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  "body": "\n## Summary\n\nTake a linear differential operator of order $m$, keep only its terms of order $m$, and replace each derivative $\\partial_j$ by a variable $\\xi_j$.\nThe resulting polynomial in $\\xi$ is the principal symbol.\nIt captures the behaviour of the operator at high frequencies.\nEllipticity, hyperbolicity and characteristic directions are all defined through the principal symbol.\nSee the [[definition|MX6D07DDAZ]].\n\n## Explanations\n\n### L1 human\n\nApply a differential operator to a fast oscillation $e^{i \\lambda x \\cdot \\xi}$ with large $\\lambda$.\nThe highest derivatives bring the largest powers of $\\lambda$, so they dominate the result.\nThe principal symbol records exactly this dominant part.\nFor the [[Laplacian|0E0B6RA8K0]] $\\partial_x^2 + \\partial_y^2$ the principal symbol is $\\xi_1^2 + \\xi_2^2$, with the sign convention that replaces $\\partial_j$ by $\\xi_j$.\n",
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