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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Curvilinear Coordinate Systems and Their Metrics.mp4

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1. Symmetric and Antisymmetric Components
2. Symmetry and the Zero Tensor
3. Independent Components and Tensor Symmetry
4. Proof of the Contravariant Nature of the Inverse Metric Tensor
5. Christoffel Symbols Geometric Change in Cylindrical Coordinates
6. Geometric Meaning of Arc Length in Spherical Coordinates
7. Curvilinear Coordinate Systems and Their Metrics
8. Impact of Non-Orthogonal Coordinates on Geometric Measurement
9. Partial vs Covariant Derivative The Core Difference </aside>