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Weird singularities/cylindrical & spherical coordinates

  1. Sep 7, 2009 #1
    If you consider the vector function (expressed in cylindrical coordinates)
    [tex]
    \frac{1}{\rho} \hat{\phi}
    [/tex]
    where [itex]\rho = \sqrt{x^2+y^2}[/itex], you notice it has a singularity at the origin ONLY. But if you express this in spherical coordinates, what you get is
    [tex]
    \frac{1}{r\sin \theta} \hat{\phi},
    [/tex]
    which is singular whenever [itex]\theta = 0[/itex] or [itex]\theta = \pi[/itex]; that is, on the entire z axis! (I got this by just substituting the identities [itex]x = r\sin \theta \cos \phi[/itex] and [itex]y = r \sin \theta \sin \phi[/itex] into the above expression for [itex]\rho[/itex].) How can this be? I'm doing something wrong, but what?
     
    Last edited: Sep 7, 2009
  2. jcsd
  3. Sep 7, 2009 #2
    Uhhh...sorry, I was being really dumb. [itex]\rho = 0[/itex] includes the [itex]z[/itex]-axis in cylindrical coordinates. FML.
     
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