Alphabeta MathClassical frames, primitive quantities and SI units
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1 · Prerequisites
- Absolute and Conditional Convergence; Rearrangement; Products
- Areas of Elementary Plane Figures
- Binary Operations, Monoids, Groups and Subgroups
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- 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
- Fubini and Change of Variables
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- 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
- Lebesgue Measure on Euclidean Space
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Measurable Densities and Radon Volume on Manifolds
- Measures and Their Basic Properties
- Metric Spaces
- 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
- Outer Measure and the Caratheodory Extension Theorem
- Polynomial Rings, the Division Algorithm and Roots
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- 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
- Sigma Algebras and Borel Sets
- Simple Field Extensions and the Construction of the Complex Numbers
- Smooth material configurations and Eulerian fields
- Smooth Partitions of Unity and Exhaustions
- Subspaces, Products, and Quotients
- Suprema and Infima
- Symmetric Groups, Cycle Decomposition and the Sign Homomorphism
- The Derivative and the Mean Value Theorems
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- The Exponential Function
- The Inverse and Implicit Function Theorems
- The Lebesgue Integral and the Convergence Theorems
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Topology of Euclidean Space
- 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
Classical frames, primitive quantities and SI units
Lemma
C1 material regions have compatible Jordan volume integrals
Lemma
Continuous residuals localize on C1 domains
Lemma
Finite continuous surface integrals on compact C1 boundaries
Postulate
Single-component nonpolar continuum and material balances
Physics theorem
Mass density reconstruction and incompressibility
Sources
- MIT OpenCourseWare 8.01SC, Chapter 11: Reference Frames
- MIT OpenCourseWare 8.09, Advanced Classical Mechanics, full lecture notes
- David Tong, Fluid Mechanics, Part I lecture notes
- BIPM, The International System of Units, 9th edition, version 4.01
- NIST SP 330-2019, The International System of Units (SI)
- Jiří Lebl, Basic Analysis II, §10.5 Jordan measurable sets
- Jiří Lebl, Basic Analysis II, §8.5 Inverse and implicit function theorems
- John K. Hunter, Notes on Partial Differential Equations
- Michael E. Taylor, Introduction to Analysis in Several Variables
- Ain A. Sonin, Fundamental Laws of Motion for Particles, Material Volumes, and Control Volumes, MIT 2.25 lecture notes
- Andrew Hazel, MATH45061 Continuum Mechanics, Chapter 3: Forces, Momentum and Stress
- Andrew Hazel, MATH45061 Continuum Mechanics, Chapter 4: Physical Conservation and Balance Laws & Thermodynamics