= Quartic surface with axes and a circle as its zero set
{title2=$F_R(x,y)=xy(x^2+y^2-R^2),\quad R>0$}
The zero <level set> of $F_R$ is the union of the coordinate axes and the <circle> of <radius> $R$. The <gradient> is
$$
\nabla F_R=\bigl(y(3x^2+y^2-R^2),\ x(x^2+3y^2-R^2)\bigr)^T.
$$
Its nine <stationary points> are the origin, the four axis points at distance $R$, and the four points with $x^2=y^2=R^2/4$. The factored <gradient> proves completeness: a zero coordinate gives the five axis cases; two nonzero coordinates give a linear system in $x^2,y^2$.
The <Hessian matrix> has diagonal entries $6xy$ and off-diagonal entries $3(x^2+y^2)-R^2$. At the origin its <eigenvalues> are $\pm R^2$; at the four unit-direction axis points they are $\pm2R^2$. These are <saddle points of a scalar function>. At the two interior points with equal coordinate signs, the <eigenvalues> are $R^2,2R^2$, giving strict <local minima> of value $-R^4/8$. Opposite coordinate signs give the negative <eigenvalues> and strict <local maxima> of value $R^4/8$.
On the zero circle, $\nabla F_R=2xy(x,y)^T$. Hence its direction is radial outward for $xy>0$, inward for $xy<0$, and zero at the four axis crossings. This simple geometric sign rule links the zero <level set> to the <gradient> sketch without requiring numerical differentiation.
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