---
tag: F31T650DKY
kind: concept
name: Principal symbol
msc: [35A99, 47F05, 35S05]
sources:
  - url: https://en.wikipedia.org/wiki/Differential_operator
---

## Summary

Take a linear differential operator of order $m$, keep only its terms of order $m$, and replace each derivative $\partial_j$ by a variable $\xi_j$.
The resulting polynomial in $\xi$ is the principal symbol.
It captures the behaviour of the operator at high frequencies.
Ellipticity, hyperbolicity and characteristic directions are all defined through the principal symbol.
See the [[definition|MX6D07DDAZ]].

## Explanations

### L1 human

Apply a differential operator to a fast oscillation $e^{i \lambda x \cdot \xi}$ with large $\lambda$.
The highest derivatives bring the largest powers of $\lambda$, so they dominate the result.
The principal symbol records exactly this dominant part.
For 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 $\xi_j$.