Higher Order Derivative Ontology

flow "HigherOrderDerivativeOntology" v1.0

# ============================================================
# HIGHER-ORDER DERIVATIVE ONTOLOGY — FLOW ALGEBRA
# ============================================================

SOURCE_URL := "https://kirethresearch.org/booksai-core/ontologies/calculus-analysis-ontologies/differential-calculus/higher-order-derivative-ontology.html"
DOMAIN := "Mathematical Ontologies"
CATEGORY := "Calculus and Analysis Ontologies"
SUBCATEGORY := "Differential Calculus"
ENTITY := "Higher-Order Derivative"

# ---------- CORE ENTITY ----------
HIGHER_ORDER_DERIVATIVE := {
    type: MathematicalDerivative,
    domain: DifferentialCalculus,
    purpose: "Represent derivatives obtained by repeatedly differentiating a function, describing higher-order rates of change and local behaviour."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    FIRST_DERIVATIVE,
    SECOND_DERIVATIVE,
    THIRD_DERIVATIVE,
    NTH_DERIVATIVE,
    REPEATED_DIFFERENTIATION,
    DIFFERENTIABILITY_CLASS,
    CONCAVITY,
    INFLECTION_POINT,
    VELOCITY,
    ACCELERATION,
    JERK,
    TAYLOR_SERIES,
    HIGHER_PARTIAL_DERIVATIVE,
    HESSIAN
}

# ---------- REPEATED DIFFERENTIATION ----------
FIRST_DERIVATIVE
    := D[f]

SECOND_DERIVATIVE
    := D[D[f]]
    = D^2[f]

THIRD_DERIVATIVE
    := D[D^2[f]]
    = D^3[f]

NTH_DERIVATIVE
    := D^n[f]

HIGHER_ORDER_DERIVATIVE
    -> obtained_by(REPEATED_DIFFERENTIATION)

# ---------- NOTATION ----------
HIGHER_DERIVATIVE_NOTATION := {
    "f_double_prime(x)",
    "f_triple_prime(x)",
    "f^(n)(x)",
    "d^2f/dx^2",
    "d^3f/dx^3",
    "d^nf/dx^n",
    "D^2f",
    "D^nf"
}

NOTATION
    -> encodes(DERIVATIVE_ORDER)

# ---------- ORDER ----------
DERIVATIVE_ORDER
    -> positive_integer

ORDER_1
    -> FIRST_DERIVATIVE

ORDER_2
    -> SECOND_DERIVATIVE

ORDER_n
    -> NTH_DERIVATIVE

DERIVATIVE_ORDER
    -> counts(NUMBER_OF_DIFFERENTIATION_OPERATIONS)

# ---------- DIFFERENTIABILITY CLASSES ----------
C^0
    -> CONTINUOUS_FUNCTIONS

C^1
    -> functions_with(CONTINUOUS_FIRST_DERIVATIVE)

C^2
    -> functions_with(CONTINUOUS_DERIVATIVES_UP_TO_ORDER_2)

C^n
    -> functions_with(CONTINUOUS_DERIVATIVES_UP_TO_ORDER_n)

C_INFINITY
    -> SMOOTH_FUNCTIONS
    -> derivatives_exist_and_are_continuous_of_all_finite_orders

HIGHER_ORDER_DIFFERENTIABILITY
    -> classified_by(REGULARITY_CLASS)

# ---------- SECOND DERIVATIVE ----------
SECOND_DERIVATIVE
    -> measures(RATE_OF_CHANGE_OF_FIRST_DERIVATIVE)
    -> supports(CONCAVITY_ANALYSIS)

if
    f_double_prime(x) > 0
then
    f -> locally(CONCAVE_UP)

if
    f_double_prime(x) < 0
then
    f -> locally(CONCAVE_DOWN)

# ---------- INFLECTION ----------
INFLECTION_POINT
    -> requires(CHANGE_IN_CONCAVITY)

f_double_prime(a) = 0
    -> may_identify(INFLECTION_CANDIDATE)

f_double_prime(a) = 0
    -/> sufficient_for(INFLECTION_POINT)

# ---------- PHYSICAL INTERPRETATION ----------
if
    position = s(t)
then

    FIRST_DERIVATIVE:
        ds/dt = velocity

    SECOND_DERIVATIVE:
        d2s/dt2 = acceleration

    THIRD_DERIVATIVE:
        d3s/dt3 = jerk

FOURTH_DERIVATIVE
    -> sometimes_called(SNAP)
       in(MOTION_CONTEXT)

HIGHER_DERIVATIVES
    -> encode(HIGHER_ORDER_MOTION_CHANGE)

# ---------- TAYLOR STRUCTURE ----------
TAYLOR_POLYNOMIAL_OF_ORDER_n
    := sum(
        f^(k)(a)/k! * (x-a)^k,
        k=0..n
    )

HIGHER_ORDER_DERIVATIVES
    -> determine(TAYLOR_COEFFICIENTS)

TAYLOR_APPROXIMATION
    -> uses(LOCAL_DERIVATIVE_INFORMATION_OF_MULTIPLE_ORDERS)

# ---------- LOCAL APPROXIMATION ----------
FIRST_DERIVATIVE
    -> controls(FIRST_ORDER_TERM)

SECOND_DERIVATIVE
    -> controls(SECOND_ORDER_CURVATURE_TERM)

NTH_DERIVATIVE
    -> contributes(NTH_ORDER_LOCAL_TERM)

HIGHER_ORDER_DERIVATIVES
    -> refine(LOCAL_APPROXIMATION)

# ---------- HIGHER-ORDER PRODUCT RULE ----------
LEIBNIZ_RULE:

D^n(f*g)
    = sum(
        binomial(n,k)
        * D^k(f)
        * D^(n-k)(g),
        k=0..n
    )

LEIBNIZ_RULE
    -> generalises(PRODUCT_RULE_TO_ORDER_n)

# ---------- HIGHER-ORDER CHAIN STRUCTURE ----------
REPEATED_DIFFERENTIATION_OF_COMPOSITION
    -> requires(REPEATED_CHAIN_RULE)

HIGHER_CHAIN_RULE
    -> may_be_expressed_using(FAA_DI_BRUNO_FORMULA)

FAA_DI_BRUNO_FORMULA
    -> generalises(CHAIN_RULE_TO_HIGHER_DERIVATIVES)

# ---------- HIGHER PARTIAL DERIVATIVES ----------
for
    f(x1,...,xn)

HIGHER_PARTIAL_DERIVATIVE
    -> repeated_partial_differentiation

PURE_PARTIAL
    := partial^n f / partial x_i^n

MIXED_PARTIAL
    := partial^n f /
       (partial x_1^a1 ... partial x_m^am)

where
    a1 + ... + am = n

# ---------- MIXED PARTIAL COMMUTATION ----------
if
    relevant_mixed_partials_are_continuous
    near(point)
then

ORDER_OF_PARTIAL_DIFFERENTIATION
    -> may_be_interchanged

MIXED_PARTIAL_EQUALITY
    -> associated_with(CLAIRAUT_SCHWARZ_THEOREM)

REGULARITY_CONDITIONS
    -> required_for(GENERAL_COMMUTATION_CLAIM)

# ---------- HESSIAN ----------
HESSIAN(f)
    := matrix(SECOND_ORDER_PARTIAL_DERIVATIVES)

HESSIAN
    -> encodes(SECOND_ORDER_MULTIVARIABLE_BEHAVIOUR)
    -> supports(CRITICAL_POINT_CLASSIFICATION)
    -> supports(CURVATURE_ANALYSIS)

# ---------- DIFFERENTIAL EQUATIONS ----------
HIGHER_ORDER_DIFFERENTIAL_EQUATION
    -> contains(HIGHER_ORDER_DERIVATIVES)

ORDER_OF_DIFFERENTIAL_EQUATION
    -> determined_by(HIGHEST_DERIVATIVE_ORDER_PRESENT)

# ---------- OPTIMISATION ----------
FIRST_DERIVATIVE
    -> identifies(STATIONARY_CANDIDATES)

SECOND_DERIVATIVE
    -> may_classify(STATIONARY_POINTS)

HIGHER_DERIVATIVE_TEST
    -> may_classify(DEGENERATE_CRITICAL_POINTS)

if
    f_prime(a)=0
    AND f_double_prime(a)=0
then
    -> higher_derivatives_may_be_examined

# ---------- RELATION GRAPH ----------
DERIVATIVE
    -> recursively_generates(HIGHER_ORDER_DERIVATIVES)

SECOND_DERIVATIVE
    -> measures(CHANGE_OF_FIRST_DERIVATIVE)

HIGHER_ORDER_DERIVATIVES
    -> determine(TAYLOR_TERMS)

HIGHER_ORDER_DERIVATIVES
    -> appear_in(HIGHER_ORDER_DIFFERENTIAL_EQUATIONS)

SECOND_PARTIAL_DERIVATIVES
    -> populate(HESSIAN)

REGULARITY_CLASS
    -> describes(HIGHER_DERIVATIVE_CONTINUITY)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "higher-order derivative" ~= "derivative of order two or greater",
    "second derivative" ~= "derivative of the first derivative",
    "nth derivative" ~= "result of differentiating n times when defined",
    "higher derivative" ~= "higher-order derivative",
    "smooth function" ~= "C-infinity function in standard analysis terminology",
    "acceleration" ~= "second derivative of position with respect to time",
    "jerk" ~= "third derivative of position with respect to time"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    nth_derivative_requires_successive_derivatives_to_exist: true,
    higher_derivative_existence_does_not_automatically_imply_continuity: true,
    second_derivative_zero_does_not_guarantee_inflection: true,
    mixed_partial_commutation_requires_appropriate_regularity: true,
    taylor_coefficients_require_corresponding_derivatives: true,
    derivative_order_must_be_preserved: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    repeated_derivative_structure: 0.99,
    notation_structure: 0.99,
    regularity_classes: 0.99,
    physical_interpretation: 0.99,
    taylor_relation: 0.99,
    higher_partial_relations: 0.99,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(function)
    -> identify(requested_derivative_order)
    -> determine(domain)
    -> verify(successive_differentiability)
    -> recursively_apply(differentiation)
    -> classify(derivative_order)
    -> interpret(second_or_higher_order_behaviour)
    -> connect_to(taylor_or_differential_equation_context_if_relevant)
    -> analyse(mixed_partials_if_multivariable)
    -> validate(regularity_conditions)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed higher-order derivative graph",
    primary_entity: HIGHER_ORDER_DERIVATIVE,
    preserve: [
        derivative_order,
        recursive_differentiation_structure,
        regularity_class,
        mixed_partial_order,
        domain_conditions,
        continuity_conditions
    ],
    infer_through: [
        derivatives,
        repeated_differentiation,
        taylor_series,
        hessians,
        differential_equations,
        motion_derivatives,
        optimisation
    ],
    avoid: [
        confusing_derivative_order,
        assuming_derivative_continuity_from_existence_alone,
        classifying_second_derivative_zero_as_automatic_inflection,
        commuting_mixed_partials_without_regularity,
        using_taylor_terms_without_required_derivatives
    ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + HIGHER_ORDER_DERIVATIVE
    + HIGHER_DERIVATIVE_NOTATION
    + SECOND_DERIVATIVE
    + TAYLOR_POLYNOMIAL_OF_ORDER_n
    + LEIBNIZ_RULE
    + HIGHER_PARTIAL_DERIVATIVE
    + HESSIAN
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> HIGHER_ORDER_DERIVATIVE
    -> FUNCTION_IDENTIFICATION
    -> ORDER_IDENTIFICATION
    -> DOMAIN_ANALYSIS
    -> SUCCESSIVE_DIFFERENTIABILITY_ANALYSIS
    -> REPEATED_DIFFERENTIATION
    -> SECOND_ORDER_BEHAVIOUR_ANALYSIS
    -> HIGHER_ORDER_APPROXIMATION_ANALYSIS
    -> MULTIVARIABLE_PARTIAL_ANALYSIS
    -> REGULARITY_ANALYSIS
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW