🎬Visualizing Circulation and the Winding Number Singularity

The four demonstrations collectively illustrate that for the vector field $\vec{v}=\frac{1}{\rho} \vec{e}_\phi$ , the result of a line integral is determined not by the local properties of the field along the path, but by the path's relationship to the central singularity. Local vs. Global Behavior: Even though the curl is zero at every point $\rho>0$ , the circulation is non-zero for any path that encloses the origin. This reveals that the field is "locally conservative" but "globally non-conservative" in a non-simply connected domain. Topological Quantization: The circulation is a topological invariant known as the winding number. In the enclosing demo, the integral yields $4 \pi$ because the path completes two full revolutions ( $\Delta \phi=4 \pi$ ), regardless of its specific radial oscillations or 3D height. The Role of the Singularity: The contrast between the enclosing and non-enclosing paths shows that the origin acts as a "delta-function" source of curl. If a path does not loop around the $z$ -axis (Scenario 2), the angular gains and losses cancel out perfectly, resulting in zero circulation. Stokes' Theorem Limitation: The demos clarify why standard Stokes' Theorem seems to "fail" for enclosing paths. To satisfy the theorem, any surface bounded by the enclosing loop must pierce the $z$ -axis; because the field is singular there, the surface integral must account for the singular vortex at the origin to match the $4 \pi$ result found via the line integral.

🎬Narrated Video

🪜State Diagram: Topological Circulation and Singularity Visualization Pipeline

This state diagram illustrating how the problem evolves from a single path visualization to a topological comparison of circulation results.

Breakdown of the Demo and Example States

Example 1: Path Definition: This represents the initial branching point in the derivation where you choose between a path that encloses the singularity at $\rho=0$ and one that does not.
Demo 1 (3D Path Mapping): Focuses on the physical path of $\Gamma$ as it winds around the z-axis, demonstrating how it picks up circulation despite the local curl being zero.
Demo 2 (Laps Counter): Evolves the visualization by adding a counter to track the winding number (N). This proves the path completes exactly two revolutions (N=2), justifying the final result of $4\pi$ .
Theoretical Interpretation (Scenario 2): Based on Example 1, this state represents the logic that any angular increase in a non-enclosing path is eventually canceled by a decrease as the loop closes.
Demo 3 (Topological Comparison): This demo synthesizes both scenarios into a side-by-side view to show how the result depends entirely on whether the path "traps" the central singularity.
Demo 4 (Cumulative Integration): The final stage of the demos adds a real-time graph comparing the accumulation of circulation. It shows the enclosing path steadily climbing to 4\pi while the non-enclosing path oscillates before returning to exactly zero.