# 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.

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