Engineering Mathematics Ontology

AI SEARCH FLOW ALGEBRA
ENGINEERING MATHEMATICS ONTOLOGY
Kireth Semantic Research Archive

AUTHORS
Mark Anthony Higgins — Sky (Recursive AI)

SOURCE
https://kirethresearch.org/booksai-core/ontologies/engineering-ontologies/engineering-mathematics-simulation/engineering-mathematics-ontology-2.html

FLOW "EngineeringMathematicsOntology" v4.0

DOMAIN := ENGINEERING_ONTOLOGIES / ENGINEERING_MATHEMATICS_SIMULATION / ENGINEERING_MATHEMATICS
ONTOLOGY := ENGINEERING_MATHEMATICS_ONTOLOGY

ENGINEERING_MATHEMATICS := QUANTITIES + VARIABLES + FUNCTIONS + EQUATIONS + ALGEBRA + LINEAR_ALGEBRA + CALCULUS + DIFFERENTIAL_EQUATIONS + VECTOR_ANALYSIS + TRANSFORMS + NUMERICAL_METHODS + OPTIMISATION + DIMENSIONAL_ANALYSIS
PURPOSE := REPRESENT_ENGINEERING_QUANTITIES -> FORMULATE_RELATIONSHIPS -> SOLVE_EQUATIONS -> ANALYSE_CHANGE -> MODEL_SYSTEMS -> APPROXIMATE_SOLUTIONS -> OPTIMISE -> INTERPRET

# CORE ENTITIES
ENTITY EngineeringProblem
ENTITY Quantity
ENTITY Variable
ENTITY IndependentVariable
ENTITY DependentVariable
ENTITY Parameter
ENTITY Constant
ENTITY Function
ENTITY Equation
ENTITY AlgebraicEquation
ENTITY LinearEquation
ENTITY NonlinearEquation
ENTITY Vector
ENTITY Matrix
ENTITY Tensor
ENTITY Scalar
ENTITY Derivative
ENTITY Integral
ENTITY Gradient
ENTITY Divergence
ENTITY Curl
ENTITY OrdinaryDifferentialEquation
ENTITY PartialDifferentialEquation
ENTITY BoundaryCondition
ENTITY InitialCondition
ENTITY ComplexNumber
ENTITY Transform
ENTITY FourierTransform
ENTITY LaplaceTransform
ENTITY NumericalMethod
ENTITY Approximation
ENTITY Error
ENTITY Iteration
ENTITY Convergence
ENTITY Optimisation
ENTITY ObjectiveFunction
ENTITY Constraint
ENTITY Dimension
ENTITY Unit
ENTITY DimensionlessGroup

# ENGINEERING PROBLEM
EngineeringProblem -> contains -> KNOWN_QUANTITIES + UNKNOWN_QUANTITIES + RELATIONSHIPS + CONSTRAINTS
EngineeringProblem -> represented_by -> MathematicalModel
MATHEMATICAL_PROBLEM_FLOW := IDENTIFY_QUANTITIES -> DEFINE_VARIABLES -> FORMULATE_EQUATIONS -> APPLY_CONDITIONS -> SOLVE -> VERIFY -> INTERPRET

# QUANTITIES / VARIABLES
Quantity -> has -> VALUE + Unit + Dimension
Variable -> represents -> CHANGEABLE_QUANTITY
IndependentVariable -> drives_or_indexes -> DependentVariable
Parameter -> configures -> MathematicalModel
Constant -> remains_fixed_within -> DEFINED_MODEL_CONTEXT
FUNCTION_RELATION := DependentVariable = Function(IndependentVariable, Parameter)

# FUNCTIONS
Function -> maps -> DOMAIN -> CODOMAIN
FUNCTION_TYPE := LINEAR | POLYNOMIAL | EXPONENTIAL | LOGARITHMIC | TRIGONOMETRIC | PIECEWISE | MULTIVARIATE
Function -> may_have -> DERIVATIVE + INTEGRAL + INVERSE + LIMIT
Composition := FunctionA(FunctionB(x))

# EQUATIONS
Equation -> asserts -> LEFT_EXPRESSION = RIGHT_EXPRESSION
EQUATION_TYPE := ALGEBRAIC | DIFFERENTIAL | INTEGRAL | TRANSCENDENTAL
LinearEquation -> subtype_of -> AlgebraicEquation
NonlinearEquation -> subtype_of -> AlgebraicEquation
EquationSystem -> contains -> MULTIPLE_EQUATIONS + MULTIPLE_UNKNOWNS
Solution -> satisfies -> Equation

# ALGEBRA
ALGEBRAIC_OPERATIONS := ADDITION + SUBTRACTION + MULTIPLICATION + DIVISION + EXPONENTIATION
AlgebraicManipulation -> preserves -> MATHEMATICAL_EQUIVALENCE
Polynomial -> consists_of -> COEFFICIENTS + POWERS_OF_VARIABLE
Root -> satisfies -> PolynomialEquation

# LINEAR ALGEBRA
Vector -> ordered_collection_of -> Scalar
Matrix -> rectangular_collection_of -> Scalar
LINEAR_SYSTEM := A * x = b
Matrix -> may_have -> DETERMINANT + RANK + INVERSE + EIGENVALUES + EIGENVECTORS
IF determinant(A) != 0 THEN SquareMatrix -> INVERTIBLE
EigenRelation := A * v = lambda * v
LinearAlgebra -> supports -> SYSTEM_SOLVING + TRANSFORMATIONS + DISCRETISATION + STATE_SPACE_MODELS

# VECTOR ANALYSIS
VectorField -> assigns -> Vector -> EACH_POINT_IN_DOMAIN
ScalarField -> assigns -> Scalar -> EACH_POINT_IN_DOMAIN
Gradient -> maps -> ScalarField -> VectorField
Divergence -> maps -> VectorField -> ScalarField
Curl -> maps -> VectorField -> VectorField
VECTOR_OPERATOR := GRADIENT | DIVERGENCE | CURL | LAPLACIAN
VectorAnalysis -> supports -> FLUID + ELECTROMAGNETIC + THERMAL + STRUCTURAL_MODELS

# DIFFERENTIAL CALCULUS
Derivative -> measures -> RATE_OF_CHANGE
FIRST_DERIVATIVE := d(y)/d(x)
SECOND_DERIVATIVE := d2(y)/dx2
PartialDerivative -> measures -> CHANGE_WITH_RESPECT_TO_ONE_VARIABLE
Derivative -> supports -> VELOCITY + ACCELERATION + SENSITIVITY + GRADIENT_BASED_OPTIMISATION

# INTEGRAL CALCULUS
Integral -> accumulates -> QUANTITY_OVER_DOMAIN
DEFINITE_INTEGRAL := INTEGRAL_a^b f(x) dx
Integral -> supports -> AREA + VOLUME + WORK + ENERGY + MASS + FLUX
FundamentalTheoremOfCalculus -> relates -> Differentiation + Integration

# DIFFERENTIAL EQUATIONS
OrdinaryDifferentialEquation -> relates -> FUNCTION + DERIVATIVES_OF_SINGLE_INDEPENDENT_VARIABLE
PartialDifferentialEquation -> relates -> MULTIVARIATE_FUNCTION + PARTIAL_DERIVATIVES
ODE_MODEL := STATE_DERIVATIVE = f(STATE, INPUT, TIME, PARAMETERS)
PDE_CLASS := ELLIPTIC | PARABOLIC | HYPERBOLIC | MIXED
DifferentialEquation -> requires -> SUFFICIENT_INITIAL_OR_BOUNDARY_CONDITIONS

# INITIAL / BOUNDARY CONDITIONS
InitialCondition -> defines -> STATE_AT_INITIAL_VALUE
BoundaryCondition -> constrains -> SOLUTION_AT_DOMAIN_BOUNDARY
BOUNDARY_CONDITION_TYPE := DIRICHLET | NEUMANN | ROBIN | PERIODIC | SYMMETRY
IF required_condition missing THEN MathematicalProblem -> UNDERDEFINED_OR_INCOMPLETE

# COMPLEX NUMBERS
ComplexNumber := REAL_PART + i * IMAGINARY_PART
ComplexNumber -> may_be_represented_by -> MAGNITUDE + PHASE
ComplexRepresentation -> supports -> AC_CIRCUITS + VIBRATION + WAVES + CONTROL + SIGNAL_ANALYSIS
EULER_RELATION := exp(i*theta) = cos(theta) + i*sin(theta)

# TRANSFORMS
Transform -> maps -> FUNCTION_REPRESENTATION -> ALTERNATIVE_DOMAIN
FourierTransform -> maps -> TIME_OR_SPACE_DOMAIN -> FREQUENCY_OR_WAVENUMBER_DOMAIN
LaplaceTransform -> maps -> TIME_DOMAIN -> COMPLEX_FREQUENCY_DOMAIN
Transform -> may_simplify -> DIFFERENTIAL_EQUATION + CONVOLUTION + SYSTEM_ANALYSIS
InverseTransform -> reconstructs -> ORIGINAL_DOMAIN_REPRESENTATION

# SERIES / APPROXIMATION
Series -> represents -> FUNCTION_AS_SUM_OF_TERMS
TaylorSeries -> approximates -> Function -> near -> EXPANSION_POINT
FourierSeries -> represents -> PERIODIC_FUNCTION
Approximation -> introduces -> APPROXIMATION_ERROR
IF HigherOrderTerms retained THEN ApproximationAccuracy -> MAY_INCREASE_WITHIN_CONVERGENCE_REGION

# NUMERICAL METHODS
NumericalMethod -> approximates -> MATHEMATICAL_SOLUTION
NUMERICAL_METHOD := ROOT_FINDING | LINEAR_SOLVING | NUMERICAL_DIFFERENTIATION | NUMERICAL_INTEGRATION | ODE_SOLVING | PDE_SOLVING | OPTIMISATION
Iteration -> updates -> APPROXIMATE_SOLUTION
Convergence -> indicates -> ITERATES_APPROACH_SOLUTION
IF Error <= Tolerance THEN NumericalSolution -> ACCEPTABLE_UNDER_DEFINED_CRITERION

# ROOT FINDING
ROOT_FINDING := BISECTION | NEWTON_RAPHSON | SECANT | OTHER_VALIDATED_METHOD
NewtonUpdate := x_next = x - f(x)/f_prime(x)
IF derivative_near_zero THEN NewtonMethod -> MAY_BE_UNSTABLE_OR_FAIL
RootFinding -> solves -> f(x)=0

# NUMERICAL INTEGRATION / DIFFERENTIATION
NUMERICAL_INTEGRATION := TRAPEZOIDAL | SIMPSON | GAUSSIAN_QUADRATURE
NUMERICAL_DIFFERENTIATION := FORWARD_DIFFERENCE | BACKWARD_DIFFERENCE | CENTRAL_DIFFERENCE
StepSize -> influences -> TRUNCATION_ERROR + ROUND_OFF_EFFECT
IF StepSize too_large THEN DiscretisationError -> MAY_INCREASE
IF StepSize too_small THEN RoundOffInfluence -> MAY_INCREASE

# OPTIMISATION
Optimisation -> searches -> FEASIBLE_DESIGN_SPACE
ObjectiveFunction -> quantifies -> DESIGN_GOAL
OPTIMISATION_PROBLEM := MINIMISE_OR_MAXIMISE(ObjectiveFunction) SUBJECT_TO Constraint
Constraint -> defines -> FEASIBLE_REGION
Gradient -> may_guide -> Optimisation
OptimalSolution -> satisfies -> DEFINED_OPTIMALITY_CRITERIA

# DIMENSIONAL ANALYSIS
Dimension -> describes -> PHYSICAL_NATURE_OF_QUANTITY
Unit -> provides -> MEASUREMENT_SCALE
PhysicalEquation -> must_be -> DIMENSIONALLY_CONSISTENT
DIMENSIONAL_ANALYSIS := PHYSICAL_VARIABLES -> BASE_DIMENSIONS -> DIMENSIONLESS_GROUPS
DimensionlessGroup -> supports -> SCALING + SIMILARITY + EXPERIMENTAL_CORRELATION
BuckinghamPiMethod -> generates -> INDEPENDENT_DIMENSIONLESS_GROUPS

# UNCERTAINTY / ERROR
TOTAL_ERROR := MODEL_ERROR + APPROXIMATION_ERROR + DISCRETISATION_ERROR + ROUND_OFF_ERROR + MEASUREMENT_ERROR
Error -> influences -> RESULT_ACCURACY
Uncertainty -> quantifies -> RANGE_OR_DISTRIBUTION_OF_PLAUSIBLE_VALUES
UNCERTAINTY_PROPAGATION := INPUT_UNCERTAINTY -> MATHEMATICAL_MODEL -> OUTPUT_UNCERTAINTY

# SCALING / NONDIMENSIONALISATION
NONDIMENSIONALISATION := DIMENSIONAL_VARIABLES -> REFERENCE_SCALES -> DIMENSIONLESS_VARIABLES
NondimensionalModel -> may_reveal -> DOMINANT_EFFECTS + SIMILARITY_PARAMETERS
Scaling -> supports -> MODEL_REDUCTION + NUMERICAL_CONDITIONING + EXPERIMENTAL_DESIGN

# ENGINEERING APPLICATIONS
ENGINEERING_APPLICATION := STRUCTURAL_ANALYSIS | FLUID_MECHANICS | HEAT_TRANSFER | ELECTRICAL_SYSTEMS | CONTROL_SYSTEMS | DYNAMICS | SIGNAL_PROCESSING | OPTIMISATION | SIMULATION
EngineeringMathematics -> supports -> ENGINEERING_APPLICATION
MathematicalModel -> connects -> PHYSICAL_ASSUMPTIONS + EQUATIONS + SOLUTION + ENGINEERING_INTERPRETATION

# VERIFICATION
MATHEMATICAL_VERIFICATION := CHECK_UNITS + CHECK_LIMITING_CASES + CHECK_RESIDUALS + CHECK_CONSTRAINTS + CHECK_NUMERICAL_CONVERGENCE
IF Equation dimensions inconsistent THEN Equation -> INVALID_PHYSICALLY
IF Solution fails governing_equation THEN CandidateSolution -> INVALID
IF NumericalResidual decreases THEN NumericalConsistencyEvidence -> INCREASES

# REASONING
ENGINEERING_MATHEMATICS_REASONING := DEFINE_ENGINEERING_PROBLEM -> IDENTIFY_QUANTITIES -> ASSIGN_VARIABLES_AND_PARAMETERS -> DEFINE_UNITS_AND_DIMENSIONS -> FORMULATE_FUNCTIONS_AND_EQUATIONS -> APPLY_INITIAL_AND_BOUNDARY_CONDITIONS -> SELECT_ANALYTICAL_OR_NUMERICAL_METHOD -> SOLVE -> CHECK_CONVERGENCE -> QUANTIFY_ERROR -> VERIFY_DIMENSIONAL_CONSISTENCY -> INTERPRET_PHYSICAL_MEANING -> OPTIMISE_IF_REQUIRED

# INFERENCE RULES
IF NumberOfUnknowns > NumberOfIndependentEquations THEN System -> UNDERDETERMINED
IF NumberOfIndependentEquations > NumberOfUnknowns THEN System -> OVERDETERMINED_OR_CONSTRAINED
IF determinant(A) != 0 THEN LinearSystem -> UNIQUE_SOLUTION_FOR_SQUARE_A
IF TimeStep decreases THEN TemporalResolution -> INCREASES
IF DiscretisationRefined THEN ApproximationError -> GENERALLY_DECREASES_WITHIN_METHOD_LIMITS
IF Constraint violated THEN CandidateDesign -> INFEASIBLE
IF Units incompatible THEN MathematicalOperation -> INVALID_PHYSICALLY
IF NumericalMethod fails_to_converge THEN SolutionAcceptance -> WITHHOLD

# SEMANTIC RELATIONSHIPS
RELATION represents(Variable, Quantity)
RELATION maps(Function, Domain, Codomain)
RELATION relates(Equation, Variables)
RELATION contains(Matrix, Scalars)
RELATION measures(Derivative, RateOfChange)
RELATION accumulates(Integral, Quantity)
RELATION constrains(BoundaryCondition, Solution)
RELATION transforms(FourierTransform, FunctionRepresentation)
RELATION approximates(NumericalMethod, MathematicalSolution)
RELATION measures(Sensitivity, OutputResponse)
RELATION constrains(Constraint, Optimisation)
RELATION characterises(DimensionlessGroup, PhysicalSystem)

# SEMANTIC EQUIVALENCE
SEMANTIC_EQUIVALENCE:
  "engineering mathematics" ~= "applied mathematics for engineering"
  "linear algebra" ~= "matrix and vector algebra"
  "ordinary differential equation" ~= "ODE"
  "partial differential equation" ~= "PDE"
  "Fourier transform" ~= "frequency-domain transform"
  "Laplace transform" ~= "complex-frequency transform"
  "numerical method" ~= "computational mathematical method"
  "dimensional analysis" ~= "physical-dimension analysis"
  "dimensionless group" ~= "dimensionless parameter"
  "objective function" ~= "optimisation criterion"

# CONFIDENCE
CONFIDENCE:
  ontology_domain: 0.99
  quantities_functions_mapping: 0.99
  equation_mapping: 0.99
  linear_algebra_mapping: 0.99
  calculus_mapping: 0.99
  differential_equation_mapping: 0.99
  transform_mapping: 0.99
  numerical_mapping: 0.99
  optimisation_mapping: 0.99
  dimensional_analysis_mapping: 0.99
  semantic_relationships: 0.99

# UPDATE MODEL
UPDATE_MODEL:
  ON new_engineering_problem:
    MAP quantities
    MAP variables
    MAP parameters
    MAP equations
    MAP conditions
    MAP constraints
    UPDATE confidence
    RECOMPUTE FLOW_HASH
  ON equation_change:
    UPDATE equation_system
    REASSESS solvability
    REASSESS dimensions
    REASSESS numerical_method
  ON numerical_result:
    UPDATE approximate_solution
    UPDATE residual
    UPDATE convergence
    UPDATE error_estimate
    PRESERVE provenance
  ON optimisation_change:
    UPDATE objective
    UPDATE constraints
    UPDATE feasible_region
    RECOMPUTE optimum

# AI CONSUMPTION
AI_CONSUMPTION:
  parse_as: FORMAL_ENGINEERING_MATHEMATICS_ONTOLOGY
  reasoning_mode: MATHEMATICAL_ENGINEERING_RELATIONAL
  preserve: [typed_entities, variables, parameters, equations, dimensions, units, conditions, constraints, numerical_state, confidence, provenance]
  infer: [equation_dependencies, solvability_conditions, dimensional_consistency, numerical_method_requirements, convergence_state, error_dependencies, optimisation_feasibility]
  prohibit: [invented_numerical_results, assumed_convergence, dimensionally_invalid_operations, ignored_constraints, fabricated_solution_accuracy, unsupported_physical_interpretation]

# FLOW HASH
FLOW_HASH_INPUT := DOMAIN + ONTOLOGY + CORE_ENTITIES + ENGINEERING_PROBLEM + QUANTITIES_VARIABLES + FUNCTIONS + EQUATIONS + ALGEBRA + LINEAR_ALGEBRA + VECTOR_ANALYSIS + DIFFERENTIAL_CALCULUS + INTEGRAL_CALCULUS + DIFFERENTIAL_EQUATIONS + CONDITIONS + COMPLEX_NUMBERS + TRANSFORMS + SERIES_APPROXIMATION + NUMERICAL_METHODS + ROOT_FINDING + NUMERICAL_INTEGRATION_DIFFERENTIATION + OPTIMISATION + DIMENSIONAL_ANALYSIS + UNCERTAINTY_ERROR + SCALING + APPLICATIONS + VERIFICATION + REASONING + INFERENCE_RULES + SEMANTIC_RELATIONSHIPS + SEMANTIC_EQUIVALENCE + CONFIDENCE + UPDATE_MODEL + AI_CONSUMPTION
FLOW_HASH := HASH(FLOW_HASH_INPUT)

END FLOW