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  "body": "\n## Summary\n\nThe principal symbol of a [[differential operator between vector bundles|CGBX1N9V8H]] records the top-order part of the operator.\nIt is a bundle map that depends on a covector.\nIt is a function on the cotangent bundle with values in linear maps between the fibres.\nIt is homogeneous in the covector, of degree the order of the operator.\nUnlike the full local expression of the operator, it does not depend on coordinates or trivialisations.\nEllipticity, characteristic directions and the index theorem are all phrased in terms of the principal symbol.\nIt is the manifold version of the [[principal symbol|F31T650DKY]] on $\\R^n$.\n\n## Explanations\n\n### L1 human\n\nIn coordinates, a differential operator is a sum of partial derivatives with coefficients.\nIf one keeps only the derivatives of highest order and replaces each $\\partial_j$ by a variable $\\xi_j$, one gets a polynomial in $\\xi$ with matrix coefficients.\nThe lower-order terms change when the coordinates change, but this top part does not, once $\\xi$ is read as a covector.\nFor example, the exterior derivative $d$ has principal symbol \"wedge with $\\xi$\": it sends a form $\\omega$ to $\\xi \\wedge \\omega$.\n",
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