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Expressing a surface in cartesian coordinates from spherical

  1. Oct 11, 2011 #1
    1. The problem statement, all variables and given/known data
    The following equation describes a surface in spherical coordinates. θ =pi/4
    Write the equation in the cartesian coordinates?

    that is, (r,θ,Ø) to (x,y,z)
    2. Relevant equations
    x=rsinθcosØ
    y=rsinθsinØ
    z=rcosθ

    r=sqrt(x^2+y^2+z^2)
    θ=cos^-1(z/r)
    Ø=tan^-1(y/x)


    3. The attempt at a solution
    I'm pretty stumped. The only start I can get is that if θ=pi/4, this means that r and phi have to work around this. Allowing r to be from 0 to infinity. I'm not really sure what happens to phi. I'm thinking that it can be anywhere from 0 to 2pi.
     
  2. jcsd
  3. Oct 11, 2011 #2
    If your surface is simply defined as

    [itex]\theta = \pi/4[/itex]

    then all you need to do is solve your conversion factor from theta of

    [itex]\theta = cos^{-1}(\frac{z}{r}) = cos^{-1}(\frac{z}{\sqrt{x^{2}+y^{2}+z^{2}}})[/itex]

    Since your value of theta is a constant, you just have

    [itex]\frac{z}{\sqrt{x^{2}+y^{2}+z^{2}}} = cos(\frac{\pi}{4})[/itex]

    Although you may want to make sure you have your notation correct. Generally I've seen

    [itex]z = r cos\phi, x = r cos\theta sin\phi, y = r sin\theta sin\phi[/itex]

    and if you did in fact get your notation mixed up (which it's possible that you did or didn't, but it's worth checking) that changes the nature of your calculations.
     
    Last edited: Oct 11, 2011
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