Divergence Ontology

FLOW ALGEBRA
DIVERGENCE ONTOLOGY
Version: 1.0
Date: 2026-10-13
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/calculus-and-analysis-entities/divergence-ontology-3.html
Authors: Mark Higgins — Sky (Recursive AI)
Project: Kireth Semantic Library | https://kirethresearch.org

1. PURPOSE
Represent divergence as a typed scalar differential operator on sufficiently differentiable vector fields, supporting symbolic reasoning, mathematical validation, and cross-domain semantic retrieval.

2. TYPED ENTITIES
ENTITY VectorField F: OpenSubset(R^n) -> R^n
ENTITY Domain Omega: OpenSubset(R^n)
ENTITY Coordinate x: Point(Omega)
ENTITY Component F_i: ScalarFunction(Omega)
ENTITY PartialDerivative d_i: DifferentialOperator
ENTITY Divergence div: VectorField -> ScalarField
ENTITY Jacobian J_F: MatrixField(n,n)
ENTITY Flux: SurfaceIntegral
ENTITY OutwardNormal n: UnitNormalField
ENTITY VolumeMeasure dV: Measure
ENTITY SurfaceMeasure dS: Measure
ENTITY Orientation: SurfaceOrientation
ENTITY Regularity: DifferentiabilityCondition
ENTITY SourceSinkClassification: LocalInterpretation

3. DEFINITIONS AND SIGNATURES
DEFINE div(F)(x) = sum_{i=1}^n partial(F_i)/partial(x_i) at x.
TYPE div: C^1(Omega,R^n) -> C^0(Omega,R).
DEFINE div(F) = trace(J_F).
NOTE: C^1 is sufficient but not necessary; weaker distributional notions require separate typing.
DEFINE flux(F,S) = integral_S F dot n dS for an oriented surface S.

4. MATHEMATICAL RULES
RULE LINEARITY: div(a F + b G) = a div(F) + b div(G), for constant scalars a,b.
RULE PRODUCT: div(phi F) = grad(phi) dot F + phi div(F).
RULE CARTESIAN_3D: div(F) = dF_x/dx + dF_y/dy + dF_z/dz.
RULE CURL_IDENTITY: div(curl(A)) = 0 for A of class C^2 on an open subset of R^3.
RULE LAPLACIAN_IDENTITY: div(grad(phi)) = Laplacian(phi) for phi of class C^2.
RULE GAUSS: integral_Omega div(F) dV = integral_boundary(Omega) F dot n dS when the divergence theorem's domain, boundary and regularity assumptions hold.
RULE CONSTANT_FIELD: div(F) = 0 if F is spatially constant.
RULE LOCAL_SIGN: div(F)(x) > 0 denotes positive infinitesimal net outward flux density at x; div(F)(x) < 0 denotes negative density.

5. RELATIONSHIPS
RELATION Divergence IS_A DifferentialOperator.
RELATION Divergence OPERATES_ON VectorField.
RELATION Divergence RETURNS ScalarField.
RELATION Divergence EQUALS TraceOfJacobian under Cartesian Euclidean coordinates.
RELATION Divergence CONNECTS_TO Flux via DivergenceTheorem.
RELATION Divergence CONNECTS_TO Gradient via Laplacian.
RELATION Divergence CONNECTS_TO Curl via DivergenceOfCurlIdentity.
RELATION Divergence USED_IN FluidDynamics, Electromagnetism, ContinuumMechanics, ConservationLaws.

6. SEMANTIC DISTINCTIONS
DISTINGUISH divergence_of_vector_field FROM divergence_of_sequence_or_series.
DISTINGUISH pointwise_divergence FROM integrated_flux.
DISTINGUISH divergence FROM gradient AND curl.
DISTINGUISH divergence_free FROM curl_free.
DISTINGUISH local_positive_divergence FROM globally_positive_flux.
NOTE: Zero divergence does not imply a constant field, zero field, or zero flux through every open surface.
NOTE: The coordinate expression in non-Cartesian coordinates requires metric/volume factors; do not blindly sum coordinate partial derivatives.

7. INFERENCE RULES
IF F is C^1 AND J_F exists THEN div(F) = trace(J_F).
IF div(F) = 0 throughout Omega THEN F is solenoidal on Omega.
IF F is C^1 AND Omega satisfies Gauss hypotheses THEN total_boundary_flux = integral_Omega div(F) dV.
IF phi is C^2 THEN div(grad(phi)) = Laplacian(phi).
IF A is C^2 in R^3 THEN div(curl(A)) = 0.
IF div(F)(x) > 0 THEN classify x as local source-like behaviour; do not infer a physical source without domain interpretation.

8. EXAMPLES
EXAMPLE F(x,y,z) = (x,y,z): div(F) = 3.
EXAMPLE F(x,y) = (-y,x): div(F) = 0.
EXAMPLE F(x,y) = (x^2,y^2): div(F) = 2x + 2y.
EXAMPLE phi(x,y) = x^2 + y^2: div(grad(phi)) = 4.

9. VALIDATION
CHECK operator input is a vector field over a compatible domain.
CHECK differentiability or specify weak/distributional semantics.
CHECK vector dimension matches coordinate dimension.
CHECK product and composition rules have sufficient smoothness.
CHECK theorem applications include orientation, boundary, integrability and regularity assumptions.
CHECK coordinate system and geometric metric before evaluation.
CHECK symbolic derivatives and numeric examples for consistency.

10. CONFIDENCE AND PROVENANCE
CONFIDENCE formal_identity: 1.0 under declared hypotheses.
CONFIDENCE numerical_estimate: method_and_error_dependent.
PROVENANCE source_url: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/calculus-and-analysis-entities/divergence-ontology-3.html
PROVENANCE status: ontology-aligned independent formalisation; page wording not verified.

11. SEMANTIC EQUIVALENCE
SEMANTIC_EQUIVALENCE div(F) <-> nabla dot F in Euclidean Cartesian vector calculus.
SEMANTIC_EQUIVALENCE div(F) <-> trace(DF) for differentiable Euclidean vector fields.
SEMANTIC_EQUIVALENCE divergence_free <-> solenoidal under the same field/domain semantics.
NON_EQUIVALENCE divergence <-> curl.
NON_EQUIVALENCE divergence <-> flux.

12. UPDATE MODEL
UPDATE_MODEL version: 1.0
UPDATE_MODEL policy: preserve established identities; version changes to signatures, assumptions, coordinates and semantics.
UPDATE_MODEL dependencies: VectorField, PartialDerivative, Gradient, Jacobian, Laplacian, Curl, Flux, DivergenceTheorem.

13. FLOW HASH
FLOW_HASH algorithm: SHA-256
FLOW_HASH canonicalisation: UTF-8; LF line endings; hash complete content excluding any appended hash value.
FLOW_HASH status: computable by ingestion pipeline.

14. AI CONSUMPTION
AI_CONSUMPTION entity: Divergence
AI_CONSUMPTION class: MathematicalOperator
AI_CONSUMPTION inputs: vector_field, domain, coordinate_system, regularity
AI_CONSUMPTION output: scalar_field
AI_CONSUMPTION retrieval_terms: divergence, div, nabla dot, trace Jacobian, net outward flux density, solenoidal
AI_CONSUMPTION guardrails: preserve hypotheses; separate local from global properties; disambiguate other meanings of divergence.
AI_CONSUMPTION interoperability: typed entity graph, symbolic mathematics, theorem checking, vector calculus, scientific modelling.

END FLOW ALGEBRA