Divergence Ontology

flow "DivergenceOntology" v1.0

# ============================================================
# DIVERGENCE ONTOLOGY — FLOW ALGEBRA
# ============================================================

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

# ---------- CORE ENTITY ----------
DIVERGENCE := {
    type: ScalarDifferentialOperatorResult,
    domain: VectorCalculus,
    purpose: "Measure the local net outward flux density of a differentiable vector field."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    VECTOR_FIELD,
    DIVERGENCE,
    NABLA_OPERATOR,
    PARTIAL_DERIVATIVE,
    SCALAR_FIELD,
    FLUX,
    SOURCE,
    SINK,
    SOLENOIDAL_FIELD,
    DIVERGENCE_THEOREM,
    VOLUME_INTEGRAL,
    SURFACE_INTEGRAL,
    JACOBIAN
}

# ---------- VECTOR FIELD ----------
VECTOR_FIELD
    := F: R^n -> R^n

in R^3:

F(x,y,z)
    := (
        P(x,y,z),
        Q(x,y,z),
        R(x,y,z)
    )

DIVERGENCE
    -> operates_on(VECTOR_FIELD)

# ---------- NABLA DOT FORM ----------
NABLA
    := ∇

DIVERGENCE(F)
    := ∇ dot F

in Cartesian R^3:

∇ dot F
    = partial_P/partial_x
      + partial_Q/partial_y
      + partial_R/partial_z

DIVERGENCE
    -> produces(SCALAR_FIELD)

# ---------- N-DIMENSIONAL FORM ----------
for
    F = (F1,...,Fn)

DIVERGENCE(F)
    := sum(
        partial_F_i/partial_x_i,
        i=1..n
    )

DIVERGENCE
    -> trace_of(JACOBIAN(F))

# ---------- JACOBIAN RELATION ----------
JACOBIAN(F)
    := matrix(
        partial_F_i/partial_x_j
    )

DIVERGENCE(F)
    = TRACE(JACOBIAN(F))

TRACE
    -> sums(DIAGONAL_LOCAL_DIRECTIONAL_EXPANSIONS)

# ---------- LOCAL FLUX INTERPRETATION ----------
DIVERGENCE(F)(p)
    -> measures(LOCAL_NET_OUTFLOW_PER_UNIT_VOLUME)
       in(EUCLIDEAN_FLUX_CONTEXT)

positive_divergence
    -> local_source_like_behaviour

negative_divergence
    -> local_sink_like_behaviour

zero_divergence
    -> locally_balanced_net_flux

# ---------- FLUX LIMIT INTERPRETATION ----------
DIVERGENCE(F)(p)
    := limit(
        outward_flux_through_boundary(V)
        / volume(V),
        V shrinks_to p
    )

under(appropriate_smoothness_and_shape_conditions)

DIVERGENCE
    -> localises(FLUX)

# ---------- SOLENOIDAL FIELD ----------
SOLENOIDAL_FIELD
    := vector_field F
       such_that
       DIVERGENCE(F) = 0

SOLENOIDAL_FIELD
    ~= DIVERGENCE_FREE_FIELD

DIVERGENCE_FREE
    -> locally_no_net_source_or_sink
       in(STANDARD_INTERPRETATION)

# ---------- DIVERGENCE THEOREM ----------
if
    V
        := volume
    AND S = boundary(V)
    AND S oriented_outward
then

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

DIVERGENCE_THEOREM
    -> connects(LOCAL_DIVERGENCE, GLOBAL_BOUNDARY_FLUX)

# ---------- CONSERVATION LAW ----------
if
    rho
        := density
    AND J
        := flux_field
then

partial_rho/partial_t
    + DIVERGENCE(J)
    = source_term

if
    source_term = 0
then

partial_rho/partial_t
    + DIVERGENCE(J)
    = 0

CONTINUITY_EQUATION
    -> expresses(LOCAL_CONSERVATION)

# ---------- FLUID MECHANICS ----------
if
    v
        := fluid_velocity_field
then

DIVERGENCE(v)
    -> measures(LOCAL_VOLUMETRIC_EXPANSION_RATE)
       under(standard_continuum_interpretation)

INCOMPRESSIBLE_FLOW
    -> commonly_satisfies(
        DIVERGENCE(v) = 0
    )

# ---------- ELECTROMAGNETISM ----------
GAUSS_LAW_DIFFERENTIAL_FORM:

DIVERGENCE(E)
    = rho / epsilon_0

ELECTRIC_FIELD_DIVERGENCE
    -> related_to(CHARGE_DENSITY)

MAGNETIC_GAUSS_LAW:

DIVERGENCE(B)
    = 0

MAGNETIC_FIELD
    -> divergence_free
       in(CLASSICAL_MAXWELL_THEORY)

# ---------- GRADIENT RELATION ----------
for scalar field phi:

DIVERGENCE(GRADIENT(phi))
    = LAPLACIAN(phi)

LAPLACIAN
    := ∇ dot ∇

DIVERGENCE
    -> combines_with(GRADIENT)
       to_form(LAPLACIAN)

# ---------- CURL RELATION ----------
under sufficient smoothness:

DIVERGENCE(CURL(F))
    = 0

CURL_FIELD
    -> divergence_free
       locally_under(SMOOTHNESS_CONDITIONS)

# ---------- PRODUCT RULE ----------
for scalar field phi
and vector field F:

DIVERGENCE(phi*F)
    = GRADIENT(phi) dot F
      + phi*DIVERGENCE(F)

DIVERGENCE_PRODUCT_RULE
    -> vector_calculus_analogue_of(PRODUCT_RULE)

# ---------- COORDINATE SYSTEMS ----------
DIVERGENCE
    -> geometric_scalar_operator

CARTESIAN_EXPRESSION
    -> simple_sum_of_component_partials

CURVILINEAR_COORDINATES
    -> require(SCALE_FACTORS_OR_METRIC_STRUCTURE)

COORDINATE_FORMULA
    -> must_match(COORDINATE_SYSTEM)

# ---------- CYLINDRICAL FORM ----------
for
    F = F_r e_r
        + F_theta e_theta
        + F_z e_z

DIVERGENCE(F)
    =
1/r * partial(r*F_r)/partial_r
+
1/r * partial(F_theta)/partial_theta
+
partial(F_z)/partial_z

# ---------- SPHERICAL FORM ----------
under standard spherical convention:

DIVERGENCE(F)
    =
1/r^2 * partial(r^2*F_r)/partial_r
+
1/(r*sin(theta))
    * partial(sin(theta)*F_theta)/partial_theta
+
1/(r*sin(theta))
    * partial(F_phi)/partial_phi

COORDINATE_CONVENTION
    -> must_be_specified

# ---------- APPLICATIONS ----------
DIVERGENCE
    -> applied_to(FLUID_DYNAMICS)
    -> applied_to(ELECTROMAGNETISM)
    -> applied_to(CONTINUUM_MECHANICS)
    -> applied_to(HEAT_TRANSFER)
    -> applied_to(CONSERVATION_LAWS)
    -> applied_to(PARTIAL_DIFFERENTIAL_EQUATIONS)

# ---------- RELATION GRAPH ----------
VECTOR_FIELD
    -> acted_on_by(DIVERGENCE_OPERATOR)

DIVERGENCE
    -> produces(SCALAR_FIELD)

JACOBIAN
    -> trace_equals(DIVERGENCE)

DIVERGENCE
    -> localises(FLUX)

DIVERGENCE_FREE
    -> defines(SOLENOIDAL_FIELD)

DIVERGENCE_THEOREM
    -> integrates(DIVERGENCE_OVER_VOLUME)
    -> equates_to(OUTWARD_BOUNDARY_FLUX)

GRADIENT
    -> followed_by(DIVERGENCE)
    -> produces(LAPLACIAN)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "divergence" ~= "scalar measure of local net outward flux density",
    "div" ~= "divergence operator",
    "nabla dot F" ~= "divergence of F",
    "divergence-free" ~= "solenoidal in standard vector-calculus terminology",
    "positive divergence" ~= "local source-like net outflow",
    "negative divergence" ~= "local sink-like net inflow"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    divergence_operates_on_vector_field: true,
    divergence_output_is_scalar_field: true,
    component_formula_depends_on_coordinate_system: true,
    divergence_free_is_local_flux_statement: true,
    divergence_theorem_requires_closed_boundary_and_outward_orientation: true,
    divergence_curl_identity_requires_sufficient_smoothness: true,
    physical_source_interpretation_depends_on_model_context: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    core_definition: 0.99,
    jacobian_trace_relation: 0.99,
    local_flux_relation: 0.99,
    divergence_theorem_relation: 0.99,
    conservation_relation: 0.99,
    coordinate_relations: 0.98,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(vector_field)
    -> determine(domain_dimension)
    -> identify(coordinate_system)
    -> verify(required_differentiability)
    -> compute(component_derivatives)
    -> apply(coordinate_appropriate_divergence_formula)
    -> classify(source_sink_or_solenoidal_behaviour_if_requested)
    -> connect_to(flux_or_conservation_law_if_relevant)
    -> apply(divergence_theorem_if_valid)
    -> validate(boundary_orientation_and_smoothness)
    -> interpret(geometric_or_physical_meaning)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed divergence semantic graph",
    primary_entity: DIVERGENCE,
    preserve: [
        vector_field,
        coordinate_system,
        component_functions,
        differentiability,
        domain_dimension,
        boundary_orientation
    ],
    infer_through: [
        jacobians,
        flux,
        solenoidal_fields,
        divergence_theorem,
        conservation_laws,
        laplacians,
        coordinate_transformations
    ],
    avoid: [
        applying_divergence_to_scalar_field,
        using_cartesian_formula_in_curvilinear_coordinates,
        interpreting_zero_divergence_as zero vector field,
        applying_divergence_theorem_to_open surface,
        ignoring_outward_orientation,
        inferring_physical_sources_without model context
    ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + DIVERGENCE
    + NABLA
    + JACOBIAN
    + SOLENOIDAL_FIELD
    + DIVERGENCE_THEOREM
    + CONTINUITY_EQUATION
    + LAPLACIAN
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> DIVERGENCE
    -> VECTOR_FIELD_IDENTIFICATION
    -> DIMENSION_ANALYSIS
    -> COORDINATE_SYSTEM_ANALYSIS
    -> DIFFERENTIABILITY_ANALYSIS
    -> COMPONENT_DERIVATIVE_ANALYSIS
    -> DIVERGENCE_CALCULATION
    -> FIELD_BEHAVIOUR_CLASSIFICATION
    -> FLUX_CONSERVATION_ANALYSIS
    -> DIVERGENCE_THEOREM_ANALYSIS
    -> GEOMETRIC_PHYSICAL_INTERPRETATION
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW