Locally write the real analytic <hypersurface> as $\Sigma=\{F=0\}$ with $dF\ne0$. The <conormal bundle> is spanned by $dF$. The surface is a <characteristic hypersurface> at $p$ precisely when the <principal symbol> vanishes on that conormal:
$$
\boxed{\sum_{i,j=1}^n a_{ij}(p)F_{x_i}(p)F_{x_j}(p)=0.}
$$
Only the matrix $(a_{ij}+a_{ji})/2$, which is a <symmetric matrix>, contributes to this expression.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 105 / 3 / b / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/iii/paper-105/3/b/i/solution