Volume Integral Ontology

flow "VolumeIntegralOntology" v1.0

# ============================================================
# VOLUME INTEGRAL ONTOLOGY — FLOW ALGEBRA
# ============================================================

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

# ---------- CORE ENTITY ----------
VOLUME_INTEGRAL := {
    type: MultipleIntegral,
    domain: MultivariableCalculus,
    purpose: "Represent accumulation of a scalar quantity throughout a three-dimensional region using a volume element."
}

# ---------- CORE CONCEPTS ----------
CONCEPTS := {
    VOLUME_INTEGRAL,
    TRIPLE_INTEGRAL,
    THREE_DIMENSIONAL_REGION,
    SCALAR_FIELD,
    VOLUME_ELEMENT,
    DENSITY,
    MASS,
    CHARGE_DENSITY,
    PROBABILITY_DENSITY,
    CHANGE_OF_VARIABLES,
    JACOBIAN_DETERMINANT,
    CYLINDRICAL_COORDINATES,
    SPHERICAL_COORDINATES
}

# ---------- BASIC STRUCTURE ----------
VOLUME_INTEGRAL
    := integral_over_V(
        f(x,y,z) dV
    )

V
    -> THREE_DIMENSIONAL_REGION

f(x,y,z)
    -> SCALAR_FIELD_OR_DENSITY

dV
    -> VOLUME_ELEMENT

VOLUME_INTEGRAL
    -> accumulates(QUANTITY_THROUGHOUT_VOLUME)

# ---------- TRIPLE INTEGRAL REPRESENTATION ----------
VOLUME_INTEGRAL
    -> commonly_represented_as(TRIPLE_INTEGRAL)

integral_over_V(f dV)
    ~= triple_integral_V(
        f(x,y,z) dx dy dz
    )
       in(CARTESIAN_COORDINATES)

TRIPLE_INTEGRAL
    -> computational_form_of(VOLUME_INTEGRAL)

# ---------- RIEMANN-SUM INTERPRETATION ----------
REGION_V
    -> partitioned_into(SMALL_VOLUME_ELEMENTS)

DELTA_V_i
    -> volume_of(SUBREGION_i)

SAMPLE_POINT_i
    -> selected_in(SUBREGION_i)

VOLUME_RIEMANN_SUM
    := sum(
        f(SAMPLE_POINT_i) * DELTA_V_i
    )

VOLUME_INTEGRAL
    := limit(VOLUME_RIEMANN_SUM)
       under(REGION_REFINEMENT)
       when(INTEGRABLE)

# ---------- CARTESIAN VOLUME ELEMENT ----------
in(CARTESIAN_COORDINATES):

dV
    = dx dy dz

ORDER
    -> may_be_permuted
       under(FUBINI_CONDITIONS_AND_CORRECT_BOUNDS)

REGION_GEOMETRY
    -> determines(NESTED_LIMITS)

# ---------- GEOMETRIC VOLUME ----------
if
    f(x,y,z) = 1
then

integral_over_V(1 dV)
    = VOLUME(V)

VOLUME_INTEGRAL
    -> generalises(GEOMETRIC_VOLUME)

# ---------- MASS ----------
if
    rho(x,y,z)
        := MASS_DENSITY
then

MASS
    = integral_over_V(
        rho dV
      )

VOLUME_INTEGRAL
    -> converts(MASS_DENSITY, TOTAL_MASS)

# ---------- CHARGE ----------
if
    rho_q(x,y,z)
        := VOLUME_CHARGE_DENSITY
then

TOTAL_CHARGE
    = integral_over_V(
        rho_q dV
      )

VOLUME_INTEGRAL
    -> converts(CHARGE_DENSITY, TOTAL_CHARGE)

# ---------- PROBABILITY ----------
if
    p(x,y,z)
        := JOINT_PROBABILITY_DENSITY
then

PROBABILITY_OF_REGION_V
    = integral_over_V(
        p dV
      )

NORMALISATION
    -> integral_over_support(p dV) = 1

# ---------- AVERAGE VALUE ----------
AVERAGE_VALUE_OVER_VOLUME
    := 1/VOLUME(V)
       * integral_over_V(
           f dV
         )

AVERAGE_VALUE
    -> normalised(VOLUME_ACCUMULATION)

# ---------- CENTRE OF MASS ----------
M
    := integral_over_V(rho dV)

x_bar
    := 1/M
       * integral_over_V(
           x*rho dV
         )

y_bar
    := 1/M
       * integral_over_V(
           y*rho dV
         )

z_bar
    := 1/M
       * integral_over_V(
           z*rho dV
         )

CENTRE_OF_MASS
    -> computed_from(WEIGHTED_VOLUME_INTEGRALS)

# ---------- MOMENTS OF INERTIA ----------
I_x
    := integral_over_V(
        (y^2 + z^2) * rho dV
    )

I_y
    := integral_over_V(
        (x^2 + z^2) * rho dV
    )

I_z
    := integral_over_V(
        (x^2 + y^2) * rho dV
    )

VOLUME_INTEGRAL
    -> supports(MOMENT_OF_INERTIA_CALCULATION)

# ---------- CHANGE OF VARIABLES ----------
TRANSFORMATION
    := T(u,v,w)
       = (x(u,v,w), y(u,v,w), z(u,v,w))

JACOBIAN_DETERMINANT
    := det(
        partial(x,y,z) /
        partial(u,v,w)
    )

dV
    = abs(JACOBIAN_DETERMINANT)
      du dv dw

CHANGE_OF_VARIABLES
    -> requires(VOLUME_SCALE_FACTOR)

# ---------- CYLINDRICAL COORDINATES ----------
x
    = r*cos(theta)

y
    = r*sin(theta)

z
    = z

dV
    = r dr dtheta dz

CYLINDRICAL_JACOBIAN
    = r

CYLINDRICAL_COORDINATES
    -> useful_for(AXIAL_SYMMETRY)

# ---------- SPHERICAL COORDINATES ----------
under common convention:

x
    = rho*sin(phi)*cos(theta)

y
    = rho*sin(phi)*sin(theta)

z
    = rho*cos(phi)

dV
    = rho^2*sin(phi)
      d_rho d_phi d_theta

SPHERICAL_JACOBIAN
    = rho^2*sin(phi)

SPHERICAL_COORDINATES
    -> useful_for(RADIAL_OR_SPHERICAL_SYMMETRY)

COORDINATE_CONVENTION
    -> must_be_preserved

# ---------- DIVERGENCE THEOREM RELATION ----------
if
    V has_boundary(S)
    AND S oriented_outward
    AND required_regularity_conditions_hold
then

integral_over_V(
    divergence(F) dV
)
    =
closed_surface_integral_S(
    F dot n_hat dS
)

DIVERGENCE_THEOREM
    -> relates(VOLUME_INTEGRAL, CLOSED_SURFACE_FLUX)

# ---------- FUBINI / TONELLI ----------
FUBINI_THEOREM
    -> allows(VOLUME_INTEGRAL_AS_ITERATED_INTEGRAL)
       under(INTEGRABILITY_CONDITIONS)

TONELLI_THEOREM
    -> supports(ITERATED_INTEGRATION_OF_NONNEGATIVE_MEASURABLE_FUNCTIONS)

ORDER_OF_INTEGRATION
    -> may_change
       only_with(CONSISTENT_REGION_BOUNDS)

# ---------- IMPROPER VOLUME INTEGRALS ----------
VOLUME_INTEGRAL
    -> may_be(IMPROPER)

IMPROPER_CASE
    -> may_have(UNBOUNDED_REGION)
    -> may_have(SINGULAR_OR_UNBOUNDED_INTEGRAND)

CONVERGENCE
    -> requires(LIMIT_OR_MEASURE_THEORETIC_ANALYSIS)

# ---------- PHYSICAL FIELDS ----------
VOLUME_INTEGRAL
    -> applied_to(MASS_DENSITY)
    -> applied_to(CHARGE_DENSITY)
    -> applied_to(ENERGY_DENSITY)
    -> applied_to(PROBABILITY_DENSITY)
    -> applied_to(MATERIAL_PROPERTIES)
    -> applied_to(CONTINUUM_MECHANICS)

# ---------- RELATION GRAPH ----------
THREE_DIMENSIONAL_REGION
    -> defines(INTEGRATION_DOMAIN)

SCALAR_FIELD
    -> accumulated_by(VOLUME_INTEGRAL)

VOLUME_ELEMENT
    -> encodes(LOCAL_VOLUME_SCALE)

CHANGE_OF_VARIABLES
    -> modifies(VOLUME_ELEMENT)

JACOBIAN_DETERMINANT
    -> determines(VOLUME_SCALE_FACTOR)

DENSITY
    -> integrated_to(TOTAL_QUANTITY)

DIVERGENCE_THEOREM
    -> connects(VOLUME_INTEGRAL, SURFACE_INTEGRAL)

# ---------- SEMANTIC EQUIVALENCE ----------
SEMANTIC_EQUIVALENCE := {
    "volume integral" ~= "integral of a scalar quantity throughout a three-dimensional region",
    "triple integral" ~= "standard coordinate representation of a volume integral in three dimensions",
    "volume element" ~= "infinitesimal three-dimensional measure element dV",
    "integral of density over volume" ~= "total amount of the density-defined quantity",
    "Cartesian volume element" ~= "dx dy dz",
    "Jacobian volume factor" ~= "absolute determinant scaling under coordinate transformation"
}

# ---------- CONSTRAINTS ----------
CONSTRAINTS := {
    volume_integral_requires_three_dimensional_domain: true,
    volume_element_depends_on_coordinate_system: true,
    coordinate_change_requires_absolute_jacobian_factor: true,
    cylindrical_coordinates_require_factor_r: true,
    spherical_coordinates_require_factor_rho_squared_sin_phi_under_stated_convention: true,
    changing_integration_order_requires_consistent_bounds: true,
    improper_volume_integrals_require_convergence_analysis: true
}

# ---------- CONFIDENCE ----------
CONFIDENCE := {
    core_definition: 0.99,
    triple_integral_relation: 0.99,
    density_accumulation_relation: 0.99,
    coordinate_change_relation: 0.99,
    cylindrical_spherical_relations: 0.99,
    divergence_theorem_relation: 0.99,
    semantic_equivalence: 0.97
}

# ---------- RECURSIVE REASONING ----------
RECURSIVE_REASONING(query):
    identify(scalar_field_or_density)
    -> identify(three_dimensional_region)
    -> determine(coordinate_system)
    -> construct(volume_element)
    -> derive(integration_bounds)
    -> test(integrability_or_convergence)
    -> apply(change_of_variables_if_useful)
    -> include(jacobian_factor_if_transformed)
    -> evaluate(iterated_integral)
    -> apply(divergence_theorem_if_advantageous_and_valid)
    -> validate(region_coordinate_and_convergence_conditions)
    -> interpret(total_quantity)
    -> return(semantic_result)

# ---------- AI CONSUMPTION ----------
AI_CONSUMPTION := {
    parse_as: "typed volume-integral semantic graph",
    primary_entity: VOLUME_INTEGRAL,
    preserve: [
        scalar_field,
        three_dimensional_region,
        coordinate_system,
        volume_element,
        integration_bounds,
        jacobian_factor,
        convergence_conditions
    ],
    infer_through: [
        triple_integrals,
        density_fields,
        coordinate_transformations,
        jacobians,
        cylindrical_coordinates,
        spherical_coordinates,
        divergence_theorem
    ],
    avoid: [
        omitting_coordinate_volume_factor,
        omitting_absolute_jacobian_in_general_transformation,
        treating_density_as_total_quantity_without integration,
        changing_order_without rebuilding bounds,
        applying_divergence_theorem_without closed boundary conditions,
        ignoring_improper_integral_convergence
    ]
}

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

# ---------- FLOW HASH ----------
FLOW_HASH := hash(
    SOURCE_URL
    + ENTITY
    + CONCEPTS
    + VOLUME_INTEGRAL
    + VOLUME_RIEMANN_SUM
    + VOLUME_ELEMENT
    + CHANGE_OF_VARIABLES
    + CYLINDRICAL_COORDINATES
    + SPHERICAL_COORDINATES
    + DIVERGENCE_THEOREM
    + SEMANTIC_EQUIVALENCE
    + CONSTRAINTS
)

# ---------- EXECUTION ----------
INPUT(query)
    -> VOLUME_INTEGRAL
    -> FIELD_IDENTIFICATION
    -> REGION_IDENTIFICATION
    -> COORDINATE_SYSTEM_ANALYSIS
    -> VOLUME_ELEMENT_CONSTRUCTION
    -> BOUND_CONSTRUCTION
    -> INTEGRABILITY_ANALYSIS
    -> CHANGE_OF_VARIABLES_ANALYSIS
    -> JACOBIAN_ANALYSIS
    -> ITERATED_INTEGRATION
    -> DIVERGENCE_THEOREM_ANALYSIS
    -> QUANTITY_INTERPRETATION
    -> RECURSIVE_REASONING(query)
    -> CONFIDENCE
    -> OUTPUT(semantic_result)

END FLOW