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Relativity: Coordinate Transformations

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:
x=sinh(nu)sin(theta)cos(pheta)
y=sinh(nu)sin(theta)sin(pheta)
z=cosh(nu)cos(pheta)

In the plane y=0, what is the coordinate transformation matrix d(x^mu)/d(x^v') relating (x,z) to (nu,theta)?

Solution Preview

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 ...

Solution Summary

This solution provides a full transformation of the Prolate sheroidal coordinates with the Jacobian matrix. All steps are shown with brief explanations.

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