Prolate sheroidal coordinates can be used to simplify the Kepler problem in celestial mechanics. They are related to the usual Cartesian coordinates of Euclidean three-space by:
In the plane y=0, what is the coordinate transformation matrix d(x^mu)/d(x^v') relating (x,z) to (nu,theta)?
In the formulae you have written there is a scale factor "a" mising (see reference web page http://mathworld.wolfram.com/ProlateSpheroidalCoordinates.html), so that the full correct ...
This solution provides a full transformation of the Prolate sheroidal coordinates with the Jacobian matrix. All steps are shown with brief explanations.