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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $\\Lambda_n = \\Lambda[\\theta_1, \\dots, \\theta_n]$ be the real [[exterior algebra|0MDMZ965VE]] on anticommuting generators $\\theta_1, \\dots, \\theta_n$, so that $\\theta_i \\theta_j = -\\theta_j \\theta_i$.\nEvery $f \\in \\Lambda_n$ has a unique expansion $f = \\sum_{I} a_I \\theta_I$, where $I = \\set{i_1 \u003c \\dots \u003c i_k}$ runs over subsets of $\\set{1, \\dots, n}$, $\\theta_I = \\theta_{i_1} \\cdots \\theta_{i_k}$ and $a_I \\in \\R$.\nThe *Berezin integral* is the linear map $\\Lambda_n \\to \\R$,\n$$\\int f \\, d^n\\theta = a_{\\set{1, \\dots, n}},$$\nthe coefficient of $\\theta_1 \\theta_2 \\cdots \\theta_n$.\nThe same definition applies with coefficients in any commutative ring, for instance $a_I \\in C^\\infty(U)$.\n\n## Remarks\n\nThe order of the symbols $d\\theta_i$ in the notation varies between authors and changes the result by a sign.\nFor $n = 1$: $\\int (a + b\\theta)\\, d\\theta = b$.\n\nThe integral kills derivatives: for each $i$, $\\int \\frac{\\partial f}{\\partial \\theta_i}\\, d^n\\theta = 0$, where $\\partial / \\partial\\theta_i$ is the odd derivation with $\\partial \\theta_j / \\partial \\theta_i = \\delta_{ij}$, since a derivative has no top term.\nEvery monomial other than $\\theta_1 \\cdots \\theta_n$ is, up to sign, such a derivative, so this property fixes the functional up to a constant factor.\nUnder a linear substitution $\\theta = A\\eta$ with $A$ an invertible real $n \\times n$ matrix, $\\theta_1 \\cdots \\theta_n = \\det(A)\\, \\eta_1 \\cdots \\eta_n$, so $\\int f(A\\eta)\\, d^n\\eta = \\det(A) \\int f(\\theta)\\, d^n\\theta$; for ordinary integrals the factor would be $\\abs{\\det A}^{-1}$.\n\nExample: for $n = 2$ and $a \\in \\R$, $e^{a\\theta_1\\theta_2} = 1 + a\\theta_1\\theta_2$, so $\\int e^{a\\theta_1\\theta_2}\\, d^2\\theta = a$.\nThis is the [[Pfaffian|NH5HWBPWS9]] of the antisymmetric matrix with entries $A_{12} = a$, $A_{21} = -a$, and in general the Gaussian integral $\\int \\exp(\\frac{1}{2} \\sum_{i,j} A_{ij}\\theta_i\\theta_j)\\, d^n\\theta$ equals $\\operatorname{Pf}(A)$ up to a sign fixed by the conventions.\n\nNon-example: the map $f \\mapsto a_\\emptyset$, which takes the constant term, is linear but is not the Berezin integral.\nFor $n = 1$ it gives $1$ on $\\partial(\\theta)/\\partial\\theta = 1$, so it does not kill derivatives.\n\nOn a [[supermanifold|Z0SB9HQBAJ]] of dimension $p|q$, consider a compactly supported function on a superdomain.\nIts integral is the Berezin integral in the odd variables followed by the ordinary integral in the even ones.\n",
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