UNNS—Maxwell Electromagnetic Module

Interactive visualizations of EM fields via recursive special functions
Where Maxwell's equations emerge as nested algebraic extensions over the rationals

Select Interface Mode

Interactive field manipulation with real-time mathematical computation

Click: Add/move sources
Drag: Reshape field boundaries
Right-click: Toggle source polarity
Cursor Position: (0, 0)
Field Magnitude: 0.00
Function Value: 0.00
Recursion Depth: 3

UNNS Function Mode

Recursion Parameters

Source Management

Visualization Settings

UNNS Recursive Structure

IMPORTANT: Heuristic vs Rigorous Mathematics

This Tool Uses HEURISTIC Approximations

  • Special functions are mathematically exact (Legendre, Bessel, etc.)
  • Field calculations are simplified for visualization only
  • NOT rigorous Maxwell equation solvers
  • Scaling factors are arbitrary for visual clarity

For Rigorous EM Calculations Use:

  • Finite Element Method (FEM) solvers
  • Finite Difference Time Domain (FDTD)
  • Proper boundary conditions and material properties
  • Actual Maxwell equations: ∇×E = -∂B/∂t, ∇×B = μ₀J + μ₀ε₀∂E/∂t
📄 READ: UNNS-Maxwell FEEC/DEC Paper (Rigorous Framework)

Visualization Controls

Documentation Mode - Legendre Polynomials Active
Value: 3
Value: 0
Value: 1.0
Value: 1.0
Value: 100
Value: 0.5

Physical Field Visualization

Recursive Structure

Legendre Polynomials → Electrostatic Fields

Pn(x) = [(2n-1)xPn-1(x) - (n-1)Pn-2(x)] / n
Through the UNNS framework, Legendre polynomials generate field extensions Q(α) where the dominant root α creates electrostatic multipole patterns. Each recursion level adds algebraic complexity, manifesting as additional field lobes in physical space.

Mathematical Foundation

Legendre polynomials solve Legendre's differential equation:

d/dx[(1-x²)dPn/dx] + n(n+1)Pn = 0

Parameter Ranges

  • Polynomial order n: 0-20
  • Evaluation range: x ∈ [-1, 1]
  • Field intensity: 0.1-5.0
  • Animation speed: 0.1-3.0

Usage Examples

Dipole Field (n=1)

Set order to 1 for classic dipole electrostatic field with two-lobe pattern

Quadrupole Field (n=2)

Set order to 2 for four-lobe field pattern with alternating positive/negative regions

Higher Multipoles (n>2)

Higher orders create complex multi-lobe patterns representing higher-order charge distributions

UNNS Interpretation: Electrostatic fields emerge from nested sequences where each polynomial order extends the rational field Q, generating multipole moments through recursive algebraic structures.

Technical Implementation

Computation

Recursive function updates employ forward recursion with coefficient caching to minimize computational overhead. Intermediate values are stored for efficient parameter sweeping and animation rendering.

Rendering

Graphics rendering uses HTML5 Canvas for field line generation and real-time updates. Optimized for smooth animation at 60fps with dynamic parameter changes.

Performance

Optimal performance achieved with recursion orders n ≤ 15 and grid resolutions below 200×200 pixels. Memory usage scales linearly with field resolution.

Mathematical Accuracy

All calculations maintain double-precision floating-point accuracy suitable for display-level visualization. Functions use analytical expressions without empirical fitting.