🎬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 ( Er,Eρ,EθE_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 EρE_\rho field and the spherical ErE_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 ρ\rho-dependent results. The spherical EθE_\theta vectors visually converge near the poles. This convergence acts as a sink and is quantified by the cot(θ)\cot (\theta) term, which becomes large as θ0\theta \rightarrow 0 or θπ\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.

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