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  "body": "\n## Definition\n\nLet $M$ be a [[smooth manifold|F45VQ47GPV]] with cotangent bundle $\\pi \\colon T^*M \\to M$, and let $E$ and $F$ be smooth [[vector bundles|P4DZM975BV]] over $M$.\nLet $P \\colon \\Gamma(E) \\to \\Gamma(F)$ be a [[differential operator|CGBX1N9V8H]] of order at most $m$.\nLet $x \\in M$, $\\xi \\in T^*_x M$ and $e \\in E_x$.\nChoose a smooth function $f$ on $M$ with $f(x) = 0$ and $df(x) = \\xi$, and a section $u \\in \\Gamma(E)$ with $u(x) = e$.\nThe *principal symbol* of order $m$ of $P$ is the bundle map $\\sigma_m(P) \\colon \\pi^* E \\to \\pi^* F$ over $T^*M$ given by\n$$\\sigma_m(P)(x, \\xi) \\, e = \\frac{1}{m!} \\, P(f^m u)(x).$$\n\n## Remarks\n\nThe right side does not depend on the choices of $f$ and $u$.\nTake a chart and trivialisations in which $P = \\sum_{\\abs{\\alpha} \\le m} A_\\alpha(x) \\partial^\\alpha$.\nThen, with $\\xi = \\sum_j \\xi_j \\, dx_j$,\n$$\\sigma_m(P)(x, \\xi) = \\sum_{\\abs{\\alpha} = m} A_\\alpha(x) \\, \\xi^\\alpha.$$\nIndeed, at $x$ only the terms in which all $m$ derivatives fall on distinct factors $f$ survive, and they give $m! \\, \\xi^\\alpha$.\nSo $\\sigma_m(P)$ agrees with the [[principal symbol|F31T650DKY]] on open subsets of $\\R^n$, and it is homogeneous of degree $m$ in $\\xi$.\nMany texts include a factor $i^m$, defining the symbol through $D_j = -i \\partial_j$; this does not change where the symbol is invertible.\nThe symbol is multiplicative: $\\sigma_{m + l}(P Q) = \\sigma_m(P) \\, \\sigma_l(Q)$.\n\nExample: for the [[exterior derivative|0ZHSTAG8ZY]] $d$ on $k$-forms, $d(f \\omega)(x) = df(x) \\wedge \\omega(x) + f(x) \\, d\\omega(x) = \\xi \\wedge \\omega(x)$.\nSo $\\sigma_1(d)(x, \\xi) \\omega = \\xi \\wedge \\omega$.\nFor the [[Laplace–Beltrami operator|F4NWZNTRMD]] on functions of a Riemannian manifold, with the sign convention $\\Delta = -\\operatorname{div} \\operatorname{grad} = -\\sum_{j,k} g^{jk} \\partial_j \\partial_k + (\\text{lower order})$, one gets $\\sigma_2(\\Delta)(x, \\xi) = -\\abs{\\xi}_g^2$, or $\\abs{\\xi}^2_g$ with the factor $i^2$.\n\nNon-example: the full expression $\\sum_{\\abs{\\alpha} \\le m} A_\\alpha(x) \\xi^\\alpha$, with the lower-order terms kept, is not the principal symbol.\nIt depends on the chart: for $P = \\partial_x^2$ on $\\R$ and the new coordinate $y = e^x$, one has $P = y^2 \\partial_y^2 + y \\partial_y$, whose first-order term is not there in the old coordinate.\n\n## Explanations\n\n### L1 human\n\nThe principal symbol keeps the top-order derivatives of an operator, with each derivative replaced by a component of a covector.\n",
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