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  "body": "\n## Definition\n\nLet $U \\subseteq \\R^n$ be an [[open set|02BH3MCBQB]] and let $0 \\le m \\le n$.\nLet $\\mathcal{D}^m(U)$ be the real vector space of smooth [[differential $m$-forms|0E3G0GRPZ6]] on $U$ with compact support in $U$.\nA sequence $(\\omega_k)$ in $\\mathcal{D}^m(U)$ *converges to $0$* if two conditions hold.\n\n1. There is a compact set $K \\subseteq U$ that contains the support of every $\\omega_k$.\n2. Every partial derivative of every coefficient of $\\omega_k$ converges to $0$ uniformly on $K$.\n\nAn *$m$-dimensional current* on $U$ is a linear map $T \\colon \\mathcal{D}^m(U) \\to \\R$ such that $T(\\omega_k) \\to 0$ whenever $\\omega_k \\to 0$ in this sense.\nThe *boundary* of an $m$-current $T$ with $m \\ge 1$ is the $(m-1)$-current $\\partial T$ given by $\\partial T(\\omega) = T(d\\omega)$.\n\n## Remarks\n\nExample: let $S \\subseteq U$ be a compact oriented $m$-dimensional $C^1$ submanifold, possibly with boundary.\nThen $[\\![S]\\!](\\omega) = \\int_S \\omega$ is linear in $\\omega$.\nIt is continuous, since $\\abs{\\int_S \\omega} \\le \\operatorname{area}(S) \\cdot \\sup_S \\abs{\\omega}$.\nSo $[\\![S]\\!]$ is an $m$-current, and Stokes' theorem gives $\\partial [\\![S]\\!] = [\\![\\partial S]\\!]$.\nNon-example: the map $\\omega \\mapsto \\abs{\\int_S \\omega}$ is continuous but not linear, so it is not a current.\nThe $0$-currents on $U$ are the distributions on $U$.\nThe *mass* of $T$ is the supremum of $T(\\omega)$ over the forms $\\omega$ with $\\abs{\\omega(x)} \\le 1$ for all $x$, for a suitable norm on $m$-covectors.\n",
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