= Solution
For a <harmonic function>, put $a=u_{xx}$ and $b=u_{xy}=u_{yx}$. Since $u_{yy}=-u_{xx}$, its <Hessian matrix> is
$$
D^2u=\begin{pmatrix}a&b\\b&-a\end{pmatrix},\qquad\det D^2u=-a^2-b^2.
$$
A nonzero determinant is therefore negative, and the two real Hessian <eigenvalues> are $\pm\sqrt{a^2+b^2}$. At a <stationary point> the quadratic term of the <Taylor expansion> has both signs. Consequently \b[every such nondegenerate <stationary point> is a <saddle point>]. This is the <nondegenerate stationary points of a planar harmonic function> criterion. The same argument applies to $v$, whose <stationary points> coincide with those of $u$. A zero Hessian determinant is outside this second-order classification.
Back to article page
