I have started working through Chapter 1 of Morse and Feshbach's "Methods of Theoretical Physics" and have encountered some difficulty replicating an expression found in the section on curvilinear coordinates (specifically, page 24, equation 1.3.4).
They take an orthogonal coordinate system defined by functions $\xi_1(x,y,z)$, $ \xi_2(x,y,z)$ and $\xi_3(x,y,z)$. Then due to the coordinate system's orthogonality, they state the scale factors $h_n$ must be such that \begin{equation} ds^2 = dx^2+dy^2+dz^2 = \sum_{n}h_n^2 \ d\xi_n^2. \end{equation}
Then they claim by 'simple substitution', the functions $h_n$ are related by
\begin{equation} h_n^2 = \left(\frac{\partial{x}}{\partial{\xi_n}}\right)^2 + \left(\frac{\partial{y}}{\partial{\xi_n}}\right)^2 + \left(\frac{\partial{z}}{\partial{\xi_n}}\right)^2 = \left[\left(\frac{\partial{\xi_n}}{\partial{x}}\right)^2+\left(\frac{\partial{\xi_n}}{\partial{y}}\right)^2+\left(\frac{\partial{\xi_n}}{\partial{z}}\right)^2\right]^{-1}. \end{equation} Although I can understand the first equality somewhat intuitively, and I can comfortably follow various derivations in other resources, I am struggling to see how the second follows. I have tried to find this final expression elsewhere in the literature but have had little success.
Could someone explain how this equality follows?