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  "body": "\n## Definition\n\nLet $U \\subseteq \\mathbb{R}^n$ be open, and let $\\omega = \\sum_I f_I \\, dx_I$ be a [[differential k-form|0QHTX7WQEY]] on $U$ whose coefficients $f_I$ are of class $C^1$.\nThe exterior derivative of $\\omega$ is the $(k + 1)$-form\n$$d\\omega = \\sum_I \\sum_{j=1}^n \\frac{\\partial f_I}{\\partial x_j} \\, dx_j \\wedge dx_I,$$\nwhere $\\wedge$ is the [[wedge product|0S166D363M]].\n\n## Remarks\n\nFor a function $f$, $df = \\sum_j \\partial_j f \\, dx_j$ is its differential, the [[one-form|0MTENFBW3Q]] that corresponds to the [[gradient|0NS2QYVC4Y]].\nThe operator $d$ is linear and satisfies $d(\\alpha \\wedge \\beta) = d\\alpha \\wedge \\beta + (-1)^k \\alpha \\wedge d\\beta$ for a $k$-form $\\alpha$. For $C^2$ forms, [[d d = 0|0X308R2DBE]].\nIn the plane, $d(L \\, dx + M \\, dy) = (\\partial_x M - \\partial_y L) \\, dx \\wedge dy$.\nIn $\\mathbb{R}^3$, $d(F_1 \\, dx + F_2 \\, dy + F_3 \\, dz)$ has the components of the [[curl|0GDNHS9J2X]] of $F$ as coefficients of $dy \\wedge dz$, $dz \\wedge dx$ and $dx \\wedge dy$. Also $d(F_1 \\, dy \\wedge dz + F_2 \\, dz \\wedge dx + F_3 \\, dx \\wedge dy) = (\\nabla \\cdot F) \\, dx \\wedge dy \\wedge dz$, which is the [[divergence|02NSMQVQWK]].\n\n## Explanations\n\n### L1 human\n\nThe exterior derivative is one operation that does the work of gradient, curl and divergence together.\nYou differentiate each coefficient in each direction and attach the new direction with a wedge product.\n",
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        "locator": "In terms of axioms",
        "url": "https://en.wikipedia.org/wiki/Exterior_derivative"
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