Cartesian coordinates $x, y, z$ and spherical polar coordinates $r, \theta, \phi$ are related by

$$x=r \sin \theta \cos \phi, \quad y=r \sin \theta \sin \phi, \quad z=r \cos \theta$$

Find scalars $h_{r}, h_{\theta}, h_{\phi}$ and unit vectors $\mathbf{e}_{r}, \mathbf{e}_{\theta}, \mathbf{e}_{\phi}$ such that

$$\mathrm{d} \mathbf{x}=h_{r} \mathbf{e}_{r} \mathrm{~d} r+h_{\theta} \mathbf{e}_{\theta} \mathrm{d} \theta+h_{\phi} \mathbf{e}_{\phi} \mathrm{d} \phi$$

Verify that the unit vectors are mutually orthogonal.

Hence calculate the area of the open surface defined by $\theta=\alpha, 0 \leqslant r \leqslant R$, $0 \leqslant \phi \leqslant 2 \pi$, where $\alpha$ and $R$ are constants.