Electromagnetic Mathematics Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Electromagnetic Mathematics Ontology within the Kireth Research Semantic Library. Electromagnetic mathematics studies the mathematical structures governing electric and magnetic fields, charge, current, potentials, electromagnetic waves, constitutive relations and field interactions. This ontology supports Maxwell's equations, vector calculus, scalar and vector potentials, gauge transformations, boundary conditions, electromagnetic energy and momentum, wave propagation, material media, electrostatics, magnetostatics, radiation, Green functions, Fourier methods, differential forms, tensor formulations and relationships with mathematical physics, partial differential equations, vector analysis, relativity and computational electromagnetics.

FLOW_ALGEBRA_RECORD {

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

IDENTITY {
 PRIMARY_ENTITY = ElectromagneticMathematics
 ENTITY_CLASS = MathematicalElectromagneticFieldFramework
 PRIMARY_CONTEXT = MathematicalPhysics
 DEFINITION = "Electromagnetic mathematics is the mathematical study of electric and magnetic fields, their sources, interactions, propagation and boundary behaviour through Maxwell's equations and related field structures."
 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 charge, current, electric and magnetic fields, potentials, electromagnetic waves, boundary conditions, material response, gauge structure, radiation and computational field analysis."
}

SCOPE {
 INCLUDE = [
  ElectromagneticMathematics,
  Electromagnetism,
  ElectricCharge,
  ChargeDensity,
  Current,
  CurrentDensity,
  ElectricField,
  MagneticField,
  ElectromagneticField,
  ElectricFlux,
  MagneticFlux,
  Permittivity,
  Permeability,
  Conductivity,
  Polarisation,
  Magnetisation,
  ElectricDisplacementField,
  MagneticFieldIntensity,
  CoulombLaw,
  BiotSavartLaw,
  AmpereLaw,
  FaradayLaw,
  GaussLawElectric,
  GaussLawMagnetic,
  MaxwellEquations,
  ContinuityEquation,
  LorentzForce,
  Electrostatics,
  Magnetostatics,
  ScalarPotential,
  VectorPotential,
  ElectromagneticPotential,
  GaugeTransformation,
  GaugeCondition,
  CoulombGauge,
  LorenzGauge,
  PoissonEquation,
  LaplaceEquation,
  WaveEquation,
  HelmholtzEquation,
  RetardedPotential,
  GreenFunction,
  BoundaryCondition,
  InterfaceCondition,
  DirichletBoundaryCondition,
  NeumannBoundaryCondition,
  PerfectElectricConductor,
  PerfectMagneticConductor,
  DielectricInterface,
  ElectromagneticWave,
  PlaneWave,
  WaveVector,
  Frequency,
  AngularFrequency,
  Wavelength,
  PhaseVelocity,
  GroupVelocity,
  PolarisationState,
  LinearPolarisation,
  CircularPolarisation,
  EllipticalPolarisation,
  PoyntingVector,
  ElectromagneticEnergyDensity,
  ElectromagneticMomentum,
  MaxwellStressTensor,
  Radiation,
  DipoleRadiation,
  MultipoleExpansion,
  NearField,
  FarField,
  SkinDepth,
  Dispersion,
  RefractiveIndex,
  Impedance,
  Reflection,
  Transmission,
  Refraction,
  SnellLaw,
  FresnelCoefficient,
  Waveguide,
  ResonantCavity,
  Mode,
  TE_Mode,
  TM_Mode,
  TEM_Mode,
  Eigenmode,
  FourierTransform,
  LaplaceTransform,
  VectorCalculus,
  Gradient,
  Divergence,
  Curl,
  DifferentialForms,
  ExteriorDerivative,
  HodgeStar,
  FieldTensor,
  ElectromagneticTensor,
  FourCurrent,
  FourPotential,
  CovariantElectromagnetism,
  NumericalElectromagnetics,
  FiniteDifferenceTimeDomain,
  FDTD,
  FiniteElementMethod,
  MethodOfMoments,
  BoundaryElementMethod,
  SpectralMethod
 ]
 EXCLUDE_AS_PRIMARY = [
  CircuitTheoryWithoutFieldRepresentation,
  QuantumElectrodynamicsAsPrimaryTheory,
  ExperimentalElectromagnetismWithoutMathematicalFieldModel,
  GeneralWavePhysicsWithoutElectromagneticFieldSemantics
 ]
 DISAMBIGUATION_RULE = "When electric and magnetic fields, charge, current, potentials or electromagnetic waves are represented through Maxwell equations or equivalent field mathematics, interpret the context as electromagnetic mathematics."
}

CORE_CONCEPTS {
 ElectromagneticMathematics = "Mathematical formulation of electric and magnetic field behaviour."
 ElectricField = "Vector field describing electric force per unit charge."
 MagneticField = "Vector field describing magnetic effects associated with currents and changing electric fields."
 MaxwellEquations = "Coupled field equations governing classical electromagnetism."
 ChargeDensity = "Spatial density of electric charge."
 CurrentDensity = "Vector density of electric current flow."
 ScalarPotential = "Scalar field whose gradient contributes to electric field representation."
 VectorPotential = "Vector field whose curl gives magnetic field."
 GaugeTransformation = "Transformation of electromagnetic potentials leaving physical fields invariant."
 PoyntingVector = "Vector representing electromagnetic energy flux."
}

RELATIONS {
 ElectromagneticMathematics --is_child_of--> MathematicalPhysicsOntology
 ElectricCharge --sources--> ElectricField
 CurrentDensity --sources--> MagneticField
 MaxwellEquations --govern--> ElectromagneticField
 GaussLawElectric --relates--> ElectricFieldAndChargeDensity
 GaussLawMagnetic --constrains--> MagneticField
 FaradayLaw --couples--> ElectricAndTimeVaryingMagneticFields
 AmpereLaw --couples--> MagneticFieldCurrentAndTimeVaryingElectricField
 ContinuityEquation --expresses--> ChargeConservation
 LorentzForce --acts_on--> ChargedParticle
 ScalarPotential --contributes_to--> ElectricField
 VectorPotential --generates--> MagneticField
 GaugeTransformation --preserves--> ElectricAndMagneticFields
 PoissonEquation --models--> ElectrostaticPotentialWithSources
 LaplaceEquation --models--> SourceFreeElectrostaticPotential
 WaveEquation --governs--> ElectromagneticWave
 PoyntingVector --represents--> ElectromagneticEnergyFlux
 MaxwellStressTensor --represents--> ElectromagneticMomentumFlux
 Reflection --occurs_at--> Interface
 Transmission --occurs_at--> Interface
 FresnelCoefficient --quantifies--> ReflectionAndTransmission
 Waveguide --supports--> ElectromagneticMode
 ResonantCavity --supports--> Eigenmode
 FieldTensor --encodes--> ElectromagneticField
 FourCurrent --sources--> CovariantElectromagneticField
 FDTD --numerically_solves--> MaxwellEquations
}

OPERATORS {
 COULOMB_FORCE(q1,q2,r_hat,r) := F=(1/(4*pi*epsilon_0))*q1*q2/r^2*r_hat
 ELECTRIC_FIELD_FORCE(F,q) := E=F/q
 LORENTZ_FORCE(q,E,v,B) := F=q*(E+v cross B)
 GAUSS_E(E,rho) := div(E)=rho/epsilon_0
 GAUSS_B(B) := div(B)=0
 FARADAY(E,B) := curl(E)=-partial B/partial t
 AMPERE_MAXWELL(B,E,J) := curl(B)=mu_0*J+mu_0*epsilon_0*partial E/partial t
 CONTINUITY(rho,J) := partial rho/partial t+div(J)=0
 POTENTIAL_E(phi,A) := E=-grad(phi)-partial A/partial t
 POTENTIAL_B(A) := B=curl(A)
 POISSON_POTENTIAL(phi,rho) := nabla^2 phi=-rho/epsilon_0
 WAVE_ELECTRIC(E,c) := nabla^2 E-(1/c^2)*partial2E/partial t2=0
 WAVE_MAGNETIC(B,c) := nabla^2 B-(1/c^2)*partial2B/partial t2=0
 POYNTING(E,H) := S=E cross H
 ENERGY_DENSITY(E,B) := u=0.5*(epsilon_0*|E|^2+|B|^2/mu_0)
 PLANE_WAVE_DISPERSION(omega,k,c) := omega=c*|k|
 SKIN_DEPTH(omega,mu,sigma) := delta=sqrt(2/(omega*mu*sigma))
}

AXIOMS {
 A1 = "Electromagnetic field variables require explicit units and coordinate conventions."
 A2 = "Maxwell equations couple sources and fields consistently with charge conservation."
 A3 = "Gauge-equivalent potentials represent the same physical electromagnetic fields."
 A4 = "Boundary and interface conditions are essential parts of electromagnetic field problems."
 A5 = "Material constitutive relations must be specified when media are present."
 A6 = "Electrostatic and magnetostatic approximations require negligible time-varying coupling."
 A7 = "Wave propagation depends on medium properties and boundary geometry."
 A8 = "Electromagnetic solutions require analytical, physical and numerical validation."
}

REASONING {
 R1_CONTEXT = identify(charges,currents,fields,media,geometry,time_dependence,boundaries)
 R2_REGIME = classify(electrostatic,magnetostatic,quasistatic,wave,radiation,covariant)
 R3_FORMULATION = construct(Maxwell_equations,potentials,constitutive_relations,boundary_conditions)
 R4_SOURCE = evaluate(charge_density,current_density,continuity)
 R5_POTENTIAL = choose(field_direct_or_potential_formulation,gauge_condition)
 R6_WAVE = analyse(dispersion,polarisation,reflection,transmission,modes)
 R7_ENERGY = compute(Poynting_flux,energy_density,stress,momentum)
 R8_RADIATION = analyse(dipole,multipole,near_field,far_field)
 R9_NUMERICAL = select(FDTD,FEM,MoM,BEM,spectral_method)
 R10_VALIDATION = check(units,divergence_constraints,continuity,boundaries,energy_balance,numerical_convergence)
}

CONSTRAINTS {
 C1 = "Electrostatic approximations are invalid when induction effects are material."
 C2 = "Magnetostatic approximations are invalid when displacement current is material."
 C3 = "Potential functions are gauge-dependent while physical fields are gauge-invariant."
 C4 = "Boundary conditions can determine allowed electromagnetic modes."
 C5 = "Constitutive parameters can depend on frequency, direction, field strength and temperature."
 C6 = "Perfect-conductor assumptions are idealisations."
 C7 = "Far-field approximations fail in the reactive near field."
 C8 = "Wave impedance and refractive index can be complex in lossy media."
 C9 = "Numerical meshes must resolve relevant wavelengths and geometric scales."
 C10 = "Discrete electromagnetic solvers must control spurious modes and conservation error."
}

SEMANTIC_EQUIVALENCE {
 "ElectromagneticMathematics" <=> "MathematicsOfElectromagnetism"
 "ElectricDisplacementField" <=> "DField"
 "MagneticFieldIntensity" <=> "HField"
 "ElectromagneticPotential" <=> "FourPotentialInCovariantContext"
 "FiniteDifferenceTimeDomain" <=> "FDTD"
 "MethodOfMoments" <=> "MoM"
}

VALIDATION {
 V1 = "Charge and current sources are explicit."
 V2 = "Field units and coordinate system are explicit."
 V3 = "Material constitutive relations are documented."
 V4 = "Charge continuity is checked."
 V5 = "Boundary and interface conditions are enforced."
 V6 = "Divergence constraints are checked."
 V7 = "Energy and momentum balance are checked where relevant."
 V8 = "Wave solutions satisfy dispersion and polarisation conditions."
 V9 = "Numerical mesh and timestep satisfy resolution and stability requirements."
 V10 = "Results are compared with analytical limits or benchmark solutions where possible."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 ELECTROMAGNETIC_MATHEMATICS_DEFINITION = 0.99
 MAXWELL_REASONING = 0.99
 POTENTIAL_REASONING = 0.99
 BOUNDARY_REASONING = 0.99
 WAVE_REASONING = 0.99
 RADIATION_REASONING = 0.98
 MATERIAL_REASONING = 0.98
 NUMERICAL_REASONING = 0.98
 POLICY = "Reduce confidence when source distributions, boundary conditions, constitutive laws, gauge choice, frequency regime, geometry or numerical resolution are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  ElectromagneticMathematicsReasoning,
  MaxwellEquationReasoning,
  ElectricFieldReasoning,
  MagneticFieldReasoning,
  ChargeConservationReasoning,
  PotentialReasoning,
  GaugeReasoning,
  ElectrostaticsReasoning,
  MagnetostaticsReasoning,
  WavePropagationReasoning,
  PolarisationReasoning,
  ReflectionTransmissionReasoning,
  RadiationReasoning,
  EnergyFluxReasoning,
  WaveguideReasoning,
  CovariantElectromagnetismReasoning,
  NumericalElectromagneticsReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]
 NORMALISE {
  "electromagnetic mathematics" -> ElectromagneticMathematics
  "mathematics of electromagnetism" -> ElectromagneticMathematics
  "electric field" -> ElectricField
  "magnetic field" -> MagneticField
  "Maxwell equations" -> MaxwellEquations
  "vector potential" -> VectorPotential
  "scalar potential" -> ScalarPotential
  "Poynting vector" -> PoyntingVector
  "FDTD" -> FiniteDifferenceTimeDomain
 }
}

EXECUTION_RULES {
 E1 = "If electric and magnetic fields are modelled mathematically, activate ElectromagneticMathematicsOntology."
 E2 = "If time variation is negligible, consider electrostatic or magnetostatic reductions."
 E3 = "If fields vary in time, retain the full Maxwell coupling unless justified otherwise."
 E4 = "If potentials are used, specify a gauge condition where needed."
 E5 = "If materials are present, define constitutive relations."
 E6 = "If interfaces are present, enforce electromagnetic boundary conditions."
 E7 = "If wave propagation is central, analyse dispersion, polarisation, reflection and transmission."
 E8 = "If radiation is central, distinguish near-field and far-field regimes."
 E9 = "If numerical solution is used, validate stability, convergence, conservation and geometric resolution."
}

RECURSION {
 ENTRY = ElectromagneticMathematics
 EXPAND(ElectromagneticMathematics) -> [
  Electromagnetism, ElectricCharge, ChargeDensity, Current, CurrentDensity,
  ElectricField, MagneticField, ElectromagneticField, ElectricFlux,
  MagneticFlux, Permittivity, Permeability, Conductivity, Polarisation,
  Magnetisation, ElectricDisplacementField, MagneticFieldIntensity,
  CoulombLaw, BiotSavartLaw, AmpereLaw, FaradayLaw, GaussLawElectric,
  GaussLawMagnetic, MaxwellEquations, ContinuityEquation, LorentzForce,
  Electrostatics, Magnetostatics, ScalarPotential, VectorPotential,
  ElectromagneticPotential, GaugeTransformation, GaugeCondition,
  CoulombGauge, LorenzGauge, PoissonEquation, LaplaceEquation, WaveEquation,
  HelmholtzEquation, RetardedPotential, GreenFunction, BoundaryCondition,
  InterfaceCondition, DirichletBoundaryCondition, NeumannBoundaryCondition,
  PerfectElectricConductor, PerfectMagneticConductor, DielectricInterface,
  ElectromagneticWave, PlaneWave, WaveVector, Frequency, AngularFrequency,
  Wavelength, PhaseVelocity, GroupVelocity, PolarisationState,
  LinearPolarisation, CircularPolarisation, EllipticalPolarisation,
  PoyntingVector, ElectromagneticEnergyDensity, ElectromagneticMomentum,
  MaxwellStressTensor, Radiation, DipoleRadiation, MultipoleExpansion,
  NearField, FarField, SkinDepth, Dispersion, RefractiveIndex, Impedance,
  Reflection, Transmission, Refraction, SnellLaw, FresnelCoefficient,
  Waveguide, ResonantCavity, Mode, TE_Mode, TM_Mode, TEM_Mode, Eigenmode,
  FourierTransform, LaplaceTransform, VectorCalculus, Gradient, Divergence,
  Curl, DifferentialForms, ExteriorDerivative, HodgeStar, FieldTensor,
  ElectromagneticTensor, FourCurrent, FourPotential, CovariantElectromagnetism,
  NumericalElectromagnetics, FiniteDifferenceTimeDomain, FiniteElementMethod,
  MethodOfMoments, BoundaryElementMethod, SpectralMethod
 ]
 EXPAND(RegimeFamilies) -> [
  Electrostatics,
  Magnetostatics,
  ElectromagneticWaves,
  RadiationTheory,
  CovariantElectromagnetism,
  NumericalElectromagnetics
 ]
 STOP_CONDITION = "Stop when sources, fields, media, geometry, boundary conditions, gauge structure, propagation regime and validation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Mathematical Physics Ontology"
 IMMEDIATE_CONTEXTS = [
  "Mathematical Physics Ontology",
  "Partial Differential Equation Ontology",
  "Vector Analysis Ontologies",
  "Wave Mathematics Ontologies"
 ]
 RELATED_ONTOLOGIES = [
  "Mathematical Physics Ontology",
  "Mathematical Physics",
  "Partial Differential Equation Ontology",
  "Vector Analysis Ontologies",
  "Tensor Analysis Ontologies",
  "Differential Geometry Ontologies",
  "Wave Mathematics Ontologies",
  "Fourier Analysis Ontology",
  "Spectral Theory Ontologies",
  "Special Relativity Ontologies",
  "Numerical Analysis Ontologies",
  "Computational Mathematics Ontologies"
 ]
 ROUTE_RULES = [
  "Electromagnetic Mathematics -> Mathematical Physics Ontology for parent semantics",
  "Electromagnetic Mathematics -> PDE Ontology for Maxwell, wave, Poisson and Helmholtz equations",
  "Electromagnetic Mathematics -> Vector Analysis for gradient, divergence and curl",
  "Electromagnetic Mathematics -> Tensor Analysis for stress and covariant field tensors",
  "Electromagnetic Mathematics -> Differential Geometry for differential-form formulations",
  "Electromagnetic Mathematics -> Wave Mathematics for propagation and modes",
  "Electromagnetic Mathematics -> Fourier Analysis for spectral field decomposition",
  "Electromagnetic Mathematics -> Spectral Theory for cavities and waveguides",
  "Electromagnetic Mathematics -> Special Relativity for covariant electromagnetism",
  "Electromagnetic Mathematics -> Numerical Analysis and Computational Mathematics for field simulation"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible electromagnetic regimes, material models or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve charge-current-source semantics.",
  "Preserve electric-versus-magnetic field distinction.",
  "Preserve gauge invariance semantics.",
  "Preserve static-versus-dynamic regime distinctions.",
  "Preserve boundary and interface semantics.",
  "Preserve material constitutive semantics.",
  "Preserve wave and radiation semantics.",
  "Preserve machine-readable electromagnetic-mathematics semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Electromagnetic mathematics describes electric and magnetic fields, their sources, interactions and propagation through Maxwell's equations and related mathematical structures."
 MAXWELL_SUMMARY = "Maxwell's equations couple electric and magnetic fields to charge and current while preserving charge conservation."
 POTENTIAL_SUMMARY = "Scalar and vector potentials provide alternative field representations with gauge freedom."
 MATERIAL_SUMMARY = "Permittivity, permeability, conductivity, polarisation and magnetisation determine electromagnetic response in media."
 WAVE_SUMMARY = "Electromagnetic waves are governed by wave and Helmholtz equations together with dispersion, polarisation and boundary conditions."
 ENERGY_SUMMARY = "Poynting vectors, energy density and Maxwell stress tensors describe electromagnetic energy and momentum transport."
 RADIATION_SUMMARY = "Radiation theory distinguishes near-field and far-field behaviour and supports multipole expansions."
 VALIDATION_SUMMARY = "Electromagnetic models should be checked for continuity, divergence constraints, boundary conditions, energy balance, limiting cases and numerical convergence."
}

FLOW {
 INPUT
 -> detect_electromagnetic_mathematics_context
 -> identify_charge_current_fields_media_geometry_time_dependence_and_boundaries
 -> classify_electrostatic_magnetostatic_quasistatic_wave_radiation_or_covariant_regime
 -> formulate_Maxwell_equations_constitutive_relations_potentials_and_boundary_conditions
 -> enforce_charge_continuity_and_divergence_constraints
 -> solve_or_characterise_fields_potentials_waves_modes_or_radiation
 -> if_wave_analyse_dispersion_polarisation_impedance_reflection_and_transmission
 -> if_radiation_analyse_near_field_far_field_and_multipole_structure
 -> compute_energy_flux_momentum_and_stress_if_relevant
 -> if_numerical_select_FDTD_FEM_MoM_BEM_or_spectral_method
 -> validate_units_sources_boundaries_conservation_energy_balance_and_numerical_convergence
 -> route_to_mathematical_physics_PDE_vector_analysis_tensor_analysis_differential_geometry_wave_mathematics_Fourier_analysis_spectral_theory_special_relativity_numerical_analysis_or_computational_mathematics_ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Electromagnetic Mathematics Ontology defines mathematical classical electromagnetism. It supports electric charge, charge density, current and current density, electric and magnetic fields, fluxes, permittivity, permeability and conductivity, polarisation and magnetisation, Coulomb, Biot-Savart, Ampere, Faraday and Gauss laws, Maxwell equations and charge continuity, Lorentz force, electrostatics and magnetostatics, scalar and vector potentials, gauge transformations and gauge conditions, Poisson, Laplace, wave and Helmholtz equations, retarded potentials and Green functions, boundary and interface conditions, electromagnetic waves, polarisation, phase and group velocity, Poynting vectors, energy density and Maxwell stress tensors, dipole and multipole radiation, near and far fields, skin depth, dispersion, refractive index and impedance, reflection, transmission and refraction, waveguides, cavities and modes, Fourier and Laplace transforms, vector calculus, differential-form and covariant tensor formulations, and numerical electromagnetics including FDTD, finite elements, method of moments, boundary elements and spectral methods."
}

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

}

END_FLOW