Volume using pseudo-spherical coordinates

[conservative]

The volume of a conic section should be the same regardless of the coordinate system used. Thus, I have attempted to calculate the volume of a two-sheeted hyperboloid in both rectangular and psuedo-spherical coordinates (q.v. attached pdf file). I am fairly confident the calculations in rectangular coordinates are correct, but much less so for the pseudo-spherical coordinates. The psuedo-spherical coordinates do not really represent real coordinates, rather they are mapped coordinates. Be that as it may, I would welcome someone who can identify the error(s) in my work.

 

[tnich]

I think you intend that $$\rho^2 = x^2 + y^2+z^2$$ but as you have defined your transformation $$\rho^2 \cosh(2\chi)= x^2 + y^2+z^2$$ because $cosh^2(\chi) + sinh^2(\chi) = cosh(2\chi)$.