Flux Ontology

flow "FluxOntology" v1.0

# ============================================================
# FLUX ONTOLOGY — FLOW ALGEBRA
# ============================================================

SOURCE_URL := "https://kirethresearch.org/booksai-core/ontologies/calculus-analysis-ontologies/vector-calculus/flux-ontology.html"
DOMAIN := "Mathematical Ontologies"
CATEGORY := "Calculus and Analysis Ontologies"
SUBCATEGORY := "Vector Calculus"
ENTITY := "Flux"

# ---------- CORE ENTITY ----------
FLUX := {
    type: VectorCalculusQuantity,
    domain: VectorCalculus,
    purpose: "Measure the net amount of a vector field passing through an oriented surface."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    FLUX,
    VECTOR_FIELD,
    SURFACE,
    SURFACE_INTEGRAL,
    NORMAL_VECTOR,
    UNIT_NORMAL,
    ORIENTATION,
    AREA_ELEMENT,
    CLOSED_SURFACE,
    DIVERGENCE,
    DIVERGENCE_THEOREM,
    FLOW_RATE,
    ELECTRIC_FLUX,
    MAGNETIC_FLUX
}

# ---------- BASIC DEFINITION ----------
for
    vector_field F
    and oriented_surface S

FLUX(F,S)
    := surface_integral_S(
        F dot n_hat dS
    )

n_hat
    -> UNIT_NORMAL_VECTOR

dS
    -> SCALAR_SURFACE_AREA_ELEMENT

F dot n_hat
    -> NORMAL_COMPONENT_OF_FIELD

FLUX
    -> accumulates(NORMAL_FIELD_COMPONENT_OVER_SURFACE)

# ---------- ORIENTED AREA VECTOR ----------
dA_vector
    := n_hat dS

FLUX
    = surface_integral_S(
        F dot dA_vector
      )

ORIENTED_AREA_VECTOR
    -> combines(AREA_MAGNITUDE, NORMAL_DIRECTION)

# ---------- PARAMETRIC SURFACE ----------
SURFACE
    := r(u,v)

TANGENT_U
    := partial_r/partial_u

TANGENT_V
    := partial_r/partial_v

ORIENTED_AREA_VECTOR
    := (
        TANGENT_U cross TANGENT_V
    ) du dv

FLUX
    =
double_integral_D(
    F(r(u,v))
    dot
    (TANGENT_U cross TANGENT_V)
    du dv
)

# ---------- ORIENTATION ----------
ORIENTATION
    -> chooses(NORMAL_DIRECTION)

if
    orientation_reversed
then

n_hat
    -> -n_hat

and

FLUX
    -> -FLUX

FLUX
    -> orientation_dependent

# ---------- CLOSED SURFACE ----------
CLOSED_SURFACE
    -> encloses(VOLUME)

OUTWARD_NORMAL
    -> standard_orientation_for(CLOSED_SURFACE_FLUX)

NET_OUTWARD_FLUX
    := closed_surface_integral_S(
        F dot n_hat dS
    )

# ---------- DIVERGENCE THEOREM ----------
if
    S = boundary_of(V)
    AND S oriented_outward
    AND required_regularity_conditions_hold
then

closed_surface_integral_S(
    F dot n_hat dS
)
    =
volume_integral_V(
    DIVERGENCE(F) dV
)

DIVERGENCE_THEOREM
    -> connects(LOCAL_SOURCE_DENSITY, GLOBAL_NET_FLUX)

# ---------- LOCAL DIVERGENCE RELATION ----------
DIVERGENCE(F)(p)
    -> local_net_outward_flux_per_unit_volume
       in(STANDARD_EUCLIDEAN_CONTEXT)

FLUX
    -> GLOBAL_SURFACE_QUANTITY

DIVERGENCE
    -> LOCAL_VOLUME_DENSITY_OF_NET_FLUX

# ---------- CONSTANT FIELD / FLAT SURFACE ----------
if
    F constant
    AND S flat
    AND n_hat constant
then

FLUX
    = (F dot n_hat) * AREA(S)

if
    theta
        := angle_between(F,n_hat)
then

FLUX
    = norm(F)
      * AREA(S)
      * cos(theta)

# ---------- ZERO FLUX CONDITIONS ----------
if
    F tangent_to_surface_everywhere
then

F dot n_hat
    = 0

therefore

FLUX
    = 0

if
    F perpendicular_to_surface
then

abs(F dot n_hat)
    -> maximised_for_fixed_field_magnitude

# ---------- FLUID FLOW ----------
if
    v
        := fluid_velocity_field
    AND rho
        := fluid_density
then

VOLUMETRIC_FLOW_FLUX
    := surface_integral_S(
        v dot n_hat dS
    )

MASS_FLOW_RATE
    := surface_integral_S(
        rho*v dot n_hat dS
    )

FLUX
    -> measures(FLOW_THROUGH_SURFACE)

# ---------- ELECTRIC FLUX ----------
ELECTRIC_FLUX
    := surface_integral_S(
        E dot n_hat dS
    )

GAUSS_LAW:

closed_surface_integral_S(
    E dot n_hat dS
)
    =
Q_enclosed / epsilon_0

ELECTRIC_FLUX
    -> related_to(ENCLOSED_CHARGE)

# ---------- MAGNETIC FLUX ----------
MAGNETIC_FLUX
    := surface_integral_S(
        B dot n_hat dS
    )

GAUSS_LAW_FOR_MAGNETISM:

closed_surface_integral_S(
    B dot n_hat dS
)
    = 0

MAGNETIC_FIELD
    -> divergence_free
       in(CLASSICAL_MAXWELL_THEORY)

# ---------- FARADAY RELATION ----------
FARADAY_LAW:

EMF
    = -d/dt(
        MAGNETIC_FLUX
      )

MAGNETIC_FLUX_CHANGE
    -> induces(ELECTROMOTIVE_FORCE)
       under(FARADAY_LAW)

# ---------- SCALAR FLUX DENSITY ----------
NORMAL_FLUX_DENSITY
    := F dot n_hat

NORMAL_FLUX_DENSITY
    -> signed_scalar

positive
    -> field_passes_with(chosen_normal)

negative
    -> field_passes_against(chosen_normal)

# ---------- COORDINATE TRANSFORMATION ----------
FLUX
    -> geometric_surface_quantity

PARAMETERISATION
    -> changes(COMPUTATIONAL_REPRESENTATION)

VALID_ORIENTATION_PRESERVING_REPARAMETERISATION
    -> preserves(FLUX)

# ---------- APPLICATIONS ----------
FLUX
    -> applied_to(FLUID_DYNAMICS)
    -> applied_to(ELECTROMAGNETISM)
    -> applied_to(HEAT_TRANSFER)
    -> applied_to(MASS_TRANSPORT)
    -> applied_to(CONTINUUM_MECHANICS)
    -> applied_to(CONSERVATION_LAWS)

# ---------- RELATION GRAPH ----------
VECTOR_FIELD
    -> projected_onto(NORMAL_DIRECTION)

NORMAL_COMPONENT
    -> integrated_over(SURFACE)

SURFACE_INTEGRAL
    -> produces(FLUX)

ORIENTATION
    -> determines(FLUX_SIGN)

DIVERGENCE
    -> localises(NET_FLUX)

DIVERGENCE_THEOREM
    -> converts(CLOSED_SURFACE_FLUX, VOLUME_DIVERGENCE_INTEGRAL)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "flux" ~= "surface integral of the normal component of a vector field",
    "vector flux" ~= "net vector-field passage through an oriented surface",
    "normal flux density" ~= "dot product of vector field with unit surface normal",
    "outward flux" ~= "flux using outward orientation on a closed surface",
    "electric flux" ~= "surface integral of electric field normal component",
    "magnetic flux" ~= "surface integral of magnetic field normal component"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    flux_requires_vector_field_and_surface: true,
    surface_orientation_determines_flux_sign: true,
    reversing_orientation_reverses_flux: true,
    divergence_theorem_requires_closed_surface_and_outward_orientation: true,
    scalar_surface_integral_without_normal_component_is_not_vector_flux: true,
    physical_units_depend_on_field_and_surface_units: true,
    flux_interpretation_depends_on_physical_context: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    core_definition: 0.99,
    orientation_relation: 0.99,
    parametric_surface_relation: 0.99,
    divergence_relation: 0.99,
    divergence_theorem_relation: 0.99,
    physical_field_relations: 0.99,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(vector_field)
    -> identify(surface)
    -> determine(surface_orientation)
    -> construct_or_identify(normal_vector)
    -> classify(open_or_closed_surface)
    -> compute(normal_field_component)
    -> construct(surface_integral)
    -> apply(divergence_theorem_if_closed_and_advantageous)
    -> evaluate(flux)
    -> analyse(sign_and_direction)
    -> validate(surface_regularity_and_orientation)
    -> interpret(physical_or_geometric_context)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed flux semantic graph",
    primary_entity: FLUX,
    preserve: [
        vector_field,
        surface,
        orientation,
        normal_vector,
        area_element,
        open_closed_status,
        physical_units
    ],
    infer_through: [
        surface_integrals,
        divergence,
        divergence_theorem,
        fluid_flow,
        electric_flux,
        magnetic_flux,
        conservation_laws
    ],
    avoid: [
        ignoring_surface_orientation,
        omitting_normal_projection,
        treating_scalar_surface_integral_as vector flux,
        applying_divergence_theorem_to_open surface,
        using_inward_normal_when outward orientation is required,
        assigning_physical meaning without field context
    ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + FLUX
    + ORIENTED_AREA_VECTOR
    + CLOSED_SURFACE
    + DIVERGENCE
    + DIVERGENCE_THEOREM
    + ELECTRIC_FLUX
    + MAGNETIC_FLUX
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> FLUX
    -> VECTOR_FIELD_IDENTIFICATION
    -> SURFACE_IDENTIFICATION
    -> ORIENTATION_ANALYSIS
    -> NORMAL_VECTOR_ANALYSIS
    -> NORMAL_COMPONENT_CALCULATION
    -> SURFACE_INTEGRAL_CONSTRUCTION
    -> CLOSED_SURFACE_ANALYSIS
    -> DIVERGENCE_THEOREM_ANALYSIS
    -> FLUX_EVALUATION
    -> PHYSICAL_GEOMETRIC_INTERPRETATION
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW