With the original aim of properly understand the ellipsoidal coordinates $ (\lambda, \mu, \nu)$ , I would like you to help me understanding the concept of “scale factors” in the contest of differential operators. Particularly, I would like you to provide me a relatively fast and intuitive explanation of such concept for someone like me who has a knowledge of differential calculus mostly coming from the courses of calculus and electromagnetism.
In order to make you understand which is my issue, I am going to copy and paste here a text from Wikipedia (https://en.wikipedia.org/wiki/Ellipsoidal_coordinates)
” Scale factors and differential operators
For brevity in the equations below, we introduce a function
\begin{eqnarray} S(\sigma) = (a^2 + \sigma)(b^2 + \sigma)(c^2 + \sigma) \end{eqnarray} where $ \sigma$ can represent any of the three variables $ (\lambda, \mu, \nu)$ . Using this function, the scale factors can be written
\begin{eqnarray} h_{\lambda} & = & \frac{1}{2}\sqrt{\frac{(\lambda – \mu)(\lambda – \nu)}{S(\lambda)}}\ h_{\mu} & = & \frac{1}{2}\sqrt{\frac{(\mu- \lambda)(\mu- \nu)}{S(\mu)}}\ h_{\nu} & = & \frac{1}{2}\sqrt{\frac{(\nu- \lambda)(\nu – \mu)}{S(\nu)}} \end{eqnarray}
Hence. the infinitesimal volume element equals \begin{eqnarray} dV = \frac{(\lambda – \mu)( \lambda – \nu )( \lambda – \nu )}{8\sqrt{ – S(\lambda)S(\mu)S(\nu)}}d\lambda d\mu d\nu \end{eqnarray} “
What I personally expect as coefficient in the previous formula is the Jacobian determinant and not the product of three different “scale factors”. Is there anybody who can provide me an intuitive explanation of this scale factors and how they are related with the Jacobian determinant?
