What is the expression for divergence of a field in spherical coordinates at $\theta=0$?
To simplify things, I'm assuming $\frac{\delta f_\phi}{\delta \phi}=0$. So it is effectively in polar coordinates
$$\nabla.\vec{f}=\frac{1}{r^2} \frac{\delta}{\delta r}(r^2 \frac{\delta f_r}{\delta r})+\frac{1}{r\sin \theta} \frac{\delta}{\delta \theta}(f_\theta \sin \theta) $$
At $\theta =0$, we can't directly evaluate $\frac{1}{r\sin \theta} \frac{\delta}{\delta \theta}(f_\theta \sin \theta)$ because both the numerator and denominator turn out to be zero.
I tried expanding the partial differential as $$\frac{1}{r\sin \theta} \frac{\delta}{\delta \theta}(f_\theta \sin \theta)=\frac{1}{r \sin \theta}\sin \theta \frac{\delta f_\theta}{\delta \theta}+\frac{\cos \theta}{r \sin\theta}f_\theta $$
which gives me $\infty$ for any non zero $f_\theta$.
Can someone explain why I'm getting this result? Am I doing something wrong in evaluating partial differential?
I got a proof from wikipedia where they were considering elements like this
One of the edges is $r\sin \theta d \phi$ which goes to zero as $\theta$ goes to zero, so that might explain it. If that's the reason, how do I pick an element at $\theta=0$? What would its dimensions be?
I'm kinda new to partial derivatives in spherical coordinates so I'd appreciate it if someone pointed out if I made a mistake in what I did.