The hyperbolic and parabolic coordinate systems, though both curvilinear, reveal different geometric properties through their metric tensors. For the hyperbolic system, the non-diagonal metric tensor indicates that the coordinate lines are not orthogonal, meaning the basis vectors at any given point are not perpendicular. The metric components, and consequently the scale factors, vary with both coordinates, highlighting a non-uniform and non-flat geometry. In contrast, the parabolic coordinate system is orthogonal, as shown by its diagonal metric tensor. While the basis vectors are perpendicular, their magnitudes (the scale factors) still depend on the coordinates. This change in scale means that while the coordinate grid is "square" in a generalized sense, the distance represented by a unit change in a coordinate varies with location. In both systems, the non-zero Christoffel symbols are a natural consequence of the changing basis vectors, which is a fundamental characteristic of curvilinear coordinates.

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✍️Mathematical Proof

$\complement\cdots$Counselor

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Hyperbolic Coordinates

• The metric tensor $g_{a b}$ is not diagonal, indicating that the coordinate system is not orthogonal. The non-zero off-diagonal components ( $g_{u v}$ and $g_{v u}$ ) reflect the fact that the coordinate lines for $u$ and $v$ are not perpendicular to each other at every point.
• The metric tensor components are functions of both coordinates, $u$ and $v$. This means that the geometry of the space, as measured by the metric, changes depending on the location.
• The Christoffel symbols are not all zero. This is expected since the coordinate basis vectors are not constant, meaning they change from point to point. The non-zero symbols describe how the basis vectors change as you move through the space.

Parabolic Coordinates

• The metric tensor $g_{a b}$ is diagonal, which confirms that the coordinate system is orthogonal. The diagonal components are equal, $g_{t t}=g_{s s}=t^2+s^2$. This means the scale factors for both coordinate directions are the same at any given point.
• The metric tensor components depend on both coordinates, $t$ and $s$, even though the system is orthogonal. The scale factor changes as you move through the space, so the "length" of a step in a given coordinate direction is not uniform.
• The Christoffel symbols are non-zero. Although the basis vectors are orthogonal to each other, their lengths and directions change from point to point, leading to non-zero Christoffel symbols. This is a common feature of curvilinear coordinate systems in general.

✍️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. ‣

🧄Proof and Derivation-1

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