🎬why the standard divergence formula requires spherical volume element

The animation visually confirms the fundamental reason for the complex terms in the spherical divergence formula: the non-uniform scaling of differential surface areas. The animation dynamically illustrates that the area of the faces perpendicular to the radial velocity ( $\tilde{v}_r$), known as the r-Area Factor ( $r^2 \sin (\theta)$), is strongly dependent on position. As the volume element sweeps toward the poles ( $\theta \rightarrow 0^{\circ}$or $\theta \rightarrow 180^{\circ}$ ), this area rapidly shrinks because the $\sin (\theta)$ factor approaches zero. This geometric change in surface area is precisely why the divergence formula must include the $r ^{ 2 }$ and $\sin (\theta)$ terms inside the derivatives, ensuring that the flux (flow) through the non-Cartesian boundaries is correctly accounted for.

Divergence in Spherical Coordinates Derivation and Verification | Proof and Derivationviadean on Notion