Principal symbol

Remarks

Example: 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. Non-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. Many 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 $i^m$. The vanishing of the principal symbol, and so the characteristic variety, is the same in all conventions. On a manifold, the principal symbol is a well-defined function on the cotangent bundle.