1. [10 Points] Let $H: \mathbb{R}^2 \to \mathbb{R}^2$, $(u, v) \mapsto H(u, v)$ be defined as
$H(u, v) = (u \cos(s(u, v)) - v \sin(s(u, v)), u \sin(s(u, v)) + v \cos(s(u, v)))$,
where $s(u, v) = -\frac{1}{2} \ln(u^2 + v^2)$ for $(u, v) \neq (0, 0)$ and $H(0, 0) = (0, 0)$. Show that $H$ conjugates the planar
systems
$u' = -u$,
$v' = -v$
and
$x' = -x - y$,
$y' = x - y$.
