= Planar first integral criterion for a centre
{title2=$F\cdot\nabla H=0,\quad D^2H(p)\text{ definite}$}
Let $F$ be a continuously differentiable planar <vector field> with an isolated <equilibrium point> $p$, and let the twice continuously differentiable <first integral> $H$ satisfy $F\cdot\nabla H=0$ and $\nabla H(p)=0$. If the <Hessian matrix> $D^2H(p)$ is positive or negative definite, then $p$ is a <centre equilibrium>.
For a <positive-definite matrix>, sufficiently short rays from $p$ have strictly increasing $H$ with radial <derivative> bounded below by a positive constant times distance. Every sufficiently small positive level difference therefore meets each ray exactly once, and the <implicit function theorem> makes these intersections a <simple closed curve>. These <regular level sets> shrink to $p$; use $-H$ in the negative-definite case. The <vector field> is tangent and nonzero on each such compact curve. Its speed is bounded away from zero, so a traversal takes finite time and each curve is a <periodic orbit>. An arbitrary strict <local extremum> without these regularity and nondegeneracy conditions does not by itself supply this argument.
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