ideal_induction_kernel

deep_causality_physics

Function ideal_induction_kernel 

Source 
pub fn ideal_induction_kernel(
    v_manifold: &Manifold<f64>,
    b_manifold: &Manifold<f64>,
) -> Result<CausalTensor<f64>, PhysicsError>
Expand description

Calculates the time evolution of the magnetic field (Frozen-in flux). $$ \frac{\partial \mathbf{B}}{\partial t} = \nabla \times (\mathbf{v} \times \mathbf{B}) $$

Geometric Algebra Implementation: In the language of differential forms/GA on a Manifold: $$ \partial_t B = -d(i_v B) $$ where $B$ is a 2-form (flux), $v$ is a vector field (represented as a 1-form), $i_v$ is interior product (contraction), and $d$ is exterior derivative.

This implementation relies on the identity: $$ i_v B = \star (v \wedge \star B) $$ (valid for 3D manifolds where $v$ and $\star B$ are 1-forms).

§Arguments

  • v_manifold - Manifold containing the velocity field $v$ (1-form).
  • b_manifold - Manifold containing the magnetic flux 2-form $B$.

§Returns

  • Result<CausalTensor<f64>, PhysicsError> - Rate of change of B (2-form), i.e., $-\partial_t B$. Wait, the equation is $\partial_t B = \dots$. The function returns $\partial_t B$.