Chapter 10.8, Problem 1P

Find $d{s}^2$ in spherical coordinates by the method used to obtain (8.5) for cylindrical coordinates. Use your result to find for spherical coordinates, the scale factors, the vector $ds,$ the volume element, the basis vectors $a_r,a_{\theta },a_{\varphi }$ and the corresponding unit basis vectors $e_r,e_{\theta },e_{\varphi }.$ Write the $g_{ij}$ matrix.