AxiomOfChoice
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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?
[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?
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