Alphabeta MathIntegration of Forms and the General Stokes Theorem
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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
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- 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
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- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
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- Partitions of Unity and Paracompactness
- Picard-Lindelöf and First-Order Ordinary Differential Equations
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- Rank Theorems and Embedded Submanifolds
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- Rⁿ as a Normed Space; Vector-Valued Functions
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- Separation Axioms: the Hierarchy
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- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Simple Field Extensions and the Construction of the Complex Numbers
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- 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
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- Topology of ℝ
- Vector Fields Flows and Lie Derivatives
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2 · Logical flowchart
3 · Items
Definition
Compact support of a differential form
Lemma
Local finiteness near compact support
Lemma
Riemann-integrable half-space extensions of chart coefficients
Definition
Chart integral with its orientation sign
Lemma
Local side-preserving extensions of half-space transitions
Theorem
Coordinate independence of chart integrals
Definition
Integral of a compactly supported top form
Theorem
Independence of atlas, partition and refinement
Lemma
Finite chart localization of compactly supported top forms, including boundary
Proposition
Linearity and additivity of the form integral
Proposition
Orientation reversal changes the integral sign
Proposition
Positivity of the oriented integral
Theorem
Change of variables on oriented manifolds
Proposition
Integration on an oriented embedded submanifold
Proposition
Computing form integrals by finite parametrizations
Definition
A signed one-density on a real vector space
Proposition
The density line and its positive cone
Definition
Density bundle and smooth density fields
Proposition
Absolute value of a top form as a density
Proposition
Existence of positive smooth densities
Proposition
Pullback of densities by local diffeomorphisms
Definition
Integral of a compactly supported smooth density
Theorem
Orientation-free density integration and its properties
Proposition
Orientation identifies top forms with signed densities
Remark
The separate measurable extension of density integration
Lemma
Form calculus extends locally across a manifold boundary
Lemma
Compact-support Stokes on Euclidean space
Lemma
Compact-support Stokes on the upper half-space
Lemma
Localization of Stokes by a partition of unity
Theorem
The general Stokes theorem
Corollary
A compactly supported primitive has zero total derivative integral
Corollary
Nonzero total integral obstructs exactness on a closed manifold
Corollary
Closed forms have zero boundary integral
Corollary
A nonzero period obstructs exactness and bounding
Corollary
Stokes agrees with the fundamental theorem of calculus
Corollary
General Stokes agrees with both planar Green formulas
Corollary
Agreement of general and classical surface Stokes
Definition
Positive volume form on an oriented manifold
Definition
Divergence relative to a volume form
Proposition
Coordinate formula and well-definedness of divergence
Proposition
Product rule for volume-form divergence
Lemma
Divergence as an exterior derivative
Theorem
Divergence theorem for a volume form
Proposition
Agreement with classical Gauss flux in Euclidean space
False statement
False: top-form integration needs no orientation
False statement
False: summing unweighted atlas integrals is valid
False statement
False: all exact forms integrate to zero everywhere
False statement
False: densities and top forms coincide on nonorientable manifolds
Sources
- Merry Definition 26.7, Lecture 26 p.2 / PDF p.214
- Merry Definition 26.11 and Lemma 26.12, PDF pp.215–216
- Lee integration of forms pp.402–404; explicit Riemann justification from cited published items
- Lee (16.1), pp.404–406
- Lee Proposition 16.3, p.404, together with the exact published boundary chain rule and inverse theorem
- Lee Propositions 16.3–16.4, pp.404–405; Merry Lemma 26.8
- Lee (16.2) and zero-dimensional paragraph, pp.405–406; Merry Definitions 26.11 and 26.15
- Lee Proposition 16.5, pp.405–406; Merry Lemma 26.12
- Lee Proposition 16.6(a), pp.407–408
- Lee Proposition 16.6(b), pp.407–408
- Lee Proposition 16.6(c), pp.407–408 (nonnegative version by the same proof)
- Lee Proposition 16.6(d), pp.407–408
- Lee submanifold paragraph p.406
- Lee Proposition 16.8 and proof, pp.408–409
- Lee (16.18) and Proposition 16.35, pp.428–429
- Lee Proposition 16.35(a,b,d), pp.428–429
- Lee Proposition 16.36, pp.429–430; Nicolaescu Definition 3.4.1
- Lee Proposition 16.35(c), p.428 and nonvanishing paragraph p.430; Nicolaescu Example 3.4.2(b) with corrected regularity
- Lee Proposition 16.37, p.430
- Lee Propositions 16.38 and 16.40, pp.430–431, with local-diffeomorphism qualification
- Lee density-integral construction pp.431–432; Nicolaescu Proposition 3.4.3
- Lee Propositions 16.41–16.42 and Exercises 16.43–16.44, pp.431–432; Nicolaescu Proposition 3.4.3
- Nicolaescu §3.4.2, p.120, paragraph from the orientation isomorphism through the gluing formula
- Nicolaescu Proposition 3.4.3(c), p.113
- Lee Stokes proof pp.412–414 and published extension/calculus dependencies
- Lee Theorem 16.11 proof pp.412–413; Merry Theorem 26.17
- Lee Theorem 16.11 proof, pp.412–413
- Lee Theorem 16.11 proof p.414; Merry Theorem 26.16 proof PDF p.218
- Lee Theorem 16.11, pp.411–414; Merry Theorem 26.16
- Lee Corollary 16.13, p.414 (compact-support version from Theorem 16.11)
- Lee Corollary 16.13; Merry Corollary 27.2 proof (nonexactness consequence without cohomology terminology)
- Lee Corollary 16.14, p.414
- Lee Corollary 16.15, pp.414–415
- Lee Example 16.12, p.414
- Lee Theorem 16.17, p.415; classical items cited for both statements
- Lee Theorem 16.34 proof, p.427
- Nicolaescu §3.4.2 orientation-isomorphism paragraph p.120; Lee Proposition 16.6(c)
- Encyclopedia of Mathematics, Divergence, Comments paragraph beginning Let M be an n-dimensional manifold
- EoM Divergence Comments; Lee defining divergence equation p.423; coordinate derivation from published Cartan formula
- EoM Divergence Comments definition; direct algebraic consequence of the coordinate formula
- Lee divergence definition p.423; EoM Divergence Comments
- Lee Theorem 16.32 proof p.424, using arbitrary positive mu in the form identity
- Lee Lemma 16.30 and Theorem 16.32, pp.423–424
- Lee Proposition 16.6(b) and Densities introduction pp.407, 428
- Lee Proposition 16.5 proof pp.405–406 (necessity of partition weights)
- Lee Theorem 16.11 and Corollary 16.13 hypotheses, pp.411–414
- Lee Proposition 16.37 and following explanation, p.430