The essential role of the metric tensor, which establishes the relationship between coordinate changes and physical distance through the differential line element ( $d s^2$ ). The general line element, $d s^2=d r^2+ r^2 d \theta^2+r^2 \sin ^2(\theta) d \varphi^2$, embeds the coordinate system's necessary scale factors ( $r^2$ and $r^2 \sin ^2(\theta)$ ) 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, $L=2 \pi R_0 \sin \left(\theta_0\right)$, 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.

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✍️Mathematical Proof

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1. Metric Tensor Defines Length: The most fundamental step is recognizing that the metric tensor ( $g_{ a b }$ ) is required to calculate distance in any non-Cartesian coordinate system. In spherical coordinates, the diagonal components $g_{r r}=1, g_{\theta \theta}=r^2$, and $g_{\varphi \varphi}=r^2 \sin ^2(\theta)$ directly encode the necessary scale factors for each coordinate.
2. The Differential Line Element ( $d s^2$ ): The line element squared, $d s^2=d r^2+r^2 d \theta^2+ r^2 \sin ^2(\theta) d \varphi^2$, is the core geometric formula. This formula dictates how small changes in each coordinate ($d r, d \theta, d \varphi$) contribute to a physical distance $d s$.
3. Path Traversal Simplifies Integration: For the specific curve given ( $r(t)=R_0, \theta(t)=\theta_0$ ), the path is simplified because the first two time derivatives, $\frac{d r}{d t}$ and $\frac{d \theta}{d t}$, are zero. This eliminated most of the terms under the square root, leaving only the $\varphi$-component.
4. Geometric Interpretation of the Result: The final answer, $L=2 \pi R_0 \sin \left(\theta_0\right)$, is physically intuitive.
   • The curve traces a circle on the surface of a sphere.
   • The term $R_0 \sin \left(\theta_0\right)$ is precisely the radius of that parallel circle (the distance from the Z-axis).
   • The total length $2 \pi$ (radius) is simply the circumference of the circle, confirming the curve made exactly one complete revolution for $0 \leq t \leq 1$.

✍️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

🧄Proof and Derivation-1

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