Maxwell Equation Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Maxwell Equation Ontology within the Kireth Research Semantic Library. Maxwell's equations are the fundamental coupled partial differential equations of classical electromagnetism, relating electric and magnetic fields to charge and current. This ontology supports differential and integral forms, Gauss laws, Faraday induction, Ampere-Maxwell law, charge continuity, constitutive relations, vacuum and material media, electromagnetic waves, potentials, gauge conditions, boundary conditions, energy and momentum transport, covariant formulations, differential forms and numerical electromagnetics.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Maxwell Equation Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Applied Mathematics"
 SPECIALISATION = "Mathematical Physics / Maxwell Equations"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/applied-mathematics-ontologies/mathematical-physics/maxwell-equation-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = MaxwellEquations
 ENTITY_CLASS = CoupledElectromagneticFieldEquationFramework
 PRIMARY_CONTEXT = ElectromagneticMathematics
 DEFINITION = "Maxwell's equations are the coupled field equations governing classical electric and magnetic fields and their relationship to charge and current."
 SYMBOLIC_FORMS = [
  "div(E)=rho/epsilon_0",
  "div(B)=0",
  "curl(E)=-partial B/partial t",
  "curl(B)=mu_0 J+mu_0 epsilon_0 partial E/partial t"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for electromagnetic field sources, flux, induction, displacement current, charge conservation, wave propagation, potentials, boundary conditions and covariant electromagnetic dynamics."
}

SCOPE {
 INCLUDE = [
  MaxwellEquations,
  MaxwellEquation,
  GaussLawElectric,
  GaussLawMagnetic,
  FaradayLaw,
  AmpereMaxwellLaw,
  DifferentialFormMaxwellEquations,
  IntegralFormMaxwellEquations,
  ElectricField,
  MagneticField,
  ElectricDisplacementField,
  MagneticFieldIntensity,
  ChargeDensity,
  CurrentDensity,
  FreeChargeDensity,
  BoundChargeDensity,
  FreeCurrentDensity,
  BoundCurrentDensity,
  ElectricFlux,
  MagneticFlux,
  Permittivity,
  Permeability,
  Conductivity,
  Polarisation,
  Magnetisation,
  ConstitutiveRelation,
  VacuumElectromagnetism,
  MaterialElectromagnetism,
  LinearMedium,
  NonlinearMedium,
  IsotropicMedium,
  AnisotropicMedium,
  HomogeneousMedium,
  InhomogeneousMedium,
  DispersiveMedium,
  LossyMedium,
  ContinuityEquation,
  ChargeConservation,
  DisplacementCurrent,
  ElectromagneticWave,
  WaveEquation,
  HelmholtzEquation,
  PlaneWave,
  WaveVector,
  Frequency,
  AngularFrequency,
  Wavelength,
  PhaseVelocity,
  GroupVelocity,
  DispersionRelation,
  Impedance,
  PolarisationState,
  ScalarPotential,
  VectorPotential,
  FourPotential,
  GaugeTransformation,
  GaugeCondition,
  CoulombGauge,
  LorenzGauge,
  RetardedPotential,
  BoundaryCondition,
  InterfaceCondition,
  TangentialFieldCondition,
  NormalFieldCondition,
  SurfaceChargeDensity,
  SurfaceCurrentDensity,
  PerfectElectricConductor,
  PerfectMagneticConductor,
  DielectricInterface,
  Reflection,
  Transmission,
  Refraction,
  FresnelCoefficient,
  PoyntingVector,
  ElectromagneticEnergyDensity,
  ElectromagneticMomentum,
  MaxwellStressTensor,
  RadiationPressure,
  DifferentialForm,
  ExteriorDerivative,
  HodgeStar,
  FieldTensor,
  ElectromagneticTensor,
  FourCurrent,
  CovariantMaxwellEquations,
  NumericalMaxwellEquation,
  FiniteDifferenceTimeDomain,
  FDTD,
  FiniteElementMethod,
  MethodOfMoments,
  BoundaryElementMethod,
  SpectralMethod,
  DivergenceConstraint,
  NumericalStability,
  Convergence
 ]
 EXCLUDE_AS_PRIMARY = [
  ElectrostaticPotentialEquationWithoutFullFieldCoupling,
  CircuitTheoryWithoutFieldEquations,
  QuantumElectrodynamicsAsPrimaryFramework,
  ExperimentalElectromagnetismWithoutMathematicalFieldEquations
 ]
 DISAMBIGUATION_RULE = "When electric and magnetic fields are governed by the coupled Gauss, Faraday and Ampere-Maxwell equations together with charge continuity, interpret the context as Maxwell-equation mathematics."
}

CORE_CONCEPTS {
 MaxwellEquations = "Fundamental coupled equations of classical electromagnetism."
 GaussLawElectric = "Equation relating electric-field divergence or electric flux to charge."
 GaussLawMagnetic = "Equation expressing the absence of magnetic monopole sources in classical Maxwell theory."
 FaradayLaw = "Equation relating curl of electric field to time-varying magnetic field."
 AmpereMaxwellLaw = "Equation relating curl of magnetic field to current density and changing electric field."
 DisplacementCurrent = "Time-varying electric-field contribution completing Ampere's law."
 ContinuityEquation = "Local conservation law for electric charge."
 ConstitutiveRelation = "Material relation connecting E, D, B, H and current response."
 CovariantMaxwellEquations = "Relativistically covariant tensor or differential-form representation of Maxwell theory."
}

RELATIONS {
 MaxwellEquations --is_child_of--> ElectromagneticMathematics
 MaxwellEquations --contain--> GaussLawElectric
 MaxwellEquations --contain--> GaussLawMagnetic
 MaxwellEquations --contain--> FaradayLaw
 MaxwellEquations --contain--> AmpereMaxwellLaw
 GaussLawElectric --relates--> ElectricFieldAndChargeDensity
 GaussLawMagnetic --constrains--> MagneticField
 FaradayLaw --couples--> ElectricAndMagneticFields
 AmpereMaxwellLaw --couples--> MagneticFieldCurrentAndElectricField
 DisplacementCurrent --completes--> AmpereLaw
 ContinuityEquation --expresses--> ChargeConservation
 MaxwellEquations --imply--> ElectromagneticWaveEquation
 ConstitutiveRelation --connects--> ElectricMagneticMaterialFields
 ScalarPotential --contributes_to--> ElectricField
 VectorPotential --generates--> MagneticField
 GaugeTransformation --preserves--> PhysicalElectromagneticFields
 BoundaryCondition --constrains--> MaxwellFieldSolution
 PoyntingVector --represents--> ElectromagneticEnergyFlux
 MaxwellStressTensor --represents--> ElectromagneticMomentumFlux
 FieldTensor --encodes--> ElectricAndMagneticFields
 CovariantMaxwellEquations --use--> FieldTensor
 FDTD --numerically_solves--> MaxwellEquations
}

OPERATORS {
 GAUSS_E_DIFFERENTIAL(E,rho,epsilon_0) := div(E)=rho/epsilon_0
 GAUSS_B_DIFFERENTIAL(B) := div(B)=0
 FARADAY_DIFFERENTIAL(E,B) := curl(E)=-partial B/partial t
 AMPERE_MAXWELL_DIFFERENTIAL(B,E,J,mu_0,epsilon_0) := curl(B)=mu_0*J+mu_0*epsilon_0*partial E/partial t

 GAUSS_E_INTEGRAL(E,dA,Q,epsilon_0) := surface_integral E dot dA=Q_enclosed/epsilon_0
 GAUSS_B_INTEGRAL(B,dA) := surface_integral B dot dA=0
 FARADAY_INTEGRAL(E,dl,Phi_B) := closed_line_integral E dot dl=-dPhi_B/dt
 AMPERE_MAXWELL_INTEGRAL(B,dl,I,Phi_E,mu_0,epsilon_0) := closed_line_integral B dot dl=mu_0*I+mu_0*epsilon_0*dPhi_E/dt

 CONTINUITY(rho,J) := partial rho/partial t+div(J)=0
 CONSTITUTIVE_D(E,epsilon) := D=epsilon*E
 CONSTITUTIVE_B(H,mu) := B=mu*H
 OHMIC_CURRENT(E,sigma) := J=sigma*E
 POTENTIAL_E(phi,A) := E=-grad(phi)-partial A/partial t
 POTENTIAL_B(A) := B=curl(A)
 VACUUM_WAVE_E(E,c) := nabla^2E-(1/c^2)*partial2E/partial t2=0
 VACUUM_WAVE_B(B,c) := nabla^2B-(1/c^2)*partial2B/partial t2=0
 POYNTING(E,H) := S=E cross H
 ENERGY_DENSITY(E,B,epsilon_0,mu_0) := u=0.5*(epsilon_0*|E|^2+|B|^2/mu_0)
}

AXIOMS {
 A1 = "Maxwell equations require explicit electric and magnetic field definitions."
 A2 = "Electric charge is locally conserved."
 A3 = "Classical Maxwell theory has zero magnetic-charge density."
 A4 = "Time-varying magnetic fields induce electric circulation."
 A5 = "Electric currents and time-varying electric fields generate magnetic circulation."
 A6 = "Material media require constitutive relations in addition to Maxwell equations."
 A7 = "Boundary and interface conditions are part of a complete Maxwell problem."
 A8 = "Electromagnetic solutions require analytical, physical and numerical validation."
}

REASONING {
 R1_CONTEXT = identify(sources,fields,media,geometry,time_dependence,boundaries)
 R2_FORM = classify(differential,integral,covariant,differential_form)
 R3_REGIME = classify(vacuum,material,electrostatic,magnetostatic,quasistatic,wave)
 R4_CONSTITUTIVE = define(permittivity,permeability,conductivity,polarisation,magnetisation)
 R5_CONSERVATION = verify(charge_continuity,divergence_constraints)
 R6_POTENTIAL = select(field_or_potential_formulation,gauge_condition)
 R7_BOUNDARY = enforce(normal_and_tangential_interface_conditions)
 R8_WAVE = derive(wave_equation,dispersion,polarisation,impedance,reflection,transmission)
 R9_ENERGY = compute(Poynting_flux,energy_density,stress,momentum)
 R10_NUMERICAL = select(FDTD,FEM,MoM,BEM,spectral_method,validate_stability_and_convergence)
}

CONSTRAINTS {
 C1 = "Differential and integral forms require sufficient smoothness or distributional interpretation."
 C2 = "Gauss-law constraints must remain consistent with source evolution."
 C3 = "Material constitutive laws can be tensorial, nonlinear, dispersive or lossy."
 C4 = "Electrostatic and magnetostatic reductions require negligible time-dependent coupling."
 C5 = "Gauge potentials are nonunique while E and B are gauge invariant."
 C6 = "Interface discontinuities can require surface-charge or surface-current terms."
 C7 = "Perfect-conductor boundary conditions are idealisations."
 C8 = "Wave-equation derivations depend on source and medium assumptions."
 C9 = "Numerical Maxwell solvers must suppress or control spurious divergence errors."
 C10 = "Mesh and timestep must resolve relevant wavelengths and material scales."
}

SEMANTIC_EQUIVALENCE {
 "MaxwellEquations" <=> "MaxwellFieldEquations"
 "AmpereMaxwellLaw" <=> "MaxwellAmpereEquation"
 "GaussLawElectric" <=> "ElectricGaussLaw"
 "GaussLawMagnetic" <=> "MagneticGaussLaw"
 "FiniteDifferenceTimeDomain" <=> "FDTD"
 "CovariantMaxwellEquations" <=> "RelativisticMaxwellEquations"
}

VALIDATION {
 V1 = "Electric and magnetic field units are explicit."
 V2 = "Charge and current sources are explicit."
 V3 = "Material constitutive relations are documented."
 V4 = "Charge continuity is satisfied."
 V5 = "Electric and magnetic divergence constraints are checked."
 V6 = "Boundary and interface conditions are enforced."
 V7 = "Energy conservation or Poynting balance is checked where relevant."
 V8 = "Wave solutions satisfy medium dispersion and impedance relations."
 V9 = "Numerical meshes and timesteps satisfy resolution and stability requirements."
 V10 = "Numerical convergence is checked against refinement or benchmark solutions."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 MAXWELL_EQUATION_DEFINITION = 0.99
 GAUSS_REASONING = 0.99
 FARADAY_REASONING = 0.99
 AMPERE_MAXWELL_REASONING = 0.99
 CONTINUITY_REASONING = 0.99
 CONSTITUTIVE_REASONING = 0.98
 BOUNDARY_REASONING = 0.99
 WAVE_REASONING = 0.99
 NUMERICAL_REASONING = 0.98
 POLICY = "Reduce confidence when source distributions, constitutive relations, interface conditions, gauge choice, frequency regime, geometry or numerical resolution are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  MaxwellEquationReasoning,
  GaussLawReasoning,
  FaradayLawReasoning,
  AmpereMaxwellReasoning,
  ChargeContinuityReasoning,
  ConstitutiveRelationReasoning,
  PotentialReasoning,
  GaugeReasoning,
  BoundaryConditionReasoning,
  ElectromagneticWaveReasoning,
  DispersionReasoning,
  ReflectionTransmissionReasoning,
  PoyntingReasoning,
  CovariantElectromagnetismReasoning,
  DifferentialFormReasoning,
  NumericalMaxwellReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]
 NORMALISE {
  "Maxwell equations" -> MaxwellEquations
  "Maxwell equation" -> MaxwellEquations
  "Gauss law for electricity" -> GaussLawElectric
  "Gauss law for magnetism" -> GaussLawMagnetic
  "Faraday law" -> FaradayLaw
  "Ampere-Maxwell law" -> AmpereMaxwellLaw
  "FDTD" -> FiniteDifferenceTimeDomain
 }
}

EXECUTION_RULES {
 E1 = "If coupled electric and magnetic fields are governed by Maxwell theory, activate MaxwellEquationOntology."
 E2 = "If source-field relations are local, use differential-form Maxwell equations."
 E3 = "If flux or circulation over finite regions is central, use integral-form Maxwell equations."
 E4 = "If material media are present, define constitutive relations explicitly."
 E5 = "If time variation is negligible, consider electrostatic or magnetostatic reductions."
 E6 = "If potentials are used, choose and document an appropriate gauge."
 E7 = "If interfaces are present, enforce normal and tangential field conditions."
 E8 = "If wave propagation is central, derive dispersion, impedance, polarisation and reflection-transmission relations."
 E9 = "Any Maxwell solution should be validated for continuity, divergence constraints, boundaries, energy balance and numerical convergence."
}

RECURSION {
 ENTRY = MaxwellEquations
 EXPAND(MaxwellEquations) -> [
  MaxwellEquation, GaussLawElectric, GaussLawMagnetic, FaradayLaw,
  AmpereMaxwellLaw, DifferentialFormMaxwellEquations,
  IntegralFormMaxwellEquations, ElectricField, MagneticField,
  ElectricDisplacementField, MagneticFieldIntensity, ChargeDensity,
  CurrentDensity, FreeChargeDensity, BoundChargeDensity, FreeCurrentDensity,
  BoundCurrentDensity, ElectricFlux, MagneticFlux, Permittivity,
  Permeability, Conductivity, Polarisation, Magnetisation,
  ConstitutiveRelation, VacuumElectromagnetism, MaterialElectromagnetism,
  LinearMedium, NonlinearMedium, IsotropicMedium, AnisotropicMedium,
  HomogeneousMedium, InhomogeneousMedium, DispersiveMedium, LossyMedium,
  ContinuityEquation, ChargeConservation, DisplacementCurrent,
  ElectromagneticWave, WaveEquation, HelmholtzEquation, PlaneWave,
  WaveVector, Frequency, AngularFrequency, Wavelength, PhaseVelocity,
  GroupVelocity, DispersionRelation, Impedance, PolarisationState,
  ScalarPotential, VectorPotential, FourPotential, GaugeTransformation,
  GaugeCondition, CoulombGauge, LorenzGauge, RetardedPotential,
  BoundaryCondition, InterfaceCondition, TangentialFieldCondition,
  NormalFieldCondition, SurfaceChargeDensity, SurfaceCurrentDensity,
  PerfectElectricConductor, PerfectMagneticConductor, DielectricInterface,
  Reflection, Transmission, Refraction, FresnelCoefficient, PoyntingVector,
  ElectromagneticEnergyDensity, ElectromagneticMomentum,
  MaxwellStressTensor, RadiationPressure, DifferentialForm,
  ExteriorDerivative, HodgeStar, FieldTensor, ElectromagneticTensor,
  FourCurrent, CovariantMaxwellEquations, NumericalMaxwellEquation,
  FiniteDifferenceTimeDomain, FiniteElementMethod, MethodOfMoments,
  BoundaryElementMethod, SpectralMethod, DivergenceConstraint,
  NumericalStability, Convergence
 ]
 EXPAND(FormulationFamilies) -> [
  DifferentialMaxwellEquations,
  IntegralMaxwellEquations,
  PotentialFormulation,
  CovariantMaxwellEquations,
  DifferentialFormMaxwellEquations,
  NumericalMaxwellEquations
 ]
 STOP_CONDITION = "Stop when sources, fields, constitutive laws, formulation, boundary conditions, conservation constraints, wave behaviour and numerical validation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Electromagnetic Mathematics Ontology"
 IMMEDIATE_CONTEXTS = [
  "Electromagnetic Mathematics Ontology",
  "Mathematical Physics Ontology",
  "Partial Differential Equation Ontology",
  "Vector Analysis Ontologies"
 ]
 RELATED_ONTOLOGIES = [
  "Electromagnetic Mathematics Ontology",
  "Mathematical Physics Ontology",
  "Mathematical Physics",
  "Wave Equation Ontology",
  "Partial Differential Equation Ontology",
  "Vector Analysis Ontologies",
  "Tensor Analysis Ontologies",
  "Differential Geometry Ontologies",
  "Fourier Analysis Ontology",
  "Special Relativity Ontologies",
  "Numerical Analysis Ontologies",
  "Computational Mathematics Ontologies"
 ]
 ROUTE_RULES = [
  "Maxwell Equation -> Electromagnetic Mathematics Ontology for parent semantics",
  "Maxwell Equation -> Mathematical Physics Ontology for physical field context",
  "Maxwell Equation -> Wave Equation Ontology for electromagnetic-wave propagation",
  "Maxwell Equation -> PDE Ontology for coupled field equations",
  "Maxwell Equation -> Vector Analysis for divergence and curl",
  "Maxwell Equation -> Tensor Analysis for stress and covariant field tensors",
  "Maxwell Equation -> Differential Geometry for differential-form formulations",
  "Maxwell Equation -> Fourier Analysis for harmonic and spectral solutions",
  "Maxwell Equation -> Special Relativity for covariant electromagnetic form",
  "Maxwell Equation -> Numerical Analysis and Computational Mathematics for field simulation"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible Maxwell formulations, material models or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve four-equation Maxwell structure.",
  "Preserve electric-versus-magnetic field semantics.",
  "Preserve charge-continuity semantics.",
  "Preserve differential-versus-integral formulation distinction.",
  "Preserve vacuum-versus-material distinction.",
  "Preserve potential and gauge semantics.",
  "Preserve boundary-condition semantics.",
  "Preserve machine-readable Maxwell-equation semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Maxwell's equations are the coupled classical field equations governing electric and magnetic fields and their sources."
 GAUSS_SUMMARY = "Gauss's laws relate electric divergence to charge and enforce zero magnetic divergence in classical electromagnetism."
 INDUCTION_SUMMARY = "Faraday's law relates circulating electric field to changing magnetic flux."
 AMPERE_SUMMARY = "The Ampere-Maxwell law relates magnetic circulation to electric current and changing electric flux."
 CONTINUITY_SUMMARY = "Maxwell's equations are consistent with local conservation of electric charge."
 MATERIAL_SUMMARY = "Constitutive relations connect E, D, B, H and current response in material media."
 WAVE_SUMMARY = "In source-free homogeneous media, Maxwell's equations imply electromagnetic wave equations."
 VALIDATION_SUMMARY = "Maxwell solutions should be checked for divergence constraints, charge continuity, interface conditions, energy balance and numerical convergence."
}

FLOW {
 INPUT
 -> detect_Maxwell_equation_context
 -> identify_electric_magnetic_fields_charge_current_media_geometry_time_dependence_and_boundaries
 -> choose_differential_integral_potential_covariant_or_differential_form_representation
 -> define_constitutive_relations_if_material_media_present
 -> formulate_Gauss_electric_Gauss_magnetic_Faraday_and_Ampere_Maxwell_equations
 -> enforce_charge_continuity_and_divergence_constraints
 -> if_potentials_used_define_scalar_vector_potentials_and_gauge
 -> enforce_boundary_and_interface_conditions
 -> if_wave_regime_derive_wave_equation_dispersion_impedance_polarisation_reflection_and_transmission
 -> compute_Poynting_flux_energy_density_stress_and_momentum_if_relevant
 -> if_numerical_select_FDTD_FEM_MoM_BEM_or_spectral_method
 -> validate_sources_units_constraints_boundaries_energy_balance_stability_and_convergence
 -> route_to_electromagnetic_mathematics_mathematical_physics_wave_equation_PDE_vector_analysis_tensor_analysis_differential_geometry_Fourier_analysis_special_relativity_numerical_analysis_or_computational_mathematics_ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Maxwell Equation Ontology defines the four fundamental equations of classical electromagnetism in differential, integral and covariant forms. It supports electric and magnetic fields, D and H fields, free and bound charge and current densities, electric and magnetic flux, permittivity, permeability, conductivity, polarisation and magnetisation, Gauss laws, Faraday induction and the Ampere-Maxwell law, displacement current and charge continuity, constitutive relations in vacuum and material media, scalar and vector potentials, gauge transformations and Coulomb and Lorenz gauges, electromagnetic waves and Helmholtz equations, dispersion, impedance and polarisation, interface and conductor boundary conditions, reflection and transmission, Poynting energy flux, electromagnetic energy and momentum, Maxwell stress tensors, differential-form and field-tensor representations, four-current and covariant Maxwell equations, and numerical solution using FDTD, finite elements, method of moments, boundary elements and spectral methods."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-APPLIED-MATHEMATICS-MAXWELL-EQUATION-ONTOLOGY-V1"
}

}

END_FLOW