🎬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 ( ) are inherently non-uniform and expanding/contracting, which is the geometric cause of the non-zero divergence results. The vectors in the cylindrical field and the spherical field are visibly spreading outward from the origin. This spatial spreading represents a source in the field, which is quantified by the positive and -dependent results. The spherical vectors visually converge near the poles. This convergence acts as a sink and is quantified by the term, which becomes large as or . 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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