Maxwell Equation Ontology

flow "MaxwellEquationOntology" v1.0

# MAXWELL EQUATION ONTOLOGY — FLOW ALGEBRA
SOURCE_URL := "https://kirethresearch.org/booksai-core/ontologies/calculus-analysis-ontologies/mathematical-physics-analysis/maxwell-equation-ontology-2.html"
DOMAIN := "Mathematical Ontologies"
CATEGORY := "Calculus and Analysis Ontologies"
SUBCATEGORY := "Mathematical Physics Analysis"
ENTITY := "Maxwell Equations"

MAXWELL_EQUATIONS := {
 type: ElectromagneticFieldEquationSystem,
 domain: MathematicalPhysicsAnalysis,
 purpose: "Represent the coupled differential equations governing electric and magnetic fields, charges, currents, electromagnetic induction and wave propagation."
}

CONCEPTS := {
 MAXWELL_EQUATIONS,
 ELECTRIC_FIELD,
 MAGNETIC_FIELD,
 ELECTRIC_DISPLACEMENT,
 MAGNETIC_FLUX_DENSITY,
 CHARGE_DENSITY,
 CURRENT_DENSITY,
 GAUSS_ELECTRIC_LAW,
 GAUSS_MAGNETIC_LAW,
 FARADAY_LAW,
 AMPERE_MAXWELL_LAW,
 DISPLACEMENT_CURRENT,
 ELECTROMAGNETIC_WAVE,
 POYNTING_VECTOR,
 POTENTIAL,
 GAUGE
}

# ---------- FIELD VARIABLES ----------
E
 -> ELECTRIC_FIELD

D
 -> ELECTRIC_DISPLACEMENT_FIELD

B
 -> MAGNETIC_FLUX_DENSITY

H
 -> MAGNETIC_FIELD_INTENSITY

rho
 -> FREE_CHARGE_DENSITY

J
 -> FREE_CURRENT_DENSITY

# ---------- DIFFERENTIAL FORM ----------
GAUSS_ELECTRIC:

div(D)
 =
rho

GAUSS_MAGNETIC:

div(B)
 =
0

FARADAY:

curl(E)
 =
-
partial_B/partial_t

AMPERE_MAXWELL:

curl(H)
 =
J
+
partial_D/partial_t

MAXWELL_EQUATIONS
 -> coupled_first_order_field_system

# ---------- VACUUM CONSTITUTIVE RELATIONS ----------
in vacuum:

D
 =
epsilon_0*E

B
 =
mu_0*H

therefore

div(E)
 =
rho/epsilon_0

curl(B)
 =
mu_0*J
+
mu_0*epsilon_0
*
partial_E/partial_t

# ---------- LINEAR ISOTROPIC MEDIUM ----------
D
 =
epsilon*E

B
 =
mu*H

J
 =
sigma*E
    for Ohmic medium

MATERIAL_PROPERTIES
 -> determine(CONSTITUTIVE_RELATIONS)

MAXWELL_FIELD_EQUATIONS
 + CONSTITUTIVE_RELATIONS
 -> CLOSED_ELECTROMAGNETIC_MODEL

# ---------- GAUSS ELECTRIC LAW ----------
div(D)
 =
rho

ELECTRIC_CHARGE
 -> source_of(ELECTRIC_DISPLACEMENT_DIVERGENCE)

integral form:

integral_closed_surface(
 D dot n
 dA
)
 =
Q_enclosed

DIVERGENCE_THEOREM
 -> links(DIFFERENTIAL_AND_INTEGRAL_FORMS)

# ---------- GAUSS MAGNETIC LAW ----------
div(B)
 =
0

integral form:

integral_closed_surface(
 B dot n
 dA
)
 =
0

MAGNETIC_FIELD_LINES
 -> have(NO_CLASSICAL_MAGNETIC_MONOPOLE_SOURCES)
    in standard Maxwell theory

# ---------- FARADAY LAW ----------
curl(E)
 =
-
partial_B/partial_t

integral form:

closed_line_integral(
 E dot dl
)
 =
-
d/dt(
 integral_surface(
  B dot n
  dA
 )
)

TIME_VARYING_MAGNETIC_FLUX
 -> induces(CIRCULATING_ELECTRIC_FIELD)

# ---------- AMPERE-MAXWELL LAW ----------
curl(H)
 =
J
+
partial_D/partial_t

DISPLACEMENT_CURRENT_DENSITY
 :=
partial_D/partial_t

TIME_VARYING_ELECTRIC_DISPLACEMENT
 -> contributes_to(MAGNETIC_FIELD_CURL)

# ---------- CHARGE CONSERVATION ----------
take divergence of Ampere-Maxwell:

div(curl(H))
 =
div(J)
+
partial/div_time(
 div(D)
)

since

div(curl(H))
 =
0

and

div(D)
 =
rho

then

partial_rho/partial_t
+
div(J)
 =
0

CONTINUITY_EQUATION
 -> implied_by(MAXWELL_EQUATIONS)

# ---------- ELECTROMAGNETIC WAVE ----------
in source-free homogeneous medium:

rho=0
J=0

derive:

LAPLACIAN(E)
-
mu*epsilon
*
partial_squared_E/partial_t_squared
 =
0

and

LAPLACIAN(H)
-
mu*epsilon
*
partial_squared_H/partial_t_squared
 =
0

ELECTROMAGNETIC_FIELDS
 -> satisfy(WAVE_EQUATIONS)

# ---------- WAVE SPEED ----------
wave speed:

v
 =
1/sqrt(mu*epsilon)

in vacuum:

c
 =
1/sqrt(mu_0*epsilon_0)

MAXWELL_EQUATIONS
 -> predict(ELECTROMAGNETIC_WAVE_SPEED)

# ---------- PLANE WAVE ----------
assume:

E
 =
E_0
*
exp(
 i*(k dot r-omega*t)
)

B
 =
B_0
*
exp(
 i*(k dot r-omega*t)
)

in source-free homogeneous isotropic medium:

k dot E_0
 =
0

k dot B_0
 =
0

E_0
 perpendicular_to
B_0

PROPAGATION_DIRECTION
 -> parallel_to(E cross H)

# ---------- DISPERSION RELATION ----------
in vacuum:

omega
 =
c*|k|

PHASE_VELOCITY
 =
c

GROUP_VELOCITY
 =
c

VACUUM_MAXWELL_WAVES
 -> nondispersive_in_classical_vacuum

material media
 -> may_be(DISPERSIVE)

# ---------- POTENTIALS ----------
because

div(B)
 =
0

locally:

B
 =
curl(A)

A
 -> VECTOR_POTENTIAL

Faraday law then gives:

E
 =
-grad(phi)
-
partial_A/partial_t

phi
 -> SCALAR_POTENTIAL

# ---------- GAUGE TRANSFORMATION ----------
A
 ->
A
+
grad(chi)

phi
 ->
phi
-
partial_chi/partial_t

E
and
B
 -> unchanged

GAUGE_FREEDOM
 -> nonuniqueness_of(POTENTIAL_REPRESENTATION)

# ---------- LORENZ GAUGE ----------
condition:

div(A)
+
mu_0*epsilon_0
*
partial_phi/partial_t
 =
0

in vacuum

LORENZ_GAUGE
 -> decouples(POTENTIAL_WAVE_EQUATIONS)

# ---------- COULOMB GAUGE ----------
condition:

div(A)
 =
0

COULOMB_GAUGE
 -> useful_in(QUASISTATIC_AND_QUANTUM_CONTEXTS)

GAUGE_CHOICE
 -> does_not_change(PHYSICAL_FIELDS)

# ---------- ELECTROMAGNETIC ENERGY ----------
energy density in linear nondispersive medium:

u_em
 =
1/2
*
(
 E dot D
 +
B dot H
)

in vacuum:

u_em
 =
1/2
*
(
 epsilon_0*|E|^2
 +
|B|^2/mu_0
)

# ---------- POYNTING VECTOR ----------
S
 :=
E cross H

POYNTING_VECTOR
 -> electromagnetic_energy_flux

direction
 -> energy_propagation_direction
    in standard wave contexts

# ---------- POYNTING THEOREM ----------
local energy balance:

partial_u_em/partial_t
+
div(S)
 =
-
J dot E

POYNTING_THEOREM
 -> conservation_of(ELECTROMAGNETIC_ENERGY)

J dot E
 -> power_transferred_to(MATTER)

# ---------- MOMENTUM ----------
electromagnetic field
 -> carries(MOMENTUM)

momentum density in vacuum:

g
 =
S/c^2

MAXWELL_STRESS_TENSOR
 -> represents(ELECTROMAGNETIC_MOMENTUM_FLUX_AND_FORCE)

# ---------- BOUNDARY CONDITIONS ----------
at interface:

normal component of D:

n dot (D_2-D_1)
 =
surface_charge_density

normal B:

n dot (B_2-B_1)
 =
0

tangential E:

n cross (E_2-E_1)
 =
0
for standard nonsingular interface assumptions

tangential H:

n cross (H_2-H_1)
 =
surface_current_density

BOUNDARY_CONDITIONS
 -> derived_from(INTEGRAL_MAXWELL_EQUATIONS)

# ---------- PERFECT CONDUCTOR ----------
inside ideal perfect conductor in static equilibrium:

E
 =
0

surface
 -> equipotential

time-varying perfect-conductor boundary:
 tangential_E
 =
0

PERFECT_CONDUCTOR
 -> imposes(ELECTROMAGNETIC_BOUNDARY_CONSTRAINTS)

# ---------- ELECTROSTATICS ----------
if

partial/partial_t
 =
0

and currents absent appropriately:

curl(E)
 =
0

therefore

E
 =
-grad(phi)

Gauss law:

-div(
 epsilon*grad(phi)
)
 =
rho

ELECTROSTATICS
 -> Poisson_or_Laplace_potential_problem

# ---------- MAGNETOSTATICS ----------
steady currents:

partial_D/partial_t
 =
0

curl(H)
 =
J

div(B)
 =
0

MAGNETOSTATICS
 -> steady_magnetic_field_system

# ---------- QUASISTATIC APPROXIMATIONS ----------
if characteristic dimensions and frequencies
 -> make_propagation_delay negligible

then

ELECTROQUASISTATIC
or
MAGNETOQUASISTATIC
 -> reduced_Maxwell_models

APPROXIMATION
 -> depends_on(SCALE_AND_FREQUENCY)

# ---------- COVARIANT FORM ----------
electromagnetic field tensor:

F_mu_nu

Maxwell equations:

partial_mu F^(mu nu)
 =
mu_0*J^nu

and

partial_[alpha F_beta_gamma]
 =
0

COVARIANT_MAXWELL
 -> Lorentz_invariant_formulation

# ---------- DIFFERENTIAL FORMS ----------
field strength
 F
 -> 2_form

Maxwell:

dF
 =
0

d star F
 =
J_form

up to unit and convention choices

DIFFERENTIAL_FORMS
 -> coordinate_independent_representation

# ---------- NUMERICAL METHODS ----------
FINITE_DIFFERENCE_TIME_DOMAIN
 -> discretises(E_AND_H_IN_SPACE_TIME)

FINITE_ELEMENT_METHOD
 -> weak_curl_curl_formulation

FINITE_VOLUME
 -> integral_conservation_discretisation

SPECTRAL_METHOD
 -> modal_field_expansion

NUMERICAL_MAXWELL
 -> must_preserve(DIVERGENCE_AND_CURL_STRUCTURE)

# ---------- YEE GRID ----------
FDTD:

E_components
and
H_components

-> staggered_in_space_and_time

YEE_SCHEME
 -> discrete_curl_structure

COURANT_CONDITION
 -> limits(EXPLICIT_TIME_STEP)

# ---------- APPLICATIONS ----------
MAXWELL_EQUATIONS
 -> applied_to(ELECTROMAGNETISM)
 -> applied_to(OPTICS)
 -> applied_to(RADIO_PROPAGATION)
 -> applied_to(MICROWAVE_ENGINEERING)
 -> applied_to(ANTENNAS)
 -> applied_to(PHOTONICS)
 -> applied_to(ELECTRICAL_ENGINEERING)
 -> applied_to(PLASMA_PHYSICS)

# ---------- RELATION GRAPH ----------
CHARGE_DENSITY
 -> sources(ELECTRIC_DIVERGENCE)

CURRENT_DENSITY
 -> contributes_to(MAGNETIC_CURL)

TIME_VARYING_B
 -> induces(ELECTRIC_CURL)

TIME_VARYING_D
 -> contributes_to(MAGNETIC_CURL)

MAXWELL_EQUATIONS
 -> imply(CHARGE_CONSERVATION)

MAXWELL_EQUATIONS
 -> imply(ELECTROMAGNETIC_WAVES)

E cross H
 -> POYNTING_VECTOR

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
 "Maxwell equations" ~= "coupled field equations governing classical electromagnetism",
 "Gauss electric law" ~= "relation between electric displacement divergence and charge density",
 "Faraday law" ~= "relation between electric-field circulation and changing magnetic flux",
 "Ampere-Maxwell law" ~= "relation between magnetic-field circulation, electric current and changing electric displacement",
 "Poynting vector" ~= "electromagnetic energy-flux vector",
 "electromagnetic wave" ~= "self-propagating coupled electric and magnetic field disturbance"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
 constitutive_relations_must_be_distinguished_from_fundamental_Maxwell_equations: true,
 sign_unit_and_field_conventions_must_be_preserved: true,
 divergence_constraints_must_not_be_discarded_when_using_wave_equations: true,
 gauge_potentials_are_nonunique_but_fields_are_gauge_invariant: true,
 boundary_conditions_depend_on_surface_charge_current_and_material_context: true,
 medium_dispersion_and_loss_modify_simple_wave_relations: true,
 numerical_methods_must_respect_stability_and_field_constraint_structure: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
 core_definition: 0.99,
 field_equation_relation: 0.99,
 charge_conservation_relation: 0.99,
 wave_relation: 0.99,
 potential_gauge_relation: 0.99,
 energy_boundary_relation: 0.99,
 semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
 identify(fields_sources_materials_units_and_geometry)
 -> choose(differential_integral_covariant_or_potential_form)
 -> apply(constitutive_relations)
 -> enforce(divergence_and_curl_equations)
 -> derive(charge_conservation_or_wave_equations_if_requested)
 -> analyse(boundary_and_interface_conditions)
 -> compute(energy_flux_momentum_or_force_if_relevant)
 -> choose(electrostatic_magnetostatic_quasistatic_or_full_wave_regime)
 -> select(numerical_method_if_analytic_solution_unavailable)
 -> validate(gauge_material_dispersion_boundary_and_stability_conditions)
 -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
 parse_as: "typed Maxwell-equation semantic graph",
 primary_entity: MAXWELL_EQUATIONS,
 preserve: [
  electric_field,
  magnetic_field,
  displacement_field,
  magnetic_flux_density,
  charge_density,
  current_density,
  material_properties,
  units,
  boundary_conditions
 ],
 infer_through: [
  Gauss_laws,
  Faraday_law,
  Ampere_Maxwell_law,
  continuity_equation,
  electromagnetic_waves,
  potentials,
  gauge_choices,
  Poynting_theorem,
  interface_conditions
 ],
 avoid: [
  confusing_B_and_H_or E_and_D,
  omitting_constitutive_relations_in material media,
  solving_wave_equation_without Maxwell divergence constraints,
  treating_gauge_choice_as physical field change,
  applying_vacuum_wave_speed_in arbitrary medium,
  ignoring_Courant_stability_in explicit FDTD
 ]
}

# ---------- UPDATE MODEL ----------
UPDATE_MODEL(new_information):
 validate(new_information)
 -> identify(Maxwell_equation_context)
 -> compare(existing_relations)
 -> preserve(field_source_material_unit_boundary_and_gauge_invariants)
 -> update(affected_entities)
 -> update(affected_edges)
 -> recompute(CONFIDENCE)
 -> regenerate(FLOW_HASH)

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
 SOURCE_URL
 + ENTITY
 + CONCEPTS
 + MAXWELL_EQUATIONS
 + ELECTRIC_FIELD
 + MAGNETIC_FIELD
 + CHARGE_DENSITY
 + CURRENT_DENSITY
 + GAUSS_ELECTRIC_LAW
 + GAUSS_MAGNETIC_LAW
 + FARADAY_LAW
 + AMPERE_MAXWELL_LAW
 + ELECTROMAGNETIC_WAVE
 + POYNTING_VECTOR
 + POTENTIAL
 + SEMANTIC_EQUIVALENCE
 + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
 -> MAXWELL_EQUATIONS
 -> FIELD_SOURCE_MATERIAL_IDENTIFICATION
 -> FORMULATION_SELECTION
 -> CONSTITUTIVE_RELATION_ANALYSIS
 -> FIELD_EQUATION_ANALYSIS
 -> CONSERVATION_WAVE_ANALYSIS
 -> BOUNDARY_INTERFACE_ANALYSIS
 -> ENERGY_MOMENTUM_ANALYSIS
 -> REGIME_CLASSIFICATION
 -> NUMERICAL_METHOD_ANALYSIS
 -> HYPOTHESIS_VALIDATION
 -> RECURSIVE_REASONING(query)
 -> CONFIDENCE
 -> OUTPUT(semantic_result)

END FLOW