# 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.

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