I am solving the Laplacian in spherical coordinates using the Method of Lines and in the textbook I am studying, it is indicated that the Laplacian term:
$$\frac{1}{r^2}\bigg(\frac{\cos\theta}{\sin\theta}\dfrac{\partial U}{\partial \theta}\bigg)$$
is singular at $r=0$ and at $\theta = 0$ and at $\theta=\pi/2$.
I definitively agree at $r=0$ and $\theta=0$, but I cannot agree with the singularity at $\theta = \pi/2$.
Can someone please confirm that my statement is correct. Otherwise, kindly explain why the term is singular at $\theta = \pi/2$.
As a commenter pointed out, I believe you mean there is a singularity at $\theta = \pi$. Spherical coordinates are not differentiable at the poles $\theta=0$ and $\theta =\pi$ just as polar coordinates are not differentiable at $\theta=0$. This is due to the fact that the coordinate transformation does not form a bijection over the "polar" points. Consider a fixed point in spherical coordinates $p = (r,\theta,\varphi) = (R,\pi,\varphi) $. The coordinate $\varphi$ can be any number on $[0,2\pi)$ and still represent the same point in $(x,y,z)$ space.
The moral is that since we can't have a bijection, we can't have an invertible Jacobian transformation and thus we can't describe vectors (and therefore derivatives) at those points.