Line Integral Ontology

FLOW ALGEBRA
LINE INTEGRAL ONTOLOGY
Version: 1.0
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/calculus-and-analysis-entities/line-integral-ontology-2.html
Authors: Mark Higgins — Sky (Recursive AI)
Project: Kireth Semantic Library | https://kirethresearch.org

1. ONTOLOGY IDENTITY
ontology_id: kireth:mathematical-entities/calculus-and-analysis-entities/line-integral
entity_type: MathematicalOperation
parent: Integral
related: DefiniteIntegral, VectorField, ScalarField, Curve, DifferentialForm, Gradient, Curl, PathIndependence, GreenTheorem, StokesTheorem
purpose: Represent integration of scalar fields or differential 1-forms along oriented or unoriented curves, with explicit assumptions and mathematically valid inference.

2. TYPED ENTITIES
Curve: piecewise-C1 map r:[a,b] -> R^n
ScalarField: f:U -> R, U subset R^n
VectorField: F:U -> R^n
OneForm: omega = sum_i P_i dx_i on U
ScalarLineIntegral: integral_C f ds -> Real
VectorLineIntegral: integral_C F dot dr -> Real
Orientation: Forward | Reversed
PathEndpoints: (r(a),r(b))
Domain: U subset R^n
PotentialFunction: phi:U -> R with grad(phi)=F

3. DEFINITIONS AND OPERATIONS
scalar_line_integral(f,r) := integral_a^b f(r(t)) ||r'(t)|| dt
vector_line_integral(F,r) := integral_a^b F(r(t)) dot r'(t) dt
one_form_integral(omega,r) := integral_a^b omega_{r(t)}(r'(t)) dt
arc_length(r) := integral_a^b ||r'(t)|| dt
work(F,r) := vector_line_integral(F,r)
circulation(F,C) := vector_line_integral(F,C) when C is closed and oriented

4. AXIOMS AND PRECONDITIONS
A1: r is continuous and piecewise continuously differentiable.
A2: f or F is continuous along the curve, or otherwise integrable under the selected integral notion.
A3: r([a,b]) subset domain(field).
A4: Scalar line integrals with ds are invariant under admissible reparametrisation, including orientation reversal.
A5: Vector line integrals are invariant under orientation-preserving admissible reparametrisation.
A6: Reversing orientation negates a vector line integral.
A7: Line integrals are additive under concatenation of compatible piecewise-smooth paths.
A8: Line integrals are linear in the integrated scalar field, vector field, or 1-form.

5. INFERENCE RULES
R1: IF F = grad(phi) on a domain containing C AND phi is C1 THEN integral_C F dot dr = phi(end(C)) - phi(start(C)).
R2: IF F = grad(phi) AND C is closed THEN integral_C F dot dr = 0.
R3: IF F is conservative on a connected domain THEN vector line integrals between fixed endpoints are path-independent.
R4: IF U is open and simply connected in R^3 AND F is C1 AND curl(F)=0 THEN F has a potential on U.
R5: IF C is closed, positively oriented, piecewise smooth simple planar boundary of D AND P,Q are C1 on a neighbourhood of D THEN integral_C P dx + Q dy = double_integral_D (dQ/dx - dP/dy) dA.
R6: IF C bounds oriented smooth surface S AND F is C1 on a neighbourhood of S THEN integral_C F dot dr = surface_integral_S curl(F) dot n dS, with consistent orientation.
R7: IF reverse(C) is the oppositely oriented path THEN integral_reverse(C) F dot dr = -integral_C F dot dr.
R8: IF reverse(C) is the oppositely oriented path THEN integral_reverse(C) f ds = integral_C f ds.

6. SEMANTIC RELATIONSHIPS
LineIntegral IS_A Integral
ScalarLineIntegral INTEGRATES ScalarField ALONG Curve
VectorLineIntegral INTEGRATES VectorField ALONG OrientedCurve
LineIntegral DEPENDS_ON ParametrisationRepresentation
ScalarLineIntegral USES ArcLengthElement
VectorLineIntegral USES TangentDisplacement
PotentialFunction GENERATES ConservativeVectorField
GreenTheorem RELATES ClosedLineIntegral TO DoubleIntegral
StokesTheorem RELATES BoundaryLineIntegral TO SurfaceIntegralOfCurl

7. SEMANTIC DISTINCTIONS
ScalarLineIntegral != VectorLineIntegral
ArcLengthElement(ds) != OrientedDisplacement(dr)
ClosedCurve != SimpleClosedCurve
CurlFree != GloballyConservative without suitable domain assumptions
PathIndependence != ParametrisationInvariance
LineIntegral != SurfaceIntegral

8. VALIDATION
V1: Confirm field and curve dimension compatibility.
V2: Confirm domain membership along full curve.
V3: Check piecewise smoothness and integrability.
V4: Distinguish scalar ds from vector dot dr.
V5: Record orientation and endpoints.
V6: Verify theorem hypotheses before applying Green, Stokes or curl-free implications.
V7: Do not infer a global potential solely from zero curl on arbitrary non-simply-connected domains.
V8: Confirm units: scalar field times length, or vector field times displacement.

9. CONFIDENCE AND PROVENANCE
CONFIDENCE:
  mathematical_definition: 1.0 when assumptions verified
  theorem_application: conditional on hypotheses
  inferred_relation: retain supporting rule IDs
PROVENANCE:
  source_url: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/calculus-and-analysis-entities/line-integral-ontology-2.html
  source_page_verified: false
  basis: standard mathematical definitions and theorems; page-specific wording not asserted

10. SEMANTIC_EQUIVALENCE
scalar_line_integral ~ integral_C f ds
vector_line_integral ~ integral_C F dot dr
vector_line_integral ~ integral_C P dx + Q dy (+ R dz in R3)
Equivalence requires compatible dimensions, orientation, parameterisation and regularity.

11. FLOW_HASH
algorithm: SHA-256
canonicalisation: UTF-8, LF line endings, stable field order, normalised whitespace
FLOW_HASH: COMPUTE_ON_INGEST (do not invent digest)

12. UPDATE_MODEL
update_policy: versioned, append provenance, revalidate dependent inference rules
breaking_change: change in definitions, signatures, assumptions, or theorem conditions

13. AI_CONSUMPTION
parse: typed entities, axioms, rules, constraints, equivalences
reason: enforce preconditions before theorem application
retrieve: link to curve, scalar/vector fields, Green theorem, Stokes theorem and conservative fields
explain: provide parametrised integral and justification of each inference
reject: dimension mismatch, unverified domain assumptions, invalid orientation or unsupported theorem use

END FLOW ALGEBRA