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  "body": "\n## Definition\n\nLet $\\Omega$ be an open set in $\\R^n$.\nLet $P = \\sum_{\\abs{\\alpha} \\le m} a_\\alpha(x) \\, \\partial^\\alpha$ be a [[linear differential operator|6DZETZ5T9G]] of order $m$ on $\\Omega$, with [[multi-indices|0PJXBR6QA3]] $\\alpha$.\nThe *principal symbol* of $P$ is the following function of $x \\in \\Omega$ and $\\xi \\in \\R^n$.\n$$\\sigma_m(P)(x, \\xi) = \\sum_{\\abs{\\alpha} = m} a_\\alpha(x) \\, \\xi^\\alpha.$$\nHere $\\xi^\\alpha = \\xi_1^{\\alpha_1} \\cdots \\xi_n^{\\alpha_n}$.\nFor each fixed $x$, it is a [[homogeneous polynomial|2C3PAQ7H50]] of degree $m$ in $\\xi$.\n\n## Remarks\n\nExample: for $P = \\partial_x^2 - \\partial_y^2 + 5 \\partial_x + 1$ on $\\R^2$, the order is $2$ and $\\sigma_2(P)(x, \\xi) = \\xi_1^2 - \\xi_2^2$; the terms $5 \\partial_x$ and $1$ do not contribute.\nNon-example: the full polynomial $\\xi_1^2 - \\xi_2^2 + 5 \\xi_1 + 1$ is not the principal symbol, because it keeps the lower-order terms.\nMany texts write $P = \\sum a_\\alpha(x) D^\\alpha$ with $D_j = -i \\partial_j$ and define the principal symbol as $\\sum_{\\abs{\\alpha} = m} a_\\alpha(x) \\xi^\\alpha$ in those coefficients; this differs from the definition above by the factor $i^m$.\nThe vanishing of the principal symbol, and so the [[characteristic variety|CQJ2E2ARRF]], is the same in all conventions.\nOn a manifold, the principal symbol is a well-defined function on the cotangent bundle.\n\n## Explanations\n\n### L1 human\n\nThe principal symbol is what remains of a differential operator when only its top-order terms are kept and each derivative is replaced by a variable.\n",
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