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  "body": "\n## Definition\n\nLet $U \\subseteq \\mathbb{R}^n$ be open. For indices $j_1, \\dots, j_m \\in \\{1, \\dots, n\\}$, let $dx_{j_1} \\wedge \\dots \\wedge dx_{j_m}$ be $0$ if two indices are equal. Otherwise it is $\\operatorname{sgn}(\\sigma) \\, dx_I$, where $I$ is the increasing rearrangement of the indices and $\\sigma$ is the permutation that sorts them.\nFor [[differential forms|0QHTX7WQEY]] $\\alpha = \\sum_I f_I \\, dx_I$ of degree $k$ and $\\beta = \\sum_J g_J \\, dx_J$ of degree $l$ on $U$, the wedge product is the $(k + l)$-form\n$$\\alpha \\wedge \\beta = \\sum_{I, J} f_I \\, g_J \\; dx_{i_1} \\wedge \\dots \\wedge dx_{i_k} \\wedge dx_{j_1} \\wedge \\dots \\wedge dx_{j_l}.$$\n\n## Remarks\n\nThe product is bilinear and associative. It is graded commutative: $\\beta \\wedge \\alpha = (-1)^{k l} \\, \\alpha \\wedge \\beta$. In particular, $dx_i \\wedge dx_j = -dx_j \\wedge dx_i$ and $dx_i \\wedge dx_i = 0$.\nFor a $0$-form $f$, the product $f \\wedge \\beta$ is the form $f \\beta$.\nIn the plane, $(a \\, dx + b \\, dy) \\wedge (c \\, dx + d \\, dy) = (a d - b c) \\, dx \\wedge dy$. The coefficient is the determinant, which is the signed area of the parallelogram spanned by $(a, b)$ and $(c, d)$.\nFor 1-forms $\\alpha$ and $\\beta$ and vectors $v, w$, $(\\alpha \\wedge \\beta)(v, w) = \\alpha(v) \\beta(w) - \\alpha(w) \\beta(v)$.\n\n## Explanations\n\n### L1 human\n\nThe wedge product is a multiplication of forms in which order matters up to sign, and a form times itself is zero.\nIt is the algebra behind determinants, signed areas and signed volumes.\n",
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