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Determinant in Transformation from spherical to cartesian space

  1. Mar 24, 2013 #1
    1. The problem statement, all variables and given/known data
    Evaluate the appropriate determinant to show that the Jacobian of the transformation from Cartesian (this is a typo, they mean spherical) pψθ-space to Cartesian xyz-space is ρ2sin(ψ).


    2. Relevant equations



    3. The attempt at a solution

    Uhm, I am lost. I'm supposed to prove that when a function F(p,ψ,θ) is transformed into a function H(x,y,z), then the jacobian is ρ2sin(ψ).

    But, to do that I am supposed to solve a determinant which involves the partial derivatives of p(x,y,z), ψ(x,y,z) and θ(x,y,z) with respect to x,y,z, namely J(x,y,z)?? That would take an hour, so I assume I am not understanding the problem properly?? As far as I know, evaluating the determinant J(p,ψ,θ) will make ρ2sin(ψ) pop out - but this is for the opposite transformation, from cartesian to spherical space.

    I'm confused. Help pls?
     
    Last edited: Mar 24, 2013
  2. jcsd
  3. Mar 24, 2013 #2
    First of all, it is not "spaces" that are transformed. It is coordinates.

    Write x = x(p, ψ, θ), y = y(p, ψ, θ), z = z(p, ψ, θ) and find the Jacobian of this.
     
  4. Mar 24, 2013 #3
    But that's the jacobian for the opposite transformation (cartesian into spherical, I want to find the jacobian of the reverse transformation)! How does that apply here?
     
  5. Mar 24, 2013 #4
    The transformation in #2 is from the spherical coordinates into Cartesian.
     
  6. Mar 24, 2013 #5
    Well, it's also the jacobian for the cartesian -> spherical transformation..

    How can that be? Can somebody explain this jacobian stuff to me?
     
  7. Mar 24, 2013 #6
    I think you are confused by the word "transform". This may be because I and whoever gave you the problem use it to mean something different. Could you restate your problem in more details without using this term?
     
  8. Mar 24, 2013 #7
    "Evaluate the appropriate determinant to show that the Jacobian of the transformation from Cartesian pψθ-space to Cartesian xyz-space is ρ2sin(ψ)."

    This is the exact wording from the book. I am not confused about transformation.

    What I am confused about is how ρ2sin(ψ) can be the jacobian determinant for both the xyz --> pψθ transformation, and the pψθ ---> xyz transformation.
     
  9. Mar 24, 2013 #8
    The Jacobian for p(x,y,z), ψ(x,y,z) and θ(x,y,z) is not ρ2sin(ψ), simply because it ought to be a function of (x, y, z). But even if expressed via (p, ψ, θ) it will be 1/(ρ2sin(ψ)).
     
  10. Mar 24, 2013 #9
    Let's say we have a sphere x^2+y^2+z^2=1.

    Its volume can be expressed as -1∫1 -1∫1 -1∫1 dxdydz. If it is transformed into spherical coordinates, then ∫∫∫x^2+y^2+z^2dxdydz --> 0∫2pi 0∫pi 0∫1 J(p,ψ,θ) dpdψdθ,

    where the Jacobian J(p,ψ,θ) = p2*sin(ψ).

    Where is the flaw in my logic? Here a xyz---> pψθ transformation happened, right?
     
    Last edited: Mar 24, 2013
  11. Mar 24, 2013 #10
    I do not understand what this means.

    This is not the volume of the sphere. The volume of the sphere is ∫∫∫dxdydz, where the integration domain is the interior of the sphere.

    In spherical coordinates, that becomes ∫∫∫J(ρ,ψ,θ)dρdψdθ, where the integration domain is the "brick" 0 ≤ ρ ≤ 1, 0 ≤ ψ ≤ ∏, 0 ≤ θ ≤ 2∏.

    You can say you "transformed" the integration problem from Cartesian to spherical coordinates. However, this uses the "coordinate transformation" from spherical coordinates to Cartesian. And the Jacobian is that of the "coordinate transformation" from spherical coordinates to Cartesian. The word transformation may mean very different things depending on what it is attached to.
     
  12. Mar 24, 2013 #11
    Oh, sorry. Youre completely right on the volume thing - my mind is exhausted after 10 hours of cramming. And yeh, i think ill return to this problem tomorrow
     
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