- #1
AxiomOfChoice
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- 1
If you consider the vector function (expressed in cylindrical coordinates)
<br /> \frac{1}{\rho} \hat{\phi}<br />
where \rho = \sqrt{x^2+y^2}, you notice it has a singularity at the origin ONLY. But if you express this in spherical coordinates, what you get is
<br /> \frac{1}{r\sin \theta} \hat{\phi},<br />
which is singular whenever \theta = 0 or \theta = \pi; that is, on the entire z axis! (I got this by just substituting the identities x = r\sin \theta \cos \phi and y = r \sin \theta \sin \phi into the above expression for \rho.) How can this be? I'm doing something wrong, but what?
<br /> \frac{1}{\rho} \hat{\phi}<br />
where \rho = \sqrt{x^2+y^2}, you notice it has a singularity at the origin ONLY. But if you express this in spherical coordinates, what you get is
<br /> \frac{1}{r\sin \theta} \hat{\phi},<br />
which is singular whenever \theta = 0 or \theta = \pi; that is, on the entire z axis! (I got this by just substituting the identities x = r\sin \theta \cos \phi and y = r \sin \theta \sin \phi into the above expression for \rho.) How can this be? I'm doing something wrong, but what?
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