🧄Divergence of Tangent Basis Vectors in Curvilinear Coordinates

The derivation shows that the divergence of any tangent basis vector $E_b$ in an orthogonal system is determined entirely by the rate of change of the metric's scale factor, $\sqrt{g}$, with respect to that coordinate, following the formula $\nabla \cdot E_b=\frac{1}{\sqrt{g}} \partial_b(\sqrt{g})$. The non-zero results- $1 / \rho$ in cylindrical coordinates and $2 / r$ and $\cot (\theta)$ 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 (physical) basis vectors often preferred in application.

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