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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✍️Mathematical Proof

$\complement\cdots$Counselor

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• Universal Formula: The divergence of a tangent basis vector $E_b$ in any orthogonal coordinate system is governed by a single, concise formula based solely on the determinant of the metric $g$.
• Geometric Significance: The divergence of a vector field ( $\nabla \cdot V$ ) measures the local expansion or contraction of that field. Since the tangent basis vectors $E_a$ define the coordinate grid lines, their non-zero divergence confirms that these coordinate grid lines spread out or converge non-uniformly as you move in space.
• Non-Zero Results:
  • Cylindrical: $\nabla \cdot E_\rho=1 / \rho$. This positive result means the radial coordinate lines (defined by $E_\rho$ ) are expanding as you move away from the $z$-axis (increasing $\rho$ ). $\nabla$. $E_\phi$ and $\nabla \cdot E_z$ are zero because the scale factors do not depend on $\phi$ or $z$.
  • Spherical: $\nabla \cdot E_r=2 / r$ and $\nabla \cdot E_\theta=\cot (\theta)$. The $2 / r$ term shows the radial lines spreading rapidly, and the $\cot (\theta)$ term shows the polar lines converging rapidly near the poles ($\theta \rightarrow 0$ or $\theta \rightarrow \pi$). $\nabla \cdot E_\phi$ is zero as there is no $\phi$ dependence in $\sqrt{g}$.
• Tangent vs. Physical Basis: This exercise demonstrates why physicists often prefer the unit (physical) basis vectors (like $\hat{r}$ or $\hat{\rho}$ ) over the non-unit tangent (coordinate) basis vectors ( $E_r$ or $E_\rho$ ). The physical basis vectors, being unit length, are non-expanding, making calculations simpler, while the tangent basis vectors intrinsically carry the complexity of the geometry.

✍️Mathematical Proof

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1. Derivation of Tensor Transformation Properties for Mixed Tensors
2. The Polar Tensor Basis in Cartesian Form
3. Verifying the Rank Two Zero Tensor
4. Tensor Analysis of Electric Susceptibility in Anisotropic Media
5. Analysis of Ohm's Law in an Anisotropic Medium
6. Verifying Tensor Transformations
7. Proof of Coordinate Independence of Tensor Contraction
8. Proof of a Tensor's Invariance Property
9. Proving Symmetry of a Rank-2 Tensor
10. Tensor Symmetrization and Anti-Symmetrization Properties
11. Symmetric and Antisymmetric Tensor Contractions
12. The Uniqueness of the Zero Tensor under Specific Symmetry Constraints
13. Counting Independent Tensor Components Based on Symmetry
14. Transformation of the Inverse Metric Tensor
15. Finding the Covariant Components of a Magnetic Field
16. Covariant Nature of the Gradient
17. Christoffel Symbol Transformation Rule Derivation
18. Contraction of the Christoffel Symbols and the Metric Determinant
19. Divergence of an Antisymmetric Tensor in Terms of the Metric Determinant
20. Calculation of the Metric Tensor and Christoffel Symbols in Spherical Coordinates
21. Christoffel Symbols for Cylindrical Coordinates
22. Finding Arc Length and Curve Length in Spherical Coordinates
23. Solving for Metric Tensors and Christoffel Symbols
24. Metric Tensor and Line Element in Non-Orthogonal Coordinates
25. Tensor vs. Non-Tensor Transformation of Derivatives
26. Verification of Covariant Derivative Identities
27. Divergence in Spherical Coordinates Derivation and Verification
28. Laplace Operator Derivation and Verification in Cylindrical Coordinates
29. Divergence of Tangent Basis Vectors in Curvilinear Coordinates

🧄Proof and Derivation-1

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