= Solution
At each regular point, the <gradient>
$$
\boxed{\nabla f=\bigl(x(1+2y),\ y+x^2-y^2\bigr)}
$$
is normal to the <level set> and points towards increasing $f$. The arrows in the <contour plot> above show this direction; their lengths have been normalized to keep the outer arrows legible. Near the origin they point outwards, as expected for a <local minimum>. Near $(0,1)$, write $\eta=y-1$; the linear approximation is $\nabla f\simeq(3x,-\eta)$. Thus the arrows point away along the horizontal direction and towards the point along the vertical direction. The other two <saddle points> have the same behaviour after rotations through $2\pi/3$. All four <stationary points> have zero <gradient>, so no direction arrow is assigned at the points themselves.
Back to article page
