Answer
$-ue^{u+v}.$
Work Step by Step
We have $$
\operatorname{Jac}(G)=\frac{\partial(x, y)}{\partial(u, v)}=\left|\begin{array}{ll}
{\frac{\partial x}{\partial u}} & {\frac{\partial x}{\partial v}} \\
{\frac{\partial y}{\partial u}} & {\frac{\partial y}{\partial v}}
\end{array}\right|=\left|\begin{array}{ll}
{ e^v} & {ue^v} \\
{e^u} & {0}
\end{array}\right| =-ue^{u+v}.
$$
