Volume using pseudo-spherical coordinates

In summary, the volume of a conic section is independent of the coordinate system used. The calculations for the volume of a two-sheeted hyperboloid have been attempted in rectangular and pseudo-spherical coordinates, with more confidence in the former. The use of pseudo-spherical coordinates is for mapping rather than representing real coordinates. There may be errors in the work and feedback on identifying them is welcome.
  • #1
conservative
1
0
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

  • pseudo-spherical problem.pdf
    80.7 KB · Views: 275
Mathematics news on Phys.org
  • #2
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)##.
 

1. What are pseudo-spherical coordinates used for?

Pseudo-spherical coordinates are used to describe three-dimensional space in a way that is more convenient for certain mathematical calculations, such as finding the volume of a shape.

2. How do you convert from Cartesian coordinates to pseudo-spherical coordinates?

To convert from Cartesian coordinates (x, y, z) to pseudo-spherical coordinates (ρ, θ, φ), you would use the following formulas: - ρ = √(x² + y² + z²)- θ = arctan(y/x)- φ = arctan(√(x² + y²)/z)

3. What is the formula for finding the volume of a shape using pseudo-spherical coordinates?

The formula for finding the volume of a shape using pseudo-spherical coordinates is: V = ∫∫∫ ρ²sin(φ) dρ dθ dφ, where ρ is the radial distance from the origin, θ is the azimuthal angle, and φ is the polar angle.

4. Can pseudo-spherical coordinates be used for any shape?

Yes, pseudo-spherical coordinates can be used for any three-dimensional shape. However, they are most useful for symmetric shapes, such as spheres or cylinders.

5. What are the advantages of using pseudo-spherical coordinates over other coordinate systems?

One advantage of using pseudo-spherical coordinates is that they have a simple relationship with the three-dimensional Laplace equation, making certain mathematical calculations easier. They also have a more intuitive representation of distance and direction compared to other coordinate systems, such as Cartesian or cylindrical coordinates.

Similar threads

Replies
6
Views
3K
  • Other Physics Topics
Replies
3
Views
1K
  • Advanced Physics Homework Help
Replies
2
Views
1K
  • Special and General Relativity
Replies
23
Views
2K
  • Calculus and Beyond Homework Help
Replies
4
Views
2K
  • Calculus and Beyond Homework Help
Replies
9
Views
8K
  • Calculus and Beyond Homework Help
Replies
7
Views
4K
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
3K
Replies
1
Views
1K
Back
Top