# Volume using pseudo-spherical coordinates

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.

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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)##.