# Weird singularities/cylindrical & spherical coordinates

#### AxiomOfChoice

If you consider the vector function (expressed in cylindrical coordinates)
$$\frac{1}{\rho} \hat{\phi}$$
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
$$\frac{1}{r\sin \theta} \hat{\phi},$$
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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#### AxiomOfChoice

Uhhh...sorry, I was being really dumb. $\rho = 0$ includes the $z$-axis in cylindrical coordinates. FML.

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