The essential role of the metric tensor, which establishes the relationship between coordinate changes and physical distance through the differential line element. The general line element embeds the coordinate system's necessary scale factors to properly measure distance. For the specific curve given, the calculation simplified significantly because the radial and polar angle derivatives were zero, isolating the integration to the azimuthal motion. The final result provides a satisfying geometric confirmation: it is precisely the circumference of the parallel circle traced out by the curve on the sphere's surface, demonstrating that the integral correctly measured one full revolution.

<aside> <img src="/icons/profile_gray.svg" alt="/icons/profile_gray.svg" width="40px" />

$\complement\cdots$Counselor

</aside>

Finding Arc Length and Curve Length in Spherical Coordinates

Finding Arc Length and Curve Length in Spherical Coordinates

<aside> 📢

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>