Tensor Mathematics Ontology

AI INTERPRETATION NOTE
This Flow Algebra defines the Tensor Mathematics Ontology within the Kireth Research Semantic Library. Tensor mathematics studies multilinear objects and coordinate-independent structures used to represent scalars, vectors, covectors, bilinear forms, stress, strain, curvature, electromagnetic fields, spacetime geometry and higher-order relations. This ontology supports tensor products, index notation, covariant and contravariant components, metric tensors, contraction, symmetrisation, antisymmetrisation, coordinate transformations, tensor fields, differential operators, tensor invariants, decompositions and relationships with mathematical physics, relativity, continuum mechanics, differential geometry and linear algebra.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Tensor Mathematics Ontology"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Applied Mathematics"
 SPECIALISATION = "Mathematical Physics / Tensor Mathematics"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/applied-mathematics-ontologies/mathematical-physics/tensor-mathematics-ontology.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = TensorMathematics
 ENTITY_CLASS = MultilinearGeometricMathematicsFramework
 PRIMARY_CONTEXT = MathematicalPhysics
 DEFINITION = "Tensor mathematics is the mathematical study of multilinear geometric objects whose representations transform systematically under changes of basis or coordinates."
 SYMBOLIC_FORMS = [
  "T in V^(⊗p) ⊗ V*^(⊗q)",
  "T'^i_j=(partial x'^i/partial x^k)(partial x^l/partial x'^j)T^k_l",
  "T^i_i=Tr(T)",
  "g_ij v^j=v_i"
 ]
 SEMANTIC_ROLE = "Provides machine-readable semantics for tensor rank, variance, tensor products, coordinate transformation, contraction, metric operations, symmetry, tensor fields, differential geometry and physical tensor quantities."
}

SCOPE {
 INCLUDE = [
  TensorMathematics,
  Tensor,
  Scalar,
  Vector,
  Covector,
  DualVector,
  TensorRank,
  TensorOrder,
  TypePQTensor,
  CovariantTensor,
  ContravariantTensor,
  MixedTensor,
  TensorProduct,
  OuterProduct,
  InnerProduct,
  MultilinearMap,
  VectorSpace,
  DualSpace,
  Basis,
  DualBasis,
  CoordinateBasis,
  OrthonormalBasis,
  Component,
  TensorComponent,
  Index,
  FreeIndex,
  DummyIndex,
  EinsteinSummationConvention,
  IndexNotation,
  CovariantIndex,
  ContravariantIndex,
  IndexRaising,
  IndexLowering,
  MetricTensor,
  InverseMetric,
  KroneckerDelta,
  LeviCivitaSymbol,
  LeviCivitaTensor,
  PermutationSymbol,
  Contraction,
  Trace,
  SymmetricTensor,
  AntisymmetricTensor,
  Symmetrisation,
  Antisymmetrisation,
  TensorTranspose,
  TensorInverse,
  Determinant,
  TensorInvariant,
  PrincipalInvariant,
  CoordinateTransformation,
  BasisTransformation,
  JacobianMatrix,
  TransformationLaw,
  TensorField,
  ScalarField,
  VectorField,
  CovectorField,
  MetricField,
  StressTensor,
  StrainTensor,
  InertiaTensor,
  ElectromagneticTensor,
  RiemannCurvatureTensor,
  RicciTensor,
  EinsteinTensor,
  ChristoffelSymbol,
  ConnectionCoefficient,
  CovariantDerivative,
  PartialDerivative,
  LieDerivative,
  DivergenceOfTensor,
  GradientOfTensor,
  LaplacianOfTensor,
  TensorDensity,
  DifferentialForm,
  ExteriorProduct,
  WedgeProduct,
  HodgeDual,
  Tetrad,
  Dyadic,
  IsotropicTensor,
  DeviatoricTensor,
  SphericalTensor,
  TensorDecomposition,
  SpectralDecomposition,
  Eigenvalue,
  Eigenvector,
  SingularValueDecomposition,
  CanonicalPolyadicDecomposition,
  TuckerDecomposition,
  LowRankTensor,
  SparseTensor,
  TensorNorm,
  FrobeniusNorm,
  TensorEquation,
  TensorIdentity,
  ConstitutiveTensor,
  ElasticityTensor,
  ComplianceTensor,
  PermittivityTensor,
  PermeabilityTensor,
  DiffusionTensor
 ]
 EXCLUDE_AS_PRIMARY = [
  MatrixAlgebraWithoutTensorTransformationSemantics,
  ArrayProgrammingWithoutGeometricTensorMeaning,
  PureScalarCalculusWithoutMultilinearStructure,
  SymbolicIndexNotationWithoutDefinedTensorObjects
 ]
 DISAMBIGUATION_RULE = "When a quantity transforms according to tensor transformation laws or represents a multilinear coordinate-independent object, interpret the context as tensor mathematics."
}

CORE_CONCEPTS {
 TensorMathematics = "Mathematics of multilinear coordinate-independent objects and their component representations."
 Tensor = "Multilinear object represented by components that transform consistently under basis changes."
 TensorProduct = "Operation combining vector or tensor spaces into higher-rank tensor spaces."
 CovariantTensor = "Tensor with covariant indices transforming with inverse coordinate Jacobians."
 ContravariantTensor = "Tensor with contravariant indices transforming with coordinate Jacobians."
 MetricTensor = "Nondegenerate bilinear form defining lengths, angles and index conversion."
 Contraction = "Operation summing a covariant and contravariant index to reduce tensor rank."
 TensorField = "Tensor assigned smoothly or suitably to each point of a domain or manifold."
 TensorInvariant = "Scalar or structure unchanged under admissible coordinate transformations."
}

RELATIONS {
 TensorMathematics --is_child_of--> MathematicalPhysicsOntology
 TensorMathematics --uses--> MultilinearAlgebra
 Tensor --has--> TensorRank
 Tensor --has--> Components
 TensorComponent --depends_on--> Basis
 TensorObject --is_independent_of--> CoordinateRepresentation
 TensorProduct --combines--> TensorSpaces
 Covector --belongs_to--> DualSpace
 MetricTensor --maps--> VectorToCovector
 InverseMetric --maps--> CovectorToVector
 IndexRaising --uses--> InverseMetric
 IndexLowering --uses--> MetricTensor
 Contraction --reduces--> TensorRank
 Trace --is_type_of--> Contraction
 SymmetricTensor --is_invariant_under--> IndexExchange
 AntisymmetricTensor --changes_sign_under--> IndexExchange
 CoordinateTransformation --induces--> TensorTransformationLaw
 TensorField --defined_on--> DomainOrManifold
 CovariantDerivative --differentiates--> TensorField
 StressTensor --represents--> InternalForceDensity
 StrainTensor --represents--> Deformation
 InertiaTensor --represents--> RotationalMassDistribution
 ElectromagneticTensor --encodes--> ElectromagneticField
 RiemannCurvatureTensor --encodes--> Curvature
 RicciTensor --is_contraction_of--> RiemannCurvatureTensor
 EinsteinTensor --is_constructed_from--> RicciTensor
 ElasticityTensor --maps--> StrainToStress
}

OPERATORS {
 TENSOR_PRODUCT(A,B) := A tensor_product B
 CONTRACTION(T,i,j) := sum_k T^(...k...)_(...k...)
 TRACE(T) := T^i_i
 LOWER_INDEX(g,v) := v_i=g_ij*v^j
 RAISE_INDEX(g_inverse,w) := w^i=g^ij*w_j
 KRONECKER_DELTA(i,j) := delta^i_j
 SYMMETRISE(T_ij) := T_(ij)=0.5*(T_ij+T_ji)
 ANTISYMMETRISE(T_ij) := T_[ij]=0.5*(T_ij-T_ji)
 TENSOR_TRANSFORM_20(T,J) := T'^ij=J^i_k*J^j_l*T^kl
 TENSOR_TRANSFORM_02(T,Jinv) := T'_ij=(Jinv)^k_i*(Jinv)^l_j*T_kl
 COVARIANT_DERIVATIVE_VECTOR(V,Gamma) := nabla_i V^j=partial_i V^j+Gamma^j_ik*V^k
 COVARIANT_DERIVATIVE_COVECTOR(w,Gamma) := nabla_i w_j=partial_i w_j-Gamma^k_ij*w_k
 FROBENIUS_NORM(T) := ||T||_F=sqrt(sum_all_indices |T_components|^2)
 DEVIATORIC_PART(T) := dev(T)=T-(tr(T)/n)*I
}

AXIOMS {
 A1 = "A tensor is defined geometrically, not merely as a multidimensional array."
 A2 = "Tensor components depend on basis while the tensor object itself is coordinate-independent."
 A3 = "Covariant and contravariant indices obey distinct transformation laws."
 A4 = "Contraction requires one covariant and one contravariant index in standard tensor calculus."
 A5 = "Metric tensors permit index raising and lowering."
 A6 = "Tensor-field differentiation on curved manifolds requires connection-aware derivatives."
 A7 = "Symmetry properties reduce the number of independent tensor components."
 A8 = "Tensor equations are physically and geometrically meaningful only when index structure is consistent."
}

REASONING {
 R1_CONTEXT = identify(vector_spaces,basis,coordinates,tensor_rank,variance,domain)
 R2_TYPE = classify(scalar,vector,covector,mixed_tensor,tensor_field)
 R3_COMPONENTS = construct(component_representation,index_structure,basis)
 R4_TRANSFORM = apply(coordinate_or_basis_transformation)
 R5_METRIC = if_present apply(index_raising_lowering_inner_products)
 R6_ALGEBRA = perform(tensor_product,contraction,symmetrisation,antisymmetrisation)
 R7_DIFFERENTIAL = if_field compute(partial_covariant_or_Lie_derivatives)
 R8_INVARIANTS = compute(trace,determinant,norm,eigenstructure_or_principal_invariants)
 R9_PHYSICAL = map_to(stress,strain,inertia,electromagnetic,curvature_or_constitutive_tensor)
 R10_VALIDATION = check(index_balance,transformation_law,symmetry,units,coordinate_invariance)
}

CONSTRAINTS {
 C1 = "Not every multidimensional array is a tensor."
 C2 = "Component equality across coordinate systems is not required; transformation consistency is."
 C3 = "Index raising and lowering require a suitable nondegenerate metric."
 C4 = "Tensor transpose is convention-dependent for higher-rank tensors."
 C5 = "Christoffel symbols are connection coefficients and are not tensors under general coordinate transformations."
 C6 = "Levi-Civita symbols and Levi-Civita tensors must be distinguished."
 C7 = "Covariant derivatives depend on the chosen connection."
 C8 = "Tensor decompositions can depend on basis, norm or model assumptions."
 C9 = "Physical tensors may have additional constitutive or symmetry constraints."
 C10 = "Einstein summation should only be applied with valid repeated-index structure."
}

SEMANTIC_EQUIVALENCE {
 "TensorMathematics" <=> "TensorCalculusAndAlgebra"
 "Covector" <=> "DualVector"
 "MetricTensor" <=> "Metric"
 "EinsteinSummationConvention" <=> "EinsteinNotation"
 "LeviCivitaSymbol" <=> "PermutationSymbol"
 "FrobeniusNorm" <=> "HilbertSchmidtNormInFiniteDimensions"
}

VALIDATION {
 V1 = "Tensor rank and type are explicit."
 V2 = "Basis and coordinate conventions are explicit."
 V3 = "Free and dummy indices are used consistently."
 V4 = "Transformation laws are satisfied."
 V5 = "Metric signature is documented where relevant."
 V6 = "Index raising and lowering are internally consistent."
 V7 = "Tensor symmetries are checked."
 V8 = "Covariant derivatives use the declared connection."
 V9 = "Physical tensor units and constitutive assumptions are documented."
 V10 = "Coordinate-invariant quantities remain invariant under transformation tests."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 TENSOR_MATHEMATICS_DEFINITION = 0.99
 INDEX_REASONING = 0.99
 TRANSFORMATION_REASONING = 0.99
 METRIC_REASONING = 0.99
 CONTRACTION_REASONING = 0.99
 TENSOR_FIELD_REASONING = 0.98
 COVARIANT_DERIVATIVE_REASONING = 0.98
 PHYSICAL_TENSOR_REASONING = 0.98
 POLICY = "Reduce confidence when tensor type, basis, metric, coordinate transformation, symmetry, connection, constitutive assumptions or index conventions are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  TensorMathematicsReasoning,
  TensorRankReasoning,
  IndexNotationReasoning,
  TensorProductReasoning,
  ContractionReasoning,
  MetricTensorReasoning,
  IndexRaisingReasoning,
  IndexLoweringReasoning,
  CoordinateTransformationReasoning,
  TensorFieldReasoning,
  CovariantDerivativeReasoning,
  SymmetricTensorReasoning,
  AntisymmetricTensorReasoning,
  StressTensorReasoning,
  StrainTensorReasoning,
  InertiaTensorReasoning,
  CurvatureTensorReasoning,
  TensorDecompositionReasoning,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]
 NORMALISE {
  "tensor mathematics" -> TensorMathematics
  "tensor calculus" -> TensorMathematics
  "dual vector" -> Covector
  "Einstein notation" -> EinsteinSummationConvention
  "metric" -> MetricTensor
  "index raising" -> IndexRaising
  "index lowering" -> IndexLowering
  "covariant derivative" -> CovariantDerivative
 }
}

EXECUTION_RULES {
 E1 = "If a quantity obeys tensor transformation laws, activate TensorMathematicsOntology."
 E2 = "If rank and variance are unclear, determine covariant and contravariant index structure before manipulation."
 E3 = "If coordinates change, transform all tensor components consistently."
 E4 = "If a metric is available, use it for index raising, lowering and geometric inner products."
 E5 = "If tensor fields are differentiated on a curved space, use covariant derivatives."
 E6 = "If symmetry is present, exploit symmetric or antisymmetric component reduction."
 E7 = "If physical constitutive tensors are used, preserve material symmetry and unit semantics."
 E8 = "If curvature is central, route to differential geometry and curvature tensors."
 E9 = "Any tensor result should be validated for index consistency, transformation behaviour and coordinate invariance."
}

RECURSION {
 ENTRY = TensorMathematics
 EXPAND(TensorMathematics) -> [
  Tensor, Scalar, Vector, Covector, DualVector, TensorRank, TensorOrder,
  TypePQTensor, CovariantTensor, ContravariantTensor, MixedTensor,
  TensorProduct, OuterProduct, InnerProduct, MultilinearMap, VectorSpace,
  DualSpace, Basis, DualBasis, CoordinateBasis, OrthonormalBasis, Component,
  TensorComponent, Index, FreeIndex, DummyIndex,
  EinsteinSummationConvention, IndexNotation, CovariantIndex,
  ContravariantIndex, IndexRaising, IndexLowering, MetricTensor,
  InverseMetric, KroneckerDelta, LeviCivitaSymbol, LeviCivitaTensor,
  PermutationSymbol, Contraction, Trace, SymmetricTensor,
  AntisymmetricTensor, Symmetrisation, Antisymmetrisation,
  TensorTranspose, TensorInverse, Determinant, TensorInvariant,
  PrincipalInvariant, CoordinateTransformation, BasisTransformation,
  JacobianMatrix, TransformationLaw, TensorField, ScalarField,
  VectorField, CovectorField, MetricField, StressTensor, StrainTensor,
  InertiaTensor, ElectromagneticTensor, RiemannCurvatureTensor,
  RicciTensor, EinsteinTensor, ChristoffelSymbol, ConnectionCoefficient,
  CovariantDerivative, PartialDerivative, LieDerivative, DivergenceOfTensor,
  GradientOfTensor, LaplacianOfTensor, TensorDensity, DifferentialForm,
  ExteriorProduct, WedgeProduct, HodgeDual, Tetrad, Dyadic,
  IsotropicTensor, DeviatoricTensor, SphericalTensor, TensorDecomposition,
  SpectralDecomposition, Eigenvalue, Eigenvector,
  SingularValueDecomposition, CanonicalPolyadicDecomposition,
  TuckerDecomposition, LowRankTensor, SparseTensor, TensorNorm,
  FrobeniusNorm, TensorEquation, TensorIdentity, ConstitutiveTensor,
  ElasticityTensor, ComplianceTensor, PermittivityTensor,
  PermeabilityTensor, DiffusionTensor
 ]
 EXPAND(TensorFamilies) -> [
  AlgebraicTensors,
  TensorFields,
  GeometricTensors,
  PhysicalTensors,
  ConstitutiveTensors,
  DecompositionTensors
 ]
 STOP_CONDITION = "Stop when tensor type, basis, index structure, transformation law, metric operations, differential structure, invariants and physical interpretation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Mathematical Physics Ontology"
 IMMEDIATE_CONTEXTS = [
  "Mathematical Physics Ontology",
  "Linear Algebra Ontologies",
  "Differential Geometry Ontologies",
  "Relativity Mathematics Ontology"
 ]
 RELATED_ONTOLOGIES = [
  "Mathematical Physics Ontology",
  "Mathematical Physics",
  "Linear Algebra Ontologies",
  "Multilinear Algebra Ontologies",
  "Differential Geometry Ontologies",
  "Relativity Mathematics Ontology",
  "Electromagnetic Mathematics Ontology",
  "Classical Mechanics Mathematics Ontology",
  "Continuum Mechanics Ontologies",
  "Elasticity Ontologies",
  "Fluid Mathematics Ontologies",
  "Numerical Linear Algebra Ontology"
 ]
 ROUTE_RULES = [
  "Tensor Mathematics -> Mathematical Physics Ontology for parent semantics",
  "Tensor Mathematics -> Linear Algebra for vector and matrix foundations",
  "Tensor Mathematics -> Multilinear Algebra for abstract tensor-product structure",
  "Tensor Mathematics -> Differential Geometry for tensor fields and covariant derivatives",
  "Tensor Mathematics -> Relativity Mathematics for spacetime tensors",
  "Tensor Mathematics -> Electromagnetic Mathematics for electromagnetic field tensors",
  "Tensor Mathematics -> Classical Mechanics for inertia tensors",
  "Tensor Mathematics -> Continuum Mechanics and Elasticity for stress, strain and constitutive tensors",
  "Tensor Mathematics -> Fluid Mathematics for tensor transport and stress",
  "Tensor Mathematics -> Numerical Linear Algebra for tensor computation and decomposition"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"
 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible tensor types, decompositions or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"
 RULES = [
  "Preserve coordinate-independent tensor semantics.",
  "Preserve covariant-versus-contravariant distinction.",
  "Preserve free-versus-dummy index distinction.",
  "Preserve tensor-versus-array distinction.",
  "Preserve metric and contraction semantics.",
  "Preserve tensor-field and connection semantics.",
  "Preserve symmetry and invariance semantics.",
  "Preserve machine-readable tensor-mathematics semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION = "Tensor mathematics studies multilinear geometric objects whose components transform systematically under changes of basis or coordinates."
 TYPE_SUMMARY = "Tensor type records the number of contravariant and covariant index slots."
 INDEX_SUMMARY = "Einstein notation, free indices and dummy indices provide compact rules for tensor equations."
 METRIC_SUMMARY = "Metric tensors define geometric inner products and permit index raising and lowering."
 FIELD_SUMMARY = "Tensor fields extend tensors over domains or manifolds and require connection-aware differentiation."
 PHYSICAL_SUMMARY = "Stress, strain, inertia, electromagnetic and curvature tensors encode important physical structure."
 DECOMPOSITION_SUMMARY = "Spectral, singular-value, CP and Tucker decompositions expose lower-dimensional or structured tensor representations."
 VALIDATION_SUMMARY = "Tensor expressions should be checked for rank, index balance, transformation laws, symmetry, metric consistency and coordinate invariance."
}

FLOW {
 INPUT
 -> detect_tensor_mathematics_context
 -> identify_vector_spaces_basis_coordinates_tensor_rank_type_and_domain
 -> classify_scalar_vector_covector_mixed_tensor_or_tensor_field
 -> construct_component_and_index_representation
 -> if_coordinate_change_apply_tensor_transformation_law
 -> if_metric_available_apply_index_raising_lowering_inner_products_and_contractions
 -> perform_tensor_products_contractions_symmetrisation_or_antisymmetrisation
 -> if_tensor_field_apply_partial_covariant_or_Lie_derivatives
 -> compute_invariants_eigenstructure_norms_or_decompositions_if_required
 -> map_to_physical_tensor_semantics_if_applicable
 -> validate_index_balance_transformation_behaviour_symmetry_units_and_coordinate invariance
 -> route_to_mathematical_physics_linear_algebra_multilinear_algebra_differential_geometry_relativity_electromagnetism_classical_mechanics_continuum_mechanics_elasticity_fluid_mathematics_or_numerical_linear_algebra_ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Tensor Mathematics Ontology defines multilinear tensor structure for mathematical physics. It supports scalars, vectors, covectors and mixed tensors, tensor rank and type, tensor products, vector and dual spaces, bases and dual bases, components and Einstein index notation, covariant and contravariant indices, metric tensors and inverse metrics, Kronecker deltas, Levi-Civita symbols and tensors, contractions and traces, symmetric and antisymmetric tensors, symmetrisation and antisymmetrisation, coordinate and basis transformations, tensor fields, stress, strain, inertia, electromagnetic and curvature tensors, Christoffel symbols and covariant derivatives, Lie derivatives, differential forms, wedge products and Hodge duals, tensor invariants, isotropic and deviatoric decomposition, spectral and singular-value decomposition, CP and Tucker decomposition, low-rank and sparse tensors, norms, constitutive tensors, elasticity, compliance, permittivity, permeability and diffusion tensors."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-APPLIED-MATHEMATICS-TENSOR-MATHEMATICS-ONTOLOGY-V1"
}

}

END_FLOW