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The volume element in spherical coordinates

A blowup of a piece of a sphere is shown below.

Using a little trigonometry and geometry, we can measure the sides of this element (as shown in the figure) and compute the volume as

\Delta V
\approx (\rho \sin \phi \Delta \theta) (\rho \Delta \phi) (\Delta \rho)\end{displaymath}

so that, in the infinitesimal limit,

dV = \rho^2 \sin \phi d\phi d\theta d\rho.\end{displaymath}

Vector Calculus