Volume using pseudo-spherical coordinates

  • #1
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.
 

Attachments

Answers and Replies

  • #2
tnich
Homework Helper
1,048
336
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)##.
 

Related Threads on Volume using pseudo-spherical coordinates

  • Last Post
Replies
4
Views
5K
  • Last Post
Replies
3
Views
2K
  • Last Post
2
Replies
26
Views
2K
  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
1
Views
1K
  • Last Post
Replies
1
Views
5K
  • Last Post
Replies
3
Views
6K
Replies
2
Views
2K
Replies
14
Views
1K
Replies
13
Views
1K
Top