Alphabeta MathIntegration of Forms and the General Stokes Theorem — Examples
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
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- 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
- Countability Axioms and Cardinal Functions
- 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
- Euclidean Ordinary Differential Equations with Smooth Dependence
- Exterior Powers, Orientation and Hodge Duality
- 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
- Hereditary and Productive Behaviour of the Separation Axioms
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Improper and Parameter-Dependent Multiple Integrals
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Integration of Forms and the General Stokes Theorem
- 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
- Manifolds with Boundary Collars and Orientations
- 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
- Partitions of Unity and Paracompactness
- Picard-Lindelöf and First-Order Ordinary Differential Equations
- Polynomial Rings, the Division Algorithm and Roots
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Rank Theorems and Embedded Submanifolds
- 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
- Separation Axioms: the Hierarchy
- 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
- Smooth Manifolds and Smooth Maps
- Smooth Partitions of Unity and Exhaustions
- Smooth Vector Bundles and Sections
- Subspaces, Products, and Quotients
- Suprema and Infima
- Symmetric Groups, Cycle Decomposition and the Sign Homomorphism
- Tangent Cotangent and the Differential
- Tensor Fields Exterior Algebra and Differential Forms
- 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 Exterior Derivative and Cartan Calculus
- The Fundamental Theorems of Calculus
- The Inverse and Implicit Function Theorems
- The Inverse Function Theorem Completed
- 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 Fields Flows and Lie Derivatives
- Vector Spaces, Linear Subspaces, Span and Direct Sums
- Volumes of Elementary Solids and Solids of Revolution
2 · Logical flowchart
3 · Items
Example
Partition weights in two overlapping charts
Example
Reflection reverses the signed form integral
Example
A density integral on the Mobius band
Example
Stokes on an interval with both endpoint chart signs
Example
Green circulation and flux on a disk
Example
Surface Stokes on a graph disk
Example
Volume-form divergence on the Euclidean ball
Example
The angular period and the obstruction to bounding
Example
An exact top form with nonzero integral on a disk
Counterexample
A noncompactly supported form whose integral diverges
Counterexample
The wrong boundary sign in the half-space computation
Example
Change of variables on an oriented circle
Sources
- Lee Proposition 16.5 proof pp.405–406
- Lee Proposition 16.6(d) and Proposition 16.42(c)
- Lee Proposition 16.37 and Exercise 16.44; Nicolaescu Definition 3.4.1
- Lee Example 16.12 and p.405 negative-chart explanation
- Lee Theorem 16.17 and Example 16.16, p.415
- Lee Theorem 16.34 proof, p.427 (explicit smooth graph specialization)
- Lee Example 16.9 pp.409–410; Nicolaescu Example 3.4.14 pp.120–121
- Lee Example 16.16 and Corollary 16.15, p.415
- Lee Theorem 16.11 and Corollary 16.14
- Lee p.407 paragraph on noncompactly supported forms and convergence
- Lee Theorem 16.11 proof pp.412–413
- Lee Proposition 16.6(d), pp.407–408 (explicit circle specialization)