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  "body": "\n## Definition\n\nLet $U \\subseteq \\mathbb{R}^n$ be open, and let $0 \\le k \\le n$.\nTake a multi-index $I = (i_1, \\dots, i_k)$ with $1 \\le i_1 \u003c \\dots \u003c i_k \\le n$. Let $dx_I$ be the function of $k$ vectors $v_1, \\dots, v_k \\in \\mathbb{R}^n$ given by\n$$dx_I(v_1, \\dots, v_k) = \\det \\big[ (v_j)_{i_l} \\big]_{l, j = 1}^{k},$$\nthe determinant of a $k \\times k$ matrix. The entry of this matrix in row $l$ and column $j$ is the $i_l$-th component of $v_j$.\nA differential $k$-form on $U$ is a sum $\\omega = \\sum_I f_I \\, dx_I$ over these multi-indices, where each $f_I \\colon U \\to \\mathbb{R}$ is a function. At $x \\in U$, it gives the function $\\omega_x(v_1, \\dots, v_k) = \\sum_I f_I(x) \\, dx_I(v_1, \\dots, v_k)$.\n\n## Remarks\n\nThe form is of class $C^r$ if all $f_I$ are of [[class C^r|0TCBZMEYGN]]. A $0$-form is a function, and a [[1-form|0MTENFBW3Q]] is $\\sum_i f_i \\, dx_i$.\nEach $\\omega_x$ is multilinear and alternating: it changes sign when two of the vectors are swapped. The forms $dx_I$ give a basis of such functions, so there are $\\binom{n}{k}$ of them, and no nonzero forms of degree $k \u003e n$.\nIn the [[wedge product|0S166D363M]] notation, $dx_I = dx_{i_1} \\wedge \\dots \\wedge dx_{i_k}$. On $\\mathbb{R}^3$, $dx \\wedge dy \\wedge dz$ gives the signed volume of the parallelepiped spanned by three vectors.\nIn $\\mathbb{R}^3$, 1-forms and 2-forms correspond to vector fields, and 0-forms and 3-forms to functions. The [[exterior derivative|004FRCXH6M]] then corresponds to the gradient, curl and divergence.\n\n## Explanations\n\n### L1 human\n\nA $k$-form measures small $k$-dimensional pieces: a 1-form measures small arrows, a 2-form small parallelograms, and so on.\nIntegrating a $k$-form over a $k$-dimensional surface adds up these measurements over the whole surface.\n",
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