Azimuth angle limit in Spherical co-ordinate system In spherical co-ordinate system $(r, \theta, \phi)$, $\theta$ can range from $0$ to $2\pi$, but $\phi$ only varies from $0$ to $\pi$. Why is that?

Look at a globe. Longitude ranges over 360 degrees, while latitude ranges over 180 degrees. In spherical coordinates, $\theta$ denotes something analogous to longitude (but with an offset domain), and $\phi$ denotes something analogous to latitude (again, with an offset domain: the south pole is "0" and the north pole is "180 degrees", because it's easier to work with positive numbers, I suppose.)

Be warned that there are many spherical polar coord systems, and in some, $\phi$ runs from $-\pi/2$ to $\pi/2,$ and in others, $\phi$ denotes the longitude while $\theta$ denotes latitude. Sigh.