Alphabeta MathFormal parameterized fluid systems and observables
About this page
Pipeline-generated
How statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
1 · Prerequisites
- Absolute and Conditional Convergence; Rearrangement; Products
- Arc Length and Rectifiable Curves
- Binary Operations, Monoids, Groups and Subgroups
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Complex Coordinate and Path Transport
- Complex Differentiability and the Cauchy–Riemann Equations
- Complex Power Series and Analytic Functions
- Connectedness
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Determinants of Matrices over a Commutative Ring
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Filters and Ultrafilters
- Finite Counting, Factorials and Binomial Coefficients
- Foundations of the Real Numbers for Analysis
- Fundamental Trigonometric Identities
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Line Integrals and the Gradient Theorem
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Metric Spaces
- Mixed Partials, Taylor Formulae, and Extrema
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Normal Subgroups and Quotient Groups
- Order, Zorn's Lemma, and the Axiom of Choice
- Polynomial Rings, the Division Algorithm and Roots
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Regular Surfaces and Surface Integrals
- Relations, Functions, and Quotients
- Rings, Subrings, Integral Domains and Fields
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Simple Field Extensions and the Construction of the Complex Numbers
- Sine, Cosine, and the Definition of Pi
- Suprema and Infima
- Symmetric Groups, Cycle Decomposition and the Sign Homomorphism
- The Complex Exponential and Euler's Formula
- The Derivative and the Mean Value Theorems
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- The Divergence Theorem and Classical Stokes
- The Exponential Function
- The Inverse and Implicit Function Theorems
- The Inverse Function Theorem Completed
- The Logarithm and General Powers
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- The ZFC Axioms and the Basic Set Constructions
- Topological Spaces and Continuity
- Topology of ℝ
- Vector Spaces, Linear Subspaces, Span and Direct Sums
2 · Logical flowchart
3 · Items
Definition
Formal parameterized fluid systems and observables
Definition
Real sine and cosine as real restrictions of complex trigonometric functions
Lemma
The vector triple-product identity in ℝ³
Lemma
Quantitative shallow and deep bounds for the hyperbolic tangent
Theorem
Exact Reynolds and Mach nondimensionalization
Lemma
Derivative identities for the real restrictions of complex sine and cosine
Theorem
Acoustic and inertial finite-mode solutions
Theorem
Finite-depth gravity-wave dispersion with quantitative limits
Sources
- Iain Stewart, MIT 8.09, Chapter 6: Fluid Mechanics
- MIT 6.013, Lecture 22: Acoustic Waves
- Falk Feddersen, UC San Diego, Linear Surface Gravity Waves 1
- Jorge Lindley, Warwick MA3D1, Fluid Dynamics Support Class 6
- Elias M. Stein and Rami Shakarchi, Complex Analysis, Chapter 1, §§2.2–2.3
- David R. Wilkins, MA232A: Euclidean and Non-Euclidean Geometry, Notes on Vector Algebra and Spherical Trigometry, §5.6
- M. E. Taylor, Introduction to Analysis in Several Variables, §1.4 Exercises on the cross product
- Eric Firing, University of Hawaiʻi OCN 620, chapter 19: Surface waves, Part 1
- Iain Stewart, MIT 8.09, Chapter 6: Fluid Mechanics, §6.7 Reynolds Number and Similarity
- MIT 16.01 Unified Engineering, F5 lecture notes: Effects of Reynolds Number and Mach Number
- Iain Stewart, MIT 8.09, Chapter 6: Fluid Mechanics, §6.5 Sound Waves
- Markus Zahn, MIT 6.013, Lecture 22: Acoustic Waves
- Jorge Lindley, Warwick MA3D1, Fluid Dynamics Support Class 6, §1 Inertial Waves in a Rotating Fluid
- Falk Feddersen, UC San Diego, Linear Surface Gravity Waves 1, §§2.1–2.4
- Eric Firing, University of Hawaiʻi OCN 620, chapter 19: Surface waves, Part 1, §§19.1–19.7