Nondegenerate stationary points of a planar harmonic function ($\det D^{2}u=-u_{xx}^2-u_{xy}^2<0$)

For a planar harmonic function, $u_{yy}=-u_{xx}$. Its Hessian matrix is therefore $\left(\begin{array}{cc}a& b\\ b& -a\end{array}\right)$, with eigenvalues $\pm \sqrt{a^2+b^2}$. At a stationary point with nonzero Hessian matrix determinant, both signs occur, so the point is a saddle point. A zero determinant requires separate treatment and is not covered by this nondegenerate test. The result is the local quadratic counterpart of the maximum principle for harmonic functions.