Struct MaxwellSolver 

pub struct MaxwellSolver;

A standard solver for Maxwell’s Equations in Geometric Algebra.

This solver provides methods to derive electromagnetic properties from potentials and fields in a coordinate-free manner, suitable for relativistic electrodynamics.

Theoretical Basis

In Geometric Algebra (Space-Time Algebra), Maxwell’s equations are unified into a single equation: $\nabla F = J$ where:

• $F = \nabla A$ is the electromagnetic field bivector.
• $A$ is the vector potential.
• $J$ is the source current density.
• $\nabla$ is the vector derivative (gradient).

Implementations

impl MaxwellSolver

pub fn calculate_field_tensor( gradient: &CausalMultiVector<f64>, potential: &CausalMultiVector<f64>, ) -> Result<CausalMultiVector<f64>, PhysicsError>

Calculates the Electromagnetic Field Tensor $F$ from the Potential $A$.

$F = \nabla \wedge A$

This represents the “Faraday bivector” containing both Electric (E) and Magnetic (B) fields.

Arguments

• gradient - The vector derivative operator $\nabla$ (or directional derivatives).
• potential - The vector potential $A$.

Returns

• Ok(CausalMultiVector) - The field bivector $F$ (Grade 2).
• Err(PhysicsError) - If dimensions or metrics mismatch.

pub fn calculate_potential_divergence( gradient: &CausalMultiVector<f64>, potential: &CausalMultiVector<f64>, ) -> Result<f64, PhysicsError>

Calculates the Lorenz Gauge scalar.

$L = \nabla \cdot A$

In the Lorenz gauge, this value should be zero (or near zero).

Returns

• Ok(f64) - The scalar divergence.
• Err(PhysicsError) - If inputs are not pure grade-1 vectors.

pub fn calculate_current_density( gradient: &CausalMultiVector<f64>, field: &CausalMultiVector<f64>, ) -> Result<CausalMultiVector<f64>, PhysicsError>

Calculates the Source Current Density $J$ from the Field Tensor $F$.

$J = \nabla \cdot F$ (Vector part of $\nabla F$)

Note: The full source equation is $\nabla F = J$. Since $F$ is a bivector, $\nabla F$ has a vector part ($\nabla \cdot F$) and a trivector part ($\nabla \wedge F$). The trivector part is zero $\nabla \wedge F = 0$ (Bianchi identity) if F comes from a potential. Thus $J$ corresponds to the vector part.

pub fn calculate_poynting_flux( e_field: &CausalMultiVector<f64>, b_field: &CausalMultiVector<f64>, ) -> Result<CausalMultiVector<f64>, PhysicsError>

Calculates the Poynting Flux Vector $S$.

$S = E \times B$

Computes the energy flux density from separated Electric and Magnetic field vectors.

Arguments

• e_field - Electric field vector E.
• b_field - Magnetic field vector B.