📢Scale Factor Dependence of Tangent Basis Vector Divergence

The derivation shows that the divergence of any tangent basis vector in an orthogonal system is determined entirely by the rate of change of the metric's scale factor with respect to that coordinate. The non-zero results in cylindrical coordinates and in spherical coordinates are a direct measure of the expansion or contraction of the coordinate grid lines in space. This confirms that these tangent basis vectors are non-unit and expanding, highlighting why the complexity of the geometry is intrinsically built into these vector fields, which contrasts with the fixed-length, nonexpanding nature of the unit basis vectors often preferred in application.

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