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Spherical co-ordinates with Implicit function thm

  1. Jul 11, 2012 #1
    So I'm asked to determine near which points of R^3 can we solve for ρ, δ, θ in terms of x,y,z:

    x = ρ sinδ cosθ
    y= ρ sinδ sinθ
    z= ρcosδ

    so the spherical co-ordinates using IFT.

    Attempt:

    Ok so in order to determine solutions, I need to first find where the determinant of the freceht derivative does not equal zero. So I set it up as so:

    \begin{bmatrix} sinδ cosθ & sinδ sinθ & cosδ\\ ρ cosδ cosθ & ρ cosδ sinθ & -ρ sinδ\\ -ρ sinδ sinθ & ρ sinδ sinθ & 0\end{bmatrix}

    so I take the determinant of that and I suppose whichever points do not make the determinant 0 are the points where the system can be solved.

    the issue is when I took the determinant, it didn't really simplify out how I hoped.....is this what I'm suppose to do or is there some trick to this?
     
  2. jcsd
  3. Jul 11, 2012 #2
    my mistake, I entered the matrix values wrong (darn latex). So it's actually the transpose of the above matrix is the one I get.
     
  4. Jul 11, 2012 #3

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    That doesn't really matter since the determinant of a matrix and its transpose are the same. However you have differentiated incorrectly. The term in the middle of the third row (as you have it) should be [itex]\rho sin(\delta) cos(\theta)[/itex], not [itex]\rho sin(\delta)sin(\theta)[/itex].
     
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