The domain $S$ in the $(x, y)$ plane is bounded by $y=x, y=a x(0 \leqslant a \leqslant 1)$ and $x y^{2}=1(x, y \geqslant 0)$. Find a transformation

$$u=f(x, y), \quad v=g(x, y)$$

such that $S$ is transformed into a rectangle in the $(u, v)$ plane.

Evaluate

$$\int_{D} \frac{y^{2} z^{2}}{x} d x d y d z$$

where $D$ is the region bounded by

$$y=x, \quad y=z x, \quad x y^{2}=1 \quad(x, y \geqslant 0)$$

and the planes

$$z=0, \quad z=1$$