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  "aliases": [
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  "body": "\n## Statement\n\nLet $U \\subseteq \\mathbb{R}^n$ be open, and let $\\omega$ be a differential $k$-form on $U$ with coefficients of class $C^2$.\nThen\n$$d(d\\omega) = 0.$$\n\n## Remarks\n\nProof: for $\\omega = f \\, dx_I$, the [[exterior derivative|004FRCXH6M]] gives $d(d\\omega) = \\sum_{j, l} \\partial_l \\partial_j f \\, dx_l \\wedge dx_j \\wedge dx_I$.\nBy [[Schwarz's theorem|0NNQY8WCW6]], $\\partial_l \\partial_j f = \\partial_j \\partial_l f$, while $dx_l \\wedge dx_j = -dx_j \\wedge dx_l$. So the terms cancel in pairs, and the terms with $j = l$ are zero.\nIn $\\mathbb{R}^3$, the cases $k = 0$ and $k = 1$ are the identities [[curl of a gradient is zero|0M2JRW1J0N]] and [[divergence of a curl is zero|0F56CMMC5P]].\nIt follows that each [[exact form|0F1AVGX26N]] with $C^2$ coefficients is closed.\n\n## Explanations\n\n### L1 human\n\nTaking the exterior derivative twice always gives zero, because mixed second partial derivatives are equal while the wedge product changes sign.\nThis one rule contains the facts that gradients have no curl and curls have no divergence.\n",
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        "locator": "In terms of axioms",
        "url": "https://en.wikipedia.org/wiki/Exterior_derivative"
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        "locator": "Closed and exact forms",
        "url": "https://en.wikipedia.org/wiki/Exterior_derivative"
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        "locator": "Curl of gradient is zero",
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  "name": "The exterior derivative squares to zero",
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