\, dx_i$ be a [[closed|0F1AVGX26N]] [[one-form|0MTENFBW3Q]] on $U$ with $C^1$ coefficients.\nThen $\\omega$ is exact. Explicitly, for a fixed $p \\in U$, the function\n$$g(x) = \\int_0^1 \\sum_{i=1}^n f_i\\big( p + t (x - p) \\big) \\, (x_i - p_i) \\, dt$$\nsatisfies $dg = \\omega$ on $U$.\n\n## Remarks\n\nProof: [[differentiate under the integral sign|0AR06W1E1B]], and use $\\partial_j f_i = \\partial_i f_j$. The integrand of $\\partial_j g$ becomes $\\frac{d}{dt} \\big( t \\, f_j(p + t (x - p)) \\big)$, which integrates to $f_j(x)$.\nIn $\\mathbb{R}^3$, this is the theorem that a [[curl-free field on a convex set is conservative|0B22P010D6]].\nThe lemma also holds for closed $k$-forms with $k \\ge 1$ on star-shaped open sets. For one-forms, it holds on [[simply connected|075C629XTH]] open sets.\nSome condition on $U$ is needed: the angle form on $\\mathbb{R}^2 \\setminus \\{0\\}$ is closed but not exact. See also the [[vortex field|0GRM3BS0DD]].\n\n## Explanations\n\n### L1 human\n\nOn a convex set, a one-form that passes the local test for being a derivative really is a derivative.\nThe proof builds the function by integrating the form along straight segments from one base point.\n",
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