Finding Arc Length and Curve Length in Spherical Coordinates

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.

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.
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 $d s$.
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.
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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🧄Proof and Derivation-1

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