🎬Visualizing Singular Radial Flux and Divergence

This project synthesizes four distinct demonstrations to explore the behavior of weighted radial vector fields. The first demo establishes the analytical foundation, proving that for $\vec{x} \neq 0$ , the surface flux of a $1/r^5$ weighted field maps to a volume integral with the scalar field $\phi(\vec{x}) = -1/r^5$ . The second demo highlights the critical role of the origin, illustrating why the $1/r^5$ weighting leads to a divergent integral unlike the standard $1/r^3$ Gauss’s Law case. The third and fourth demos provide interactive 3D and high-contrast 2D visualizations, respectively, demonstrating how the mathematical identity holds true as long as the volume of integration maintains an "exclusion zone" around the singularity. Together, these demos bridge the gap between abstract vector calculus identities and the physical intuition of potential wells and flux conservation.

🎬Narrated Video

🪜Analysis of the Scalar Field $\phi(x)$

Description

Analytical Derivation: The process begins by using the Divergence Theorem component-wise to transform the surface integral into a volume integral. This stage establishes the identity where the interaction between the radial field and coordinate weights results in the scalar field $\phi(x) = -1/r^5$ .
Origin Check: A critical decision point where the physical validity of the theorem is tested.
- Included: If the origin is within the volume, the field becomes singular (blows up), and the standard Divergence Theorem cannot be applied directly.
- Excluded: If the volume V exists entirely away from the origin, the field remains smooth and well-defined.
Divergence State (Demo 1): This state visualizes why the $1/r^5$ integral fails when the origin is included. It compares the "slow" growth of a standard $1/r^3$ field (which yields a finite flux via a Delta function) against the "aggressive" growth of the $1/r^5$ field, which results in a divergent (infinite) integral.
Valid Simulation 3D (Demos 2 & 3): In this scenario, the volume V is modeled as a moving region that navigates around the "Forbidden Zone" at the origin. It transitions into a more detailed visualization by adding a color bar and Scalar Mappable, allowing for real-time tracking of the field magnitude as the volume moves through the potential well.
Dimensional Translation (Demo 4): The problem is adapted to a 2D system to demonstrate how geometric changes affect divergence. The threshold for divergence shifts from $1/r^3$ (in 3D) to $1/r^2$ (in 2D), necessitating a shift to an approximately $1/r^4$ field for similar "extreme" behavior.
High-Contrast Visibility: The final state of the demonstration involves technical refinements to ensure the scalar field changes are visible, using high-contrast color maps (Inferno vs. Greys) and real-time numerical labels to display the exact value of $\phi(x)$ as the area moves.