Alphabeta MathRadiation asymptotics and controlled multipole expansion — Examples
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1 · Prerequisites
- Absolute and Conditional Convergence; Rearrangement; Products
- Analytic Semigroups and Linear Evolution Equations
- Areas of Elementary Plane Figures
- Banach Valued Integration and the Radon Nikodym Property
- Banach-Space Differential Calculus and Banach Manifolds
- Binary Operations, Monoids, Groups and Subgroups
- Bounded Linear Operators and Quotient Spaces
- Causal kernels, gauge and point fields after wave existence
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Complex Norms and Realification
- Complexification, Realification and Real Structures
- 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
- Density Separability and Convolution in Lᵖ
- Determinants of Matrices over a Commutative Ring
- Distributions Test Functions and Differentiation
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Electrostatic multipoles
- Electrostatics in vacuum
- Em Elementary Mathematical Prerequisites
- Em Expanded Analysis Prerequisites
- Em Wave Equation Prerequisites
- Energy momentum angular balance
- Euclidean Surface Measure, Divergence, and Green Identities
- Exterior Powers, Orientation and Hodge Duality
- Filters and Ultrafilters
- Finite Counting, Factorials and Binomial Coefficients
- Finite Dimensional Normed Spaces and Riesz Lemma
- Finite Probability and the Probabilistic Method
- Foundations of the Real Numbers for Analysis
- Free Modules, Exact Sequences, Projective and Injective Modules
- Fubini and Change of Variables
- Fundamental Trigonometric Identities
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Harmonic Functions and Mean Values in Rn
- Hausdorff via the Diagonal
- Hereditary and Productive Behaviour of the Separation Axioms
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Improper and Parameter-Dependent Multiple Integrals
- Improper Integrals
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Integral Maxwell and constraints
- Lebesgue Measure on Euclidean Space
- 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
- Lorentz Transformations
- Manifolds with Boundary Collars and Orientations
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Measurable Densities and Radon Volume on Manifolds
- Measurable Functions and Simple Approximation
- Measures and Their Basic Properties
- Metric Spaces
- Mixed Partials, Taylor Formulae, and Extrema
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Normal Subgroups and Quotient Groups
- Normed and Banach Spaces
- Norming and Separation under Hahn–Banach
- Order, Zorn's Lemma, and the Axiom of Choice
- Outer Measure and the Caratheodory Extension Theorem
- Partial Differential Equations and Characteristics
- Pointwise Extended-Real Function Products
- Polynomial Rings, the Division Algorithm and Roots
- Potentials and gauge
- Power Series and Real-Analytic Functions
- Primitives and postulates
- Product Measures and the Fubini Tonelli Theorems
- Properties of the Integral and the Working FTC
- Radiation asymptotics and controlled multipole expansion
- Radon Measures and the Riesz Markov Kakutani Theorem
- Rank Theorems and Embedded Submanifolds
- Regular Surfaces and Surface Integrals
- Relations, Functions, and Quotients
- Retarded fields and radiation
- Rings, Subrings, Integral Domains and Fields
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Schwartz Space and the Plancherel Theorem
- Separation Axioms: the Hierarchy
- 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
- Sine, Cosine, and the Definition of Pi
- Smooth Manifolds and Smooth Maps
- Smooth Partitions of Unity and Exhaustions
- Spacetime Primitives
- Strongly Continuous Semigroups and Hille Yosida
- 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
- Tensor Products of Modules
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- 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 Fundamental Theorems of Calculus
- The Inverse and Implicit Function Theorems
- The Inverse Function Theorem Completed
- The Lebesgue and Riemann Integrals Compared
- The Lebesgue Integral and the Convergence Theorems
- The Logarithm and General Powers
- The Lᵖ Spaces Holder Minkowski and Riesz Fischer
- The Maximal Function and Lebesgue Differentiation
- The Real Gamma and Beta Functions
- 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
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
- Volumes of Elementary Solids and Solids of Revolution
- Wave Energy, Finite Propagation and Huygens' Principle
- Wave Equation Representation Formulas
- Worldlines And Clocks
2 · Logical flowchart
3 · Items
Example
A dipole node does not control the relative multipole error
Example
The radiation limit depends on the observation-time path
Sources
- Niklas Beisert, Elektrodynamik, Vorlesungsskript, ETH Zürich 2024 FS, complete 176-page course script
- Richard Fitzpatrick, Classical Electromagnetism: An intermediate level course, University of Texas at Austin, complete author-hosted HTML lecture notes
- Steven Errede, UIUC Physics 436 EM Fields & Sources II, Lecture Notes 14.5, complete 44-page PDF