Question

Question #e4aa5

Answer 1

A point `P` is defined in spherical coordinated as a function of `(r, theta, phi)`. Where `r` is the distance from the origin to the point, `phi` is the angle that we need to rotate down from the positive `z`-axis to get to the point and `theta` is angle we need to rotate around the `z`-axis to get to the point.

As you have rightly posted the figure, infinitesimal volume element `dV` in spherical coordinates is given by multiplication of three unit coordinates

1. Infinitesimal radial element `=dr`
2. Recall formula which relates the arc length `s` of a circle of radius `r` to the central angle `theta` is given by
   `s=rtheta`, where angle is in radians.
   Hence, arc length subtended by angle `d theta` at a distance `r` would be `=rd theta`
3. Consider area element as shown in the figure below
   
   We see that area element is located at a perpendicular distance `=r sin theta` from the `z`-axis.
   Using the same formula as used in step 2. above we see that arc length subtended by angle `d phi` at a distance of `rsin theta` would be `=rsin theta dphi`
   Therefore we get

   `dV=drcdot rd theta cdot rsinthetadphi`
   `=>dV=r^2 sin theta cdot dr cdot d theta cdot dphi`

`r` ranges from `0` to `r`
`theta` from `0` to `pi` and
`phi` from `0` to `2pi`

A word of caution : Sometimes `angle theta and angle phi` are interchanged in Physics and Mathematics.