🎬the divergence of the tangent basis vectors illustrates why these coordinate systems have non-Cartes

The visualization explicitly shows that the tangent basis vector fields ( $E_r, E_{\rho,} E_\theta$ ) are inherently non-uniform and expanding/contracting, which is the geometric cause of the non-zero divergence results. The vectors in the cylindrical $E_\rho$ field and the spherical $E_r$ field are visibly spreading outward from the origin. This spatial spreading represents a source in the field, which is quantified by the positive and $\rho$-dependent results. The spherical $E_\theta$ vectors visually converge near the poles. This convergence acts as a sink and is quantified by the $\cot (\theta)$ term, which becomes large as $\theta \rightarrow 0$ or $\theta \rightarrow \pi$. This means the tangent vectors themselves are not suitable as invariant, physical reference vectors because their properties (length and spacing) change with position. The non-zero divergence is a direct measure of the expansion of the coordinate system's grid lines.

Divergence of Tangent Basis Vectors in Curvilinear Coordinates | Animated Resultsviadean on Notion