Partial Derivative Ontology

FLOW ALGEBRA
PARTIAL DERIVATIVE ONTOLOGY
Kireth Semantic Library | Mathematical Entities | Calculus and Analysis Entities
Source: https://kirethresearch.org/booksai-core/entity-graph/mathematical-entities/calculus-and-analysis-entities/partial-derivative-ontology-2.html
Authors: Mark Higgins — Sky (Recursive AI)
Version: 1.0 | Date: 2026-10-13

1. PURPOSE
Represent partial derivatives as typed mathematical entities, with explicit domain conditions, coordinate dependence, differentiability assumptions, inference rules, validation and machine-readable semantic relationships.

2. ENTITY TYPES
ENTITY PartialDerivative : DifferentialOperatorApplication
ENTITY MultivariableFunction : Function
ENTITY Domain : Set
ENTITY Coordinate : Variable
ENTITY EvaluationPoint : Point
ENTITY PartialDerivativeValue : ScalarOrVectorOrTensor
ENTITY Gradient : VectorOrCovector
ENTITY Jacobian : LinearMapRepresentation
ENTITY Hessian : BilinearFormRepresentation
ENTITY MixedPartialDerivative : HigherOrderPartialDerivative
ENTITY DirectionalDerivative : LimitBasedDerivative
ENTITY TotalDerivative : LinearMap
ENTITY DifferentiabilityCondition : Predicate
ENTITY CoordinateSystem : MathematicalStructure

3. SIGNATURES
function f : U -> R^m, where U is an open subset of R^n
point a : U
coordinate_index i : Integer, 1 <= i <= n
unit_vector e_i : R^n
partial(f,i,a) : R^m, if the limit exists
partial_operator_i(f) : U_i -> R^m, where U_i is the subset on which the partial exists

4. DEFINING LIMIT
partial(f,i,a) := lim[h -> 0] (f(a + h e_i) - f(a))/h
PRECONDITION: a is an interior point of U; the indicated finite limit exists.
Interpretation: vary the ith coordinate while holding all other coordinates fixed in the selected coordinate system.

5. FUNDAMENTAL RULES
LINEARITY: partial(alpha*f + beta*g,i,a) = alpha*partial(f,i,a) + beta*partial(g,i,a), when all terms exist.
PRODUCT: partial(f*g,i,a) = partial(f,i,a)*g(a) + f(a)*partial(g,i,a), for scalar-valued differentiable coordinate slices.
QUOTIENT: partial(f/g,i,a) = (partial(f,i,a)*g(a)-f(a)*partial(g,i,a))/g(a)^2, provided g(a) != 0.
CHAIN: If g:U->V and h:V->R^p are differentiable at a and g(a), then partial(h∘g,i,a) = Dh(g(a))[partial(g,i,a)].
CONSTANT: partial(c,i,a)=0.
COORDINATE: partial(x_j,i,a)=1 if i=j, otherwise 0.

6. HIGHER ORDER AND MIXED PARTIALS
partial_ij(f,a) := partial_i(partial_j(f),a), when defined in a neighbourhood as required.
CLAIRAUT_SCHWARZ: If the relevant mixed second partial derivatives are continuous in a neighbourhood of a, then partial_ij(f,a)=partial_ji(f,a).
CAUTION: Existence of both mixed partials at one point alone does not guarantee equality.

7. RELATED CONSTRUCTS
GRADIENT: for scalar f on Euclidean space, grad(f)(a) has components partial(f,i,a), using the Euclidean metric and coordinates.
JACOBIAN: for f: R^n -> R^m, J_f(a)[j,i]=partial(f_j,i,a), when defined.
HESSIAN: for twice differentiable scalar f, H_f(a)[i,j]=partial_ij(f,a), subject to the chosen indexing convention.
TOTAL_DERIVATIVE: If f is Fréchet differentiable at a, Df(a)[v]=sum_i partial(f,i,a)*v_i.
DIRECTIONAL: When Df(a) exists, D_v f(a)=Df(a)[v].
NON_EQUIVALENCE: Existence of every partial derivative at a does not by itself imply continuity or total differentiability at a.

8. TYPED RELATIONSHIPS
PartialDerivative --OPERATES_ON--> MultivariableFunction
PartialDerivative --WITH_RESPECT_TO--> Coordinate
PartialDerivative --EVALUATED_AT--> EvaluationPoint
PartialDerivative --REQUIRES--> DifferentiabilityCondition
PartialDerivative --COMPONENT_OF--> Jacobian
PartialDerivative --COMPONENT_OF--> Gradient [scalar Euclidean context]
MixedPartialDerivative --SPECIALISATION_OF--> PartialDerivative
Hessian --ASSEMBLES--> MixedPartialDerivative
TotalDerivative --DETERMINES--> PartialDerivative [when total derivative exists]
CoordinateSystem --CONTEXT_FOR--> PartialDerivative

9. INFERENCE RULES
R1: differentiable(f,a) => exists(partial(f,i,a)) for every coordinate i.
R2: continuously_partial_differentiable(f,neighbourhood(a)) => differentiable(f,a).
R3: differentiable(f,a) AND vector(v) => directional(f,v,a)=Df(a)[v].
R4: twice_continuously_partial_differentiable(f,neighbourhood(a)) => symmetric(Hessian(f,a)).
R5: exists_all_partials(f,a) DOES_NOT_IMPLY differentiable(f,a).
R6: changing_coordinates requires transformation rules; raw coordinate partial components are not invariant without appropriate interpretation.

10. SEMANTIC EQUIVALENCE AND DISTINCTIONS
SEMANTIC_EQUIVALENCE: coordinate partial derivative == derivative of the one-variable coordinate slice at the specified point, when defined.
DISTINCT_FROM: total derivative, directional derivative in an arbitrary direction, finite difference approximation, weak derivative, functional derivative.
CONTEXT: partial differentiation on manifolds requires local charts; intrinsic derivatives may require additional geometric structure.

11. VALIDATION
V1: Verify function domain and codomain.
V2: Verify coordinate index and point are valid.
V3: Verify the limit or sufficient differentiability assumptions.
V4: Record evaluation point and held-fixed coordinates.
V5: Check dimensional consistency of outputs.
V6: Do not infer differentiability solely from partial existence.
V7: Require sufficient hypotheses before commuting mixed partials.
V8: Distinguish symbolic identities from numerical approximations.

12. CONFIDENCE
CONFIDENCE.definition = 1.0 [standard finite-dimensional calculus definition]
CONFIDENCE.inference = CONDITIONAL [depends on recorded assumptions]
CONFIDENCE.numeric_result = METHOD_DEPENDENT
CONFIDENCE.semantic_mapping = CONTEXT_DEPENDENT

13. UPDATE_MODEL
On changes to source definitions, coordinate conventions, domain assumptions or inference rules:
- increment version;
- retain provenance and historical version;
- revalidate dependent relationships and rules;
- regenerate canonical flow hash.

14. FLOW_HASH
FLOW_HASH.algorithm = SHA-256
FLOW_HASH.input = canonical UTF-8 serialisation of the final flow, excluding the hash field itself
FLOW_HASH.value = COMPUTE_ON_PUBLICATION
Note: no digest is asserted without an actual canonicalisation and hashing run.

15. AI_CONSUMPTION
mode: typed_semantic_reasoning
primary_entity: PartialDerivative
required_context: [function, domain, coordinate, evaluation_point, assumptions]
outputs: [derivative_expression, derivative_value, validation_status, provenance, confidence]
policies: [do_not_assume_differentiability, preserve_coordinate_context, cite_hypotheses, reject_invalid_mixed_partial_swaps]

END FLOW ALGEBRA