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.