Let $\theta$ and $\phi$ be the polar coordinates. Introduce the complex numbers $z$ and $\bar{z}$, where

$$\begin{equation*}
z=e^{i \phi} \tan (\theta / 2) \equiv \xi+i \eta \tag{5.393}
\end{equation*}$$

and $\xi$ and $\eta$ are real numbers. Show that the metric of the two-sphere transforms as

$$\begin{align*}
d s^{2} & =d \theta \otimes d \theta+\sin ^{2} \theta d \phi \otimes d \phi \\
& =\frac{2}{\left(1+|z|^{2}\right)^{2}}(d \bar{z} \otimes d z+d z \otimes d \bar{z}) \\
& =\frac{2}{\left(1+\xi^{2}+\eta^{2}\right)^{2}}(d \xi \otimes d \xi+d \eta \otimes d \eta) \tag{5.394}
\end{align*}$$

and the area two-form $\omega$ transforms as

$$\begin{align*}
\omega & =\sin \theta d \theta \wedge d \phi \\
& =\frac{2 i}{\left(1+|z|^{2}\right)^{2}}(d z \wedge d \bar{z}) \\
& =\frac{4}{\left(1+\xi^{2}+\eta^{2}\right)^{2}}(d \xi \wedge d \eta) \tag{5.395}
\end{align*}$$