If $z = \left(x+y\right)e^{y}$ and $x = u^{2}+v^{2}$ and $y = u^{2}-v^{2}$, find the following partial derivatives using the chain rule. Enter your answers as functions of $u$ and $v$.

$\displaystyle \frac{\partial z}{\partial u} =$

$\displaystyle \frac{\partial z}{\partial v} =$