Principal symbol of a partial differential equation ($p(x,\xi )$)

For a scalar differential operator of order $k$, the principal symbol replaces each order-$k$ derivative ${\partial }^{\alpha }$ by ${\xi }^{\alpha }$ and discards lower-order terms. For a quasilinear partial differential equation, the coefficients are evaluated at the prescribed lower-order data.

A hypersurface $\Sigma =\{\varphi =0\}$ is characteristic for a differential equation at $x\in \Sigma$ when its principal symbol vanishes on the conormal $d\varphi (x)$:It is non-characteristic where this quantity is nonzero.

A non-characteristic hypersurface satisfies $p(x,d\varphi (x))\ne 0$. This lets the equation solve for the highest derivative normal to the hypersurface.