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Let $\mathbf{f}$ be a $C^{1}$ vector field on an open set $W \subset \mathbb{R}^{2}$ and $H: W \rightarrow \mathbb{R}$ a $C^{1}$ function such that
$D H(\mathbf{u}) \mathbf{f}(\mathbf{u})=0$
for all $u$. Prove that:
(a) $H$ is constant on solution curves of $d \mathbf{u} / d t=f(\mathbf{u})$
(b) $D H(\mathbf{u})=0$ if $\mathbf{u}$ belongs to a limit cycle;
(c) If $\mathbf{u}$ belongs to a compact invariant set on which $D H$ is never 0 , then $\mathbf{u}$ lies on a closed orbit.
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