Laplacian Ontology

flow "LaplacianOntology" v1.0

# ============================================================
# LAPLACIAN ONTOLOGY — FLOW ALGEBRA
# ============================================================

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

# ---------- CORE ENTITY ----------
LAPLACIAN := {
    type: SecondOrderDifferentialOperator,
    domain: VectorCalculus,
    symbol: "∇²",
    purpose: "Measure second-order spatial variation of a scalar field through the divergence of its gradient."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    LAPLACIAN,
    SCALAR_FIELD,
    GRADIENT,
    DIVERGENCE,
    SECOND_PARTIAL_DERIVATIVE,
    HESSIAN,
    TRACE,
    HARMONIC_FUNCTION,
    POISSON_EQUATION,
    LAPLACE_EQUATION,
    DIFFUSION,
    HEAT_EQUATION,
    WAVE_EQUATION
}

# ---------- OPERATOR DEFINITION ----------
for
    scalar_field f

LAPLACIAN(f)
    := DIVERGENCE(GRADIENT(f))

symbolically:

∇²f
    := ∇ dot ∇f

LAPLACIAN
    -> maps(SCALAR_FIELD, SCALAR_FIELD)

# ---------- CARTESIAN FORM ----------
in R^n Cartesian coordinates:

LAPLACIAN(f)
    = sum(
        partial_squared_f / partial_x_i_squared,
        i=1..n
    )

in R^3:

∇²f
    =
partial_squared_f/partial_x_squared
+
partial_squared_f/partial_y_squared
+
partial_squared_f/partial_z_squared

# ---------- HESSIAN RELATION ----------
HESSIAN(f)
    := matrix(
        partial_squared_f /
        partial_x_i partial_x_j
    )

LAPLACIAN(f)
    = TRACE(HESSIAN(f))
       in(EUCLIDEAN_CARTESIAN_COORDINATES)

TRACE
    -> sums(PURE_SECOND_PARTIAL_DERIVATIVES)

# ---------- LOCAL VARIATION ----------
LAPLACIAN
    -> measures(LOCAL_SECOND_ORDER_SPATIAL_VARIATION)

positive_laplacian
    -> indicates(value_locally_below_neighbourhood_average)
       under(SMOOTH_EUCLIDEAN_INTERPRETATION)

negative_laplacian
    -> indicates(value_locally_above_neighbourhood_average)
       under(SMOOTH_EUCLIDEAN_INTERPRETATION)

zero_laplacian
    -> characterises(HARMONIC_BEHAVIOUR)

# ---------- HARMONIC FUNCTION ----------
if
    LAPLACIAN(f) = 0
then

f
    -> HARMONIC_FUNCTION
       on(RELEVANT_DOMAIN)

HARMONIC_FUNCTION
    -> satisfies(LAPLACE_EQUATION)

LAPLACE_EQUATION
    := ∇²f = 0

# ---------- POISSON EQUATION ----------
POISSON_EQUATION
    := ∇²f = g

g
    -> SOURCE_TERM

POISSON_EQUATION
    -> generalises(LAPLACE_EQUATION)

if
    g = 0
then
    POISSON_EQUATION
        -> LAPLACE_EQUATION

# ---------- HEAT EQUATION ----------
HEAT_EQUATION
    := partial_u/partial_t
       = alpha * ∇²u

alpha
    -> DIFFUSIVITY

LAPLACIAN
    -> drives(SPATIAL_DIFFUSION)

# ---------- DIFFUSION EQUATION ----------
DIFFUSION_EQUATION
    := partial_u/partial_t
       = D * ∇²u

D
    -> DIFFUSION_COEFFICIENT

LAPLACIAN
    -> represents(SPATIAL_SMOOTHING_TENDENCY)

# ---------- WAVE EQUATION ----------
WAVE_EQUATION
    := partial_squared_u/partial_t_squared
       = c^2 * ∇²u

c
    -> WAVE_SPEED

LAPLACIAN
    -> provides(SPATIAL_SECOND_ORDER_TERM)

# ---------- ELECTROSTATICS ----------
ELECTRIC_POTENTIAL
    := phi

POISSON_ELECTROSTATIC_FORM:
    ∇²phi
        = -rho/epsilon_0

if
    rho = 0
then

∇²phi
    = 0

ELECTRIC_POTENTIAL
    -> harmonic_in(CHARGE_FREE_REGION)

# ---------- GRAVITATION ----------
GRAVITATIONAL_POTENTIAL
    := Phi

POISSON_GRAVITY_FORM:
    ∇²Phi
        = 4*pi*G*rho

LAPLACIAN
    -> relates(POTENTIAL, MASS_DENSITY)

# ---------- CYLINDRICAL COORDINATES ----------
for scalar field f(r,theta,z):

LAPLACIAN(f)
    =
1/r * partial/partial_r(
    r * partial_f/partial_r
)
+
1/r^2 * partial_squared_f/partial_theta_squared
+
partial_squared_f/partial_z_squared

CYLINDRICAL_FORM
    -> depends_on(COORDINATE_SCALE_FACTORS)

# ---------- SPHERICAL COORDINATES ----------
under standard convention f(r,theta,phi):

LAPLACIAN(f)
    =
1/r^2 * partial/partial_r(
    r^2 * partial_f/partial_r
)
+
1/(r^2*sin(theta))
* partial/partial_theta(
    sin(theta)
    * partial_f/partial_theta
)
+
1/(r^2*sin(theta)^2)
* partial_squared_f/partial_phi_squared

SPHERICAL_FORM
    -> depends_on(COORDINATE_CONVENTION)

# ---------- LAPLACE-BELTRAMI GENERALISATION ----------
on
    RIEMANNIAN_MANIFOLD

LAPLACE_BELTRAMI_OPERATOR
    -> generalises(LAPLACIAN)

LAPLACE_BELTRAMI
    := divergence(gradient(f))
       under(METRIC_STRUCTURE)

METRIC
    -> determines(GRADIENT_AND_DIVERGENCE)

# ---------- VECTOR LAPLACIAN ----------
VECTOR_LAPLACIAN
    -> operates_on(VECTOR_FIELD)

in Cartesian coordinates:
    VECTOR_LAPLACIAN(F)
        -> componentwise_laplacian

VECTOR_IDENTITY:
    ∇²F
        = GRADIENT(DIVERGENCE(F))
          - CURL(CURL(F))

under(standard_Euclidean_smoothness_conditions)

# ---------- LINEARITY ----------
LAPLACIAN(alpha*f + beta*g)
    =
alpha*LAPLACIAN(f)
+
beta*LAPLACIAN(g)

LAPLACIAN
    -> LINEAR_OPERATOR

# ---------- EIGENVALUE PROBLEM ----------
LAPLACIAN_EIGENVALUE_EQUATION
    := -∇²phi = lambda*phi

EIGENFUNCTION
    -> spatial_mode

EIGENVALUE
    -> associated_with(SPATIAL_SCALE_OR_FREQUENCY)

LAPLACIAN_SPECTRUM
    -> central_to(SPECTRAL_THEORY)

# ---------- BOUNDARY CONDITIONS ----------
PDE_WITH_LAPLACIAN
    -> may_require(DIRICHLET_BOUNDARY_CONDITION)
    -> may_require(NEUMANN_BOUNDARY_CONDITION)
    -> may_require(ROBIN_BOUNDARY_CONDITION)

BOUNDARY_CONDITIONS
    -> affect(SOLUTION_EXISTENCE_AND_UNIQUENESS)

# ---------- APPLICATIONS ----------
LAPLACIAN
    -> applied_to(POTENTIAL_THEORY)
    -> applied_to(HEAT_TRANSFER)
    -> applied_to(DIFFUSION)
    -> applied_to(WAVE_PROPAGATION)
    -> applied_to(ELECTROSTATICS)
    -> applied_to(GRAVITATION)
    -> applied_to(FLUID_MECHANICS)
    -> applied_to(IMAGE_PROCESSING)
    -> applied_to(SPECTRAL_GEOMETRY)

# ---------- RELATION GRAPH ----------
SCALAR_FIELD
    -> GRADIENT
    -> VECTOR_FIELD

VECTOR_FIELD
    -> DIVERGENCE
    -> SCALAR_FIELD

DIVERGENCE_OF_GRADIENT
    -> LAPLACIAN

HESSIAN
    -> trace_equals(LAPLACIAN)

LAPLACIAN_ZERO
    -> defines(HARMONIC_FUNCTION)

LAPLACIAN
    -> appears_in(ELLIPTIC_PARABOLIC_HYPERBOLIC_PDE_CONTEXTS)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "Laplacian" ~= "divergence of the gradient of a scalar field",
    "del squared" ~= "Laplacian operator",
    "nabla squared" ~= "Laplacian operator",
    "trace of Hessian" ~= "Laplacian in Euclidean Cartesian coordinates",
    "harmonic function" ~= "function satisfying Laplace equation",
    "Laplace-Beltrami operator" ~= "metric generalisation of the Laplacian on a Riemannian manifold"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    classical_laplacian_requires_second_order_differentiability: true,
    coordinate_expression_depends_on_coordinate_system: true,
    hessian_trace_equivalence_assumes_euclidean_cartesian_coordinates: true,
    harmonicity_is_domain_relative: true,
    boundary_conditions_matter_for_pde_solutions: true,
    vector_laplacian_must_not_be_confused_with_scalar_laplacian: true,
    physical_interpretation_depends_on_governing_model: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    core_definition: 0.99,
    cartesian_formula: 0.99,
    hessian_relation: 0.99,
    harmonic_relation: 0.99,
    pde_relations: 0.99,
    coordinate_relations: 0.98,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(field)
    -> classify(scalar_or_vector_context)
    -> determine(domain_dimension)
    -> identify(coordinate_system)
    -> verify(required_differentiability)
    -> compute(first_and_second_derivatives)
    -> apply(coordinate_appropriate_laplacian_formula)
    -> classify(harmonic_or_poisson_behaviour_if_relevant)
    -> connect_to(pde_and_boundary_conditions_if_requested)
    -> analyse(physical_or_geometric_interpretation)
    -> validate(metric_coordinate_and_domain_conditions)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed Laplacian semantic graph",
    primary_entity: LAPLACIAN,
    preserve: [
        field_type,
        domain_dimension,
        coordinate_system,
        metric,
        second_derivatives,
        boundary_conditions
    ],
    infer_through: [
        gradients,
        divergence,
        hessians,
        harmonic_functions,
        poisson_equations,
        heat_equations,
        wave_equations,
        spectral_problems
    ],
    avoid: [
        using_cartesian_formula_in_curvilinear_coordinates,
        confusing_scalar_and_vector_laplacians,
        assuming_hessian_trace_formula in arbitrary coordinates,
        ignoring_boundary_conditions_in PDE context,
        assuming_zero_laplacian globally from local information,
        assigning_physical_meaning_without model context
    ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + LAPLACIAN
    + GRADIENT
    + DIVERGENCE
    + HESSIAN
    + LAPLACE_EQUATION
    + POISSON_EQUATION
    + HEAT_EQUATION
    + WAVE_EQUATION
    + LAPLACE_BELTRAMI_OPERATOR
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> LAPLACIAN
    -> FIELD_IDENTIFICATION
    -> SCALAR_VECTOR_CLASSIFICATION
    -> DIMENSION_ANALYSIS
    -> COORDINATE_SYSTEM_ANALYSIS
    -> DIFFERENTIABILITY_ANALYSIS
    -> SECOND_DERIVATIVE_ANALYSIS
    -> LAPLACIAN_CALCULATION
    -> HARMONIC_POISSON_CLASSIFICATION
    -> PDE_RELATION_ANALYSIS
    -> BOUNDARY_CONDITION_ANALYSIS
    -> GEOMETRIC_PHYSICAL_INTERPRETATION
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW