🎬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 ( ), known as the r-Area Factor ( ), is strongly dependent on position. As the volume element sweeps toward the poles ( or ), 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 and terms inside the derivatives, ensuring that the flux (flow) through the non-Cartesian boundaries is correctly accounted for.
Previousthe covariant derivative is indispensable in non-Cartesian or curved systems that distinguishes it fNextAudios
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