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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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