# Question #e4aa5

Jan 24, 2017

A point $P$ is defined in spherical coordinated as a function of $\left(r , \theta , \phi\right)$. 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 $\mathrm{dV}$ in spherical coordinates is given by multiplication of three unit coordinates

1. Infinitesimal radial element $= \mathrm{dr}$
2. Recall formula which relates the arc length $s$ of a circle of radius $r$ to the central angle $\theta$ is given by
$s = r \theta$, where angle is in radians.
Hence, arc length subtended by angle $d \theta$ at a distance $r$ would be $= r d \theta$
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 $r \sin \theta$ would be $= r \sin \theta \mathrm{dp} h i$
Therefore we get

$\mathrm{dV} = \mathrm{dr} \cdot r d \theta \cdot r \sin \theta \mathrm{dp} h i$
$\implies \mathrm{dV} = {r}^{2} \sin \theta \cdot \mathrm{dr} \cdot d \theta \cdot \mathrm{dp} h i$

$r$ ranges from $0$ to $r$
$\theta$ from $0$ to $\pi$ and
$\phi$ from $0$ to $2 \pi$

A word of caution : Sometimes $\angle \theta \mathmr{and} \angle \phi$ are interchanged in Physics and Mathematics.