Surface Integral Ontology

FLOW ALGEBRA
SURFACE INTEGRAL ONTOLOGY
Kireth Semantic Library | Entity Graph | Mathematical Entities | Calculus and Analysis Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/calculus-and-analysis-entities/surface-integral-ontology-2.html
Authors: Mark Higgins — Sky (Recursive AI)
Version: 1.0 | Date: 2026-10-13 | Locale: en-GB

1. PURPOSE
Represent surface integrals as typed mathematical entities with explicit domains, parametrisations, orientation, integration measures, preconditions, transformation rules and verifiable inference pathways.

2. ENTITY TYPES
SurfaceIntegral : IntegralEntity
ScalarSurfaceIntegral : SurfaceIntegral
FluxSurfaceIntegral : SurfaceIntegral
Surface : GeometricEntity
ParametrisedSurface : Surface
SurfacePatch : Surface
SurfaceParameterDomain : SubsetOfR2
SurfaceParametrisation : Map(R2,R3)
ScalarField : Map(R3,R)
VectorField : Map(R3,R3)
TangentVector : VectorEntity
NormalVector : VectorEntity
UnitNormal : NormalVector
SurfaceAreaElement : DifferentialMeasure
OrientedAreaElement : VectorDifferentialMeasure
SurfaceBoundary : CurveEntity
Orientation : GeometricAttribute
RegularityCondition : Constraint
DivergenceTheorem : MathematicalTheorem
StokesTheorem : MathematicalTheorem

3. TYPED RELATIONSHIPS
HAS_DOMAIN(SurfaceIntegral, Surface)
HAS_INTEGRAND(ScalarSurfaceIntegral, ScalarField)
HAS_INTEGRAND(FluxSurfaceIntegral, VectorField)
PARAMETRISED_BY(Surface, SurfaceParametrisation)
HAS_PARAMETER_DOMAIN(SurfaceParametrisation, SurfaceParameterDomain)
HAS_TANGENT_VECTOR(SurfaceParametrisation, TangentVector)
HAS_NORMAL(Surface, NormalVector)
HAS_ORIENTATION(Surface, Orientation)
HAS_BOUNDARY(Surface, SurfaceBoundary)
USES_MEASURE(ScalarSurfaceIntegral, SurfaceAreaElement)
USES_MEASURE(FluxSurfaceIntegral, OrientedAreaElement)
REQUIRES(SurfaceIntegral, RegularityCondition)
RELATED_TO(SurfaceIntegral, DivergenceTheorem)
RELATED_TO(SurfaceIntegral, StokesTheorem)

4. MATHEMATICAL DEFINITIONS
Let r(u,v) = (x(u,v),y(u,v),z(u,v)), (u,v) in D, be a piecewise C1 regular parametrisation of a surface S in R3.
ru = partial r / partial u; rv = partial r / partial v.
Surface Jacobian J_S = ||ru x rv||.
Scalar surface integral: integral_S f dS = integral_D f(r(u,v)) ||ru x rv|| du dv.
For a chosen consistent orientation n = (ru x rv)/||ru x rv|| on a regular patch:
Flux integral: integral_S F . n dS = integral_D F(r(u,v)) . (ru x rv) du dv.
Reversing orientation changes the sign of flux, but not scalar surface area integration.
Area(S) = integral_S 1 dS when area is finite.

5. CONDITIONS AND INVARIANTS
C1 regularity requires ru x rv != 0 on each regular patch; singular points need separate handling.
Scalar integrals are invariant under valid regular reparametrisations, accounting for absolute Jacobian.
Flux integrals are invariant under orientation-preserving reparametrisations; orientation reversal negates flux.
Additivity applies to suitable piecewise surfaces with overlapping patches only on measure-zero boundaries.
Linearity: integral_S (a f + b g) dS = a integral_S f dS + b integral_S g dS, when integrable.
Linearity also holds for vector-field flux with a fixed orientation.

6. THEOREM RELATIONSHIPS
Divergence theorem: for a sufficiently regular bounded volume V with outward-oriented closed boundary S, integral_S F . n dS = integral_V div(F) dV.
Stokes' theorem: for a suitably oriented smooth surface S with positively induced boundary C, integral_S (curl F) . n dS = integral_C F . dr.
Do not infer the divergence theorem for an open surface without closing it and accounting for additional boundary flux.

7. INFERENCE RULES
RULE SI01: IF surface parametrisation is regular AND scalar field is integrable THEN compute scalar integral by J_S-weighted double integration.
RULE SI02: IF orientation is specified AND vector field is integrable THEN compute flux using the oriented cross-product area vector.
RULE SI03: IF surface orientation reverses THEN flux value is negated.
RULE SI04: IF surface is closed, outward oriented, and divergence theorem hypotheses hold THEN replace total flux with volume divergence integral.
RULE SI05: IF field and surface satisfy Stokes' theorem hypotheses THEN replace curl flux with oriented boundary circulation.
RULE SI06: IF integrand equals 1 AND surface has finite area THEN scalar surface integral equals area.
RULE SI07: IF a parametrisation changes THEN verify Jacobian and orientation before declaring integral equivalence.

8. SEMANTIC DISTINCTIONS
Scalar surface integral measures accumulated scalar quantity over surface area.
Flux surface integral measures signed normal passage of a vector field through an oriented surface.
Surface integral != line integral: integration domain dimensions differ.
Surface integral != volume integral: geometric measures and domains differ.
Flux != divergence: flux is an integral quantity; divergence is a local scalar field.
Surface normal is not necessarily outward unless a closed surface and outward orientation are specified.

9. VALIDATION MODEL
CHECK V01: Domain and integrand types are compatible.
CHECK V02: Parametrisation dimension and image are correct.
CHECK V03: Surface Jacobian is nonnegative and well-defined almost everywhere.
CHECK V04: Orientation is explicit for signed flux.
CHECK V05: Surface patches are not double-counted except on measure-zero overlaps.
CHECK V06: Convergence and integrability are established, including improper cases.
CHECK V07: Theorem-specific smoothness, boundary and orientation assumptions are satisfied.
CHECK V08: Output dimensions/units match integrand multiplied by area units.

10. CONFIDENCE AND PROVENANCE
CONFIDENCE: theorem-backed | conditional | unresolved.
Assign theorem-backed only where hypotheses and calculation steps are verified.
Track source, definitions, assumptions, parametrisation, orientation and derivation provenance.

11. SEMANTIC_EQUIVALENCE
Two scalar surface integral expressions are equivalent when they represent the same integrable scalar field on the same geometric surface with correctly transformed area measure.
Two flux expressions are equivalent when their vector field, surface and orientation agree, including correct transformation of oriented area elements.
Equivalence must not be asserted from symbolic resemblance alone.

12. FLOW_HASH
FLOW_HASH_INPUT = canonicalised(ontology_id, version, entity_types, relationships, definitions, constraints, inference_rules, validation_rules).
FLOW_HASH = SHA256(UTF8(FLOW_HASH_INPUT)).
Hash is a computation specification; no digest is asserted without running canonicalisation and hashing.

13. UPDATE_MODEL
On a definition, theorem or relationship change: increment version; revalidate affected inference rules; regenerate canonical representation and FLOW_HASH; retain prior provenance.

14. AI_CONSUMPTION
Parse entities and relationships into a typed graph.
Preserve distinctions between scalar area integrals and oriented flux integrals.
Require explicit orientation and theorem preconditions before applying transformation rules.
Return derivations with hypotheses, confidence status, and provenance.
Reject unsupported equivalence, invalid domains and unverified theorem applications.

END FLOW ALGEBRA