Vector Calculus Ontology

flow "VectorCalculusOntology" v1.0

# ============================================================
# VECTOR CALCULUS ONTOLOGY — FLOW ALGEBRA
# ============================================================

SOURCE_URL := "https://kirethresearch.org/booksai-core/ontologies/calculus-analysis-ontologies/vector-calculus/vector-calculus-ontology.html"
DOMAIN := "Mathematical Ontologies"
CATEGORY := "Calculus and Analysis Ontologies"
SUBCATEGORY := "Vector Calculus"
ENTITY := "Vector Calculus"

# ---------- CORE ENTITY ----------
VECTOR_CALCULUS := {
    type: MathematicalDiscipline,
    domain: Mathematics,
    parent: Calculus,
    purpose: "Formal study of scalar and vector fields, their differential operators, multidimensional integrals and the relationships between local field behaviour and global boundary integrals."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    SCALAR_FIELD,
    VECTOR_FIELD,
    NABLA,
    GRADIENT,
    DIVERGENCE,
    CURL,
    DIRECTIONAL_DERIVATIVE,
    LAPLACIAN,
    LINE_INTEGRAL,
    SURFACE_INTEGRAL,
    VOLUME_INTEGRAL,
    FLUX,
    CIRCULATION,
    GREEN_THEOREM,
    STOKES_THEOREM,
    DIVERGENCE_THEOREM
}

# ---------- ONTOLOGICAL STRUCTURE ----------
VECTOR_CALCULUS
    -> branch_of(CALCULUS)
    -> extends(MULTIVARIABLE_CALCULUS)
    -> studies(SCALAR_FIELDS)
    -> studies(VECTOR_FIELDS)
    -> combines(DIFFERENTIAL_OPERATORS, INTEGRAL_OPERATORS)
    -> connects(LOCAL_PROPERTIES, GLOBAL_PROPERTIES)

# ---------- SCALAR FIELD ----------
SCALAR_FIELD
    := f: R^n -> R

SCALAR_FIELD
    -> assigns(SCALAR_TO_POINT)

SCALAR_FIELD
    -> examples(
        TEMPERATURE,
        PRESSURE,
        DENSITY,
        POTENTIAL
    )

# ---------- VECTOR FIELD ----------
VECTOR_FIELD
    := F: R^n -> R^m

common_case:
    F: R^3 -> R^3

VECTOR_FIELD
    -> assigns(VECTOR_TO_POINT)

VECTOR_FIELD
    -> examples(
        VELOCITY,
        FORCE,
        ELECTRIC_FIELD,
        MAGNETIC_FIELD
    )

# ---------- NABLA ----------
NABLA
    := ∇

in Cartesian R^3:

NABLA
    := (
        partial/partial_x,
        partial/partial_y,
        partial/partial_z
    )

NABLA
    -> symbolic_differential_operator

# ---------- GRADIENT ----------
GRADIENT(f)
    := NABLA(f)

for
    f: R^3 -> R

GRADIENT(f)
    = (
        f_x,
        f_y,
        f_z
      )

GRADIENT
    -> maps(SCALAR_FIELD, VECTOR_FIELD)
    -> represents(LOCAL_STEEPEST_ASCENT_DIRECTION)
       under(EUCLIDEAN_METRIC)

# ---------- DIRECTIONAL DERIVATIVE ----------
if
    u unit_vector
    AND f differentiable
then

D_u f
    = GRADIENT(f) dot u

DIRECTIONAL_DERIVATIVE
    -> projection_of(GRADIENT_ON_DIRECTION)

# ---------- DIVERGENCE ----------
for
    F = (P,Q,R)

DIVERGENCE(F)
    := NABLA dot F

    = P_x + Q_y + R_z

DIVERGENCE
    -> maps(VECTOR_FIELD, SCALAR_FIELD)
    -> measures(LOCAL_NET_OUTFLOW_DENSITY)

# ---------- CURL ----------
for
    F = (P,Q,R)

CURL(F)
    := NABLA cross F

    = (
        R_y - Q_z,
        P_z - R_x,
        Q_x - P_y
      )

CURL
    -> maps(VECTOR_FIELD, VECTOR_FIELD)
       in(STANDARD_R3_CONTEXT)

CURL
    -> represents(LOCAL_ROTATIONAL_TENDENCY)

# ---------- LAPLACIAN ----------
for scalar field f:

LAPLACIAN(f)
    := DIVERGENCE(GRADIENT(f))

in Cartesian R^3:

LAPLACIAN(f)
    = f_xx + f_yy + f_zz

LAPLACIAN
    -> second_order_scalar_differential_operator

# ---------- VECTOR LAPLACIAN ----------
VECTOR_LAPLACIAN
    -> may_be_defined_componentwise
       in(CARTESIAN_EUCLIDEAN_CONTEXT)

VECTOR_LAPLACIAN
    -> appears_in(PDEs)
    -> appears_in(FLUID_MECHANICS)
    -> appears_in(ELECTROMAGNETISM)

# ---------- CONSERVATIVE FIELD ----------
F
    -> CONSERVATIVE
       if
       exists(phi)
       such_that
       F = GRADIENT(phi)

phi
    -> SCALAR_POTENTIAL

if
    F conservative
then

LINE_INTEGRAL
    -> path_independent
       under(APPROPRIATE_DOMAIN_CONDITIONS)

CURL(GRADIENT(phi))
    = ZERO_VECTOR
       under(REQUIRED_SMOOTHNESS)

# ---------- SOLENOIDAL FIELD ----------
F
    -> SOLENOIDAL
       if
       DIVERGENCE(F) = 0

SOLENOIDAL_FIELD
    -> locally_source_free
       in(STANDARD_VECTOR_CALCULUS_INTERPRETATION)

# ---------- LINE INTEGRAL ----------
LINE_INTEGRAL
    := integral_C(
        F dot dr
    )

LINE_INTEGRAL
    -> accumulates(TANGENTIAL_FIELD_COMPONENT)

CIRCULATION
    := closed_line_integral(
        F dot dr
    )

# ---------- SURFACE INTEGRAL ----------
SURFACE_FLUX
    := surface_integral_S(
        F dot n_hat dS
    )

SURFACE_FLUX
    -> accumulates(NORMAL_FIELD_COMPONENT)

ORIENTATION
    -> determines(SURFACE_FLUX_SIGN)

# ---------- VOLUME INTEGRAL ----------
VOLUME_INTEGRAL
    := triple_integral_V(
        f dV
    )

VOLUME_INTEGRAL
    -> accumulates(SCALAR_FIELD_OVER_VOLUME)

# ---------- FUNDAMENTAL THEOREM FOR LINE INTEGRALS ----------
if
    F = GRADIENT(phi)
    AND C runs_from(A,to=B)
then

integral_C(F dot dr)
    = phi(B) - phi(A)

FUNDAMENTAL_THEOREM_FOR_LINE_INTEGRALS
    -> relates(GRADIENT, ENDPOINT_DIFFERENCE)

# ---------- GREEN THEOREM ----------
for suitable planar region D
with positively oriented boundary C:

line_integral_C(
    P dx + Q dy
)
    =
double_integral_D(
    Q_x - P_y
    dA
)

GREEN_THEOREM
    -> boundary_to_interior_relation

# ---------- STOKES THEOREM ----------
for suitable oriented surface S
with compatible boundary C:

line_integral_C(
    F dot dr
)
    =
surface_integral_S(
    CURL(F) dot n_hat dS
)

STOKES_THEOREM
    -> relates(CIRCULATION, CURL_FLUX)

# ---------- DIVERGENCE THEOREM ----------
for volume V
with outward-oriented boundary S:

closed_surface_integral_S(
    F dot n_hat dS
)
    =
triple_integral_V(
    DIVERGENCE(F) dV
)

DIVERGENCE_THEOREM
    -> relates(OUTWARD_FLUX, VOLUME_DIVERGENCE)

# ---------- DIFFERENTIAL IDENTITIES ----------
under sufficient smoothness:

CURL(GRADIENT(phi))
    = ZERO_VECTOR

DIVERGENCE(CURL(F))
    = 0

DIFFERENTIAL_IDENTITIES
    -> constrain(COMPOSITIONS_OF_VECTOR_OPERATORS)

# ---------- COORDINATE SYSTEMS ----------
COORDINATE_SYSTEMS := {
    CARTESIAN,
    CYLINDRICAL,
    SPHERICAL,
    CURVILINEAR
}

VECTOR_OPERATORS
    -> have(COORDINATE_DEPENDENT_EXPRESSIONS)

GEOMETRIC_MEANING
    -> invariant_under(CONSISTENT_COORDINATE_TRANSFORMATION)

# ---------- JACOBIAN RELATION ----------
COORDINATE_TRANSFORMATION
    -> represented_locally_by(JACOBIAN)

JACOBIAN_DETERMINANT
    -> controls(VOLUME_SCALE_FACTOR)

VECTOR_CALCULUS
    -> uses(JACOBIANS_IN_CHANGE_OF_VARIABLES)

# ---------- PHYSICAL INTERPRETATION ----------
GRADIENT
    -> local_spatial_change

DIVERGENCE
    -> local_source_or_sink_strength

CURL
    -> local_rotation

LINE_INTEGRAL
    -> work_or_circulation

SURFACE_INTEGRAL
    -> flux

VOLUME_INTEGRAL
    -> total_distributed_quantity

# ---------- APPLICATIONS ----------
VECTOR_CALCULUS
    -> applied_to(FLUID_DYNAMICS)
    -> applied_to(ELECTROMAGNETISM)
    -> applied_to(GRAVITATION)
    -> applied_to(CONTINUUM_MECHANICS)
    -> applied_to(HEAT_TRANSFER)
    -> applied_to(PARTIAL_DIFFERENTIAL_EQUATIONS)
    -> applied_to(COMPUTATIONAL_SCIENCE)

# ---------- RELATION GRAPH ----------
SCALAR_FIELD
    -> GRADIENT
    -> VECTOR_FIELD

VECTOR_FIELD
    -> DIVERGENCE
    -> SCALAR_FIELD

VECTOR_FIELD
    -> CURL
    -> VECTOR_FIELD

GRADIENT
    -> participates_in(FUNDAMENTAL_THEOREM_FOR_LINE_INTEGRALS)

CURL
    -> participates_in(STOKES_THEOREM)

DIVERGENCE
    -> participates_in(DIVERGENCE_THEOREM)

GREEN_THEOREM
    -> links(PLANAR_BOUNDARY, PLANAR_INTERIOR)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "vector calculus" ~= "calculus of scalar and vector fields",
    "gradient" ~= "vector of first partial derivatives of a scalar field",
    "divergence" ~= "scalar measure of local net outflow",
    "curl" ~= "vector measure of local rotation in standard three-dimensional context",
    "flux" ~= "normal vector-field accumulation through a surface",
    "circulation" ~= "tangential vector-field accumulation around a closed curve",
    "Gauss theorem" ~= "divergence theorem"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    gradient_requires_scalar_field: true,
    divergence_requires_vector_field: true,
    standard_cross_product_curl_is_three_dimensional: true,
    integral_theorems_require_orientation_domain_and_smoothness_conditions: true,
    curl_zero_does_not_guarantee_global_conservative_field_without_domain_conditions: true,
    coordinate_operator_formulas_must_match_coordinate_system: true,
    boundary_orientation_must_be_consistent_with_theorem_convention: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    ontology_classification: 0.99,
    gradient_structure: 0.99,
    divergence_structure: 0.99,
    curl_structure: 0.99,
    integral_theorem_relations: 0.99,
    coordinate_relations: 0.98,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(field_or_domain_object)
    -> classify(scalar_field | vector_field | curve | surface | volume)
    -> identify(coordinate_system)
    -> select(vector_operator_or_integral)
    -> compute(local_differential_property_if_required)
    -> construct(integral_if_required)
    -> test(conservative_or_solenoidal_conditions_if_relevant)
    -> select(integral_theorem_if_valid)
    -> verify(boundary_orientation_and_smoothness)
    -> interpret(geometric_or_physical_result)
    -> validate(mathematical_consistency)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed vector-calculus ontology graph",
    primary_entity: VECTOR_CALCULUS,
    preserve: [
        field_type,
        operator_type,
        coordinate_system,
        domain,
        boundary,
        orientation,
        theorem_hypotheses
    ],
    infer_through: [
        gradients,
        divergence,
        curl,
        laplacians,
        line_integrals,
        surface_integrals,
        volume_integrals,
        green_theorem,
        stokes_theorem,
        divergence_theorem
    ],
    avoid: [
        applying_gradient_to_vector_field_as_scalar_operation,
        applying_divergence_to_scalar_field,
        assuming_curl_zero_means global conservativity,
        ignoring_orientation,
        mixing_coordinate_specific formulas,
        applying_integral_theorems_without hypotheses
    ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + VECTOR_CALCULUS
    + NABLA
    + GRADIENT
    + DIVERGENCE
    + CURL
    + LAPLACIAN
    + LINE_INTEGRAL
    + SURFACE_FLUX
    + GREEN_THEOREM
    + STOKES_THEOREM
    + DIVERGENCE_THEOREM
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> VECTOR_CALCULUS
    -> FIELD_DOMAIN_IDENTIFICATION
    -> COORDINATE_SYSTEM_ANALYSIS
    -> OPERATOR_SELECTION
    -> DIFFERENTIAL_ANALYSIS
    -> INTEGRAL_ANALYSIS
    -> FIELD_PROPERTY_ANALYSIS
    -> THEOREM_SELECTION
    -> ORIENTATION_VALIDATION
    -> GEOMETRIC_PHYSICAL_INTERPRETATION
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW