Alphabeta MathR2 De Rham Currents And Source Cycles
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Free Modules, Exact Sequences, Projective and Injective Modules
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Tensor Products of Modules
Algebraic Topology
- Cup Cap Cross Products and Cohomology Rings
- Cw Complexes and Cellular Homology
- Homology Axioms Degree and Classical Applications
- Orientations Poincare Lefschetz and Alexander Duality
- Relative Homology Excision and Mayer Vietoris
- Simplicial Complexes and Simplicial Homology
- Simplicial Subdivision and Simplicial Approximation
- Singular Chains and Singular Homology
- Singular Cohomology and Coefficient Theorems
Category Theory
- Abelian Categories
- Categories, Functors and Natural Transformations
- Exactness and the Member Calculus
- Limits and Colimits
- Preadditive and Additive Categories and Biproducts
- Reflective Subcategories and the Adjoint Functor Theorems
- Subobject Lattices Generators and the Grothendieck Axioms
- The Diagram Lemmas in an Abelian Category
- Universal Properties, Representables and the Yoneda Lemma
Differential Geometry
- Integration of Forms and the General Stokes Theorem
- Manifolds with Boundary Collars and Orientations
- Rank Theorems and Embedded Submanifolds
- Smooth Manifolds and Smooth Maps
- Smooth Partitions of Unity and Exhaustions
- Smooth Vector Bundles and Sections
- Tangent Cotangent and the Differential
- Tensor Fields Exterior Algebra and Differential Forms
- The De Rham Complex Homotopy and Mayer Vietoris
- The de Rham Theorem and Degree
- The Exterior Derivative and Cartan Calculus
- Vector Fields Flows and Lie Derivatives
Homological Algebra
- Chain Complexes and Homology
- Chain Homotopy and the Homotopy Category
- Derived Functors
- Ext and Balanced Resolutions
- Kunneth Exactness and Splittings over Principal Ideal Domains
- Long Exact Sequences in Homology
- Projective and Injective Resolutions
- Tor Flatness and Global Dimension
- Universal Coefficients and Kunneth Theorems
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Exterior Powers, Orientation and Hodge Duality
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Constant Rank, Submersions, Immersions and Regular Level Sets
- Construction of the Real Numbers via Cauchy Sequences
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- Fubini and Change of Variables
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- limsup, liminf, and Subsequential Limits
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- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
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- Series: Convergence and the Nonnegative Tests
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- The Derivative and the Mean Value Theorems
- The Exponential Function
- 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 Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Set Theory
- Cardinal Arithmetic, Cofinality and the Alephs
- Construction of the Natural Numbers
- Filters and Ultrafilters
- Order, Zorn's Lemma, and the Axiom of Choice
- Ordinal Arithmetic and the First Uncountable Ordinal
- Ordinals, Cardinals, and Transfinite Recursion
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Topology
- Compactness
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- Connectedness
- Countability Axioms and Cardinal Functions
- Function Space Topologies and the Exponential Law
- Hereditary and Productive Behaviour of the Separation Axioms
- Homotopy and Homotopy Equivalence
- Metric Spaces
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Topology of Euclidean Space
- Topological Spaces and Continuity
2 · Logical flowchart
3 · Items
Definition
De Rham currents of compact oriented source cycles
Lemma
Transverse complementary submanifolds meet in finitely many points
Lemma
The current differential is continuous and squares to zero
Definition
Current-compatible signed intersection number
Lemma
Smooth forms define currents compatibly with the differential
Lemma
A compact oriented source cycle defines a closed current
Lemma
An exact top-degree current has zero total charge on a closed manifold
Lemma
The current-compatible intersection number is well-defined
Lemma
Normalized normal-coordinate forms converge to the source current
Lemma
A source-cycle current pairs with a transverse dual cycle by intersection number
Sources
- Tien-Cuong Dinh and Nessim Sibony, Introduction to the Theory of Currents, §4.4
- Victor Guillemin and Peter J. Haine, Differential Forms, MIT 18.952 course book, §5.5
- Joel W. Robbin and Dietmar A. Salamon, Introduction to Differential Topology, §4.3
- Tien-Cuong Dinh and Nessim Sibony, Introduction to the Theory of Currents, §§3.1–3.2
- Joel W. Robbin and Dietmar A. Salamon, Introduction to Differential Topology, Definition 4.3.1 and §7.2.3
- Joel W. Robbin and Dietmar A. Salamon, Introduction to Differential Topology, Definition 4.3.1
- Joel W. Robbin and Dietmar A. Salamon, Introduction to Differential Topology, §7.2
- Tien-Cuong Dinh and Nessim Sibony, Introduction to the Theory of Currents, §§3.1–3.2 and §4.4
- Joel W. Robbin and Dietmar A. Salamon, Introduction to Differential Topology, §§4.3 and 7.2.3