Physics 4321 / 7305 Syllabus - Spring 2024
Fourier series: vector dot-product analogy; orthogonality and closure; expansions (even, odd, neither); Fourier transforms.
Reading - Boaz: chapter 3 section 10, and chapter 7; Arfken and Weber: chapter 4 section 10, and chapter 14.
Relationships between various basic mathematical structures by Max Tegmark, https://arxiv.org/pdf/gr-qc/9704009.pdf
Space
Vector Space
Hilbert Space
Peter Olver's notes
Dirichlet conditions for a Fourier Series to converge to a function
Convergence of Fourier series
Convergence of Fourier series
Divergence of Fourier series
Weierstrass function - made of cosines, differentiable nowhere
Gibbs phenomenon
Image Resolution
Homer's orbit - Mathologer, complex Fourier series, epicycles
But what is a Fourier series? From heat flow to circle drawings | DE4 - 3Blue1Brown series S4 . E4
Aleph number
Georg Cantor's diagonalization
How many infinities are there?
How To Count Past Infinity - Vsauce
A Hierarchy of Infinities | Infinite Series | PBS Digital Studios
How Big are All Infinities Combined? (Cantor's Paradox) | Infinite Series | PBS Digital Studios
Defining Infinity | Infinite Series | PBS Digital Studios
Continuum hypothesis
Jan 16 T Lecture notes
Jan 18 R Lecture notes
Jan 23 T Lecture notes
Jan 25 R Lecture notes
Whiteboard photos: 1, 2, 3, 4.
Jan 30 T Lecture notes
Square wave Mathematica example notebook, PDF format;
Triangle wave Mathematica example notebook, PDF format.
Feb 01 R Lecture notes
Generalized Functions / Distributions: delta function, theta step Heavyside
function, and their derivatives and Laplace transforms;
applications to electric charge distributions.
Reading - Boaz: chapter 8 section 11; Arfken and Weber: chapter 1 section 15.
Generalized function a.k.a. Distribution
Dirac delta function from Wikipedia
Dirac delta function from Wolfram MathWorld
Feb 06 T Lecture notes
Feb 08 R Lecture notes
Ordinary Differential Equations: order, linearity, homogeneity;
examples simple harmonic oscillator, damped SHO,
damped driven SHO, resonance, Green functions.
Reading - Boaz: Chapter 8; Arfken and Weber: chapter 9.
Feb 13 T Lecture notes
Feb 15 R Lecture notes, Differential Equation Mathematica notebook
Feb 20 T Lecture notes
Wronskian - used to check for linear independence of solutions
Feb 22 R No class meeting today.
Monte Carlo Methods, video Watch this asynchronously at your convenience.
See also Introduction to Monte Carlo methods by Stefan Weinzierl; Buffon's Needle
Feb 27 T Lecture notes, Zoom video, Whiteboard video;
Green functions, Green function for heat equation; Green function example problem; Mathematica notebook.
Feb 29 R
Mar 05 T Numerical Approximations to Solutions of Differential Equations
video for Numerical Approximations to Solutions of Differential Equations
When I was your age, we programmed in BASIC. And we liked it. Not really.
Mathematica notebook for the quantum harmonic oscillator, PDF version
First-order (forward) Euler method for solving differential equations
Runge-Kutta method
Mar 07 R Midterm exam - Open book, open notes, open Mathematica, closed internet; online, during class time.
Green function example problem
Nonlinear Simple Pendulum
Spring Break 11-17 March
Mar 19 T
Coordinate Systems; Scale functions;
Differential operators: Divergence, Gradient, Curl, Laplacian.
Reading - Boaz: Chapter 6; Arfken and Weber: chapters 1 and 2.
Mar 21 R Lecture notes
Mar 26 T Lecture notes, Lecture notes
Mar 28 R Lecture notes, Lecture notes
Apr 02 T Divergence Theorem, Stokes' Theorem; Lecture notes
Why π is in the normal distribution - by 3Blue1Brown
Divergence theorem
Stokes' theorem
Gaussian integrals
Fundamental theorem of calculus
Partial Differential Equations: Separation of variables, Cartesian
coordinates in 1, 2, and 3 dimensions, Cylindrical
Polar coordinates, Spherical Polar coordinates,
example Laplace's Equation.
Reading - Boaz: Chapter 13; Arfken and Weber: chapter 9.
Separation of variables
Lebesgue integration, Lebesgue measure
Laplace's equation is separable in these coordinate systems from Mathworld
An example where separation of variables fails
Apr 04 R Lecture notes, Lecture notes
soap film equation
See also my notes from PHYS 7311/7312 graduate Electricity and Magnetism: Separation of variables 1/8, 2/8, 3/8, 4/8, 5/8, 6/8, 7/8, 8/8.
Solutions to the two-dimensional wave equation: Particle in a two-dimensional rectangular box, Particle in a two-dimensional circular box.
Vibrational Modes of a Circular Membrane
Bessel function (first kind) Jn (complete, like sin and cos but for cylindrical coord's),
Weber function, Neumann function, Bessel (second kind) Yn (infinite at s=0),
modified Bessel (first kind) In (exponential growth for cylindrical coord's),
modified Bessel (second kind) Kn (exponential decay for cylindrical coord's),
Legendre Polynomials - Pl(cos θ) (complete in polar angle for spherical coord's)
Spherical harmonics - Ylm(θ, φ) (complete in polar and azimuthal angles for spherical coord's)
Spherical Bessel (first kind) jn (complete in radius r for spherical coord's)
Apr 09 T Lecture notes
Apr 11 R Intro to Chaos - YouTube, Veritasium;
Group theory, abstraction, and the 196,883-dimensional monster - YouTube, 3Blue1Brown;
Complex Analysis: Complex numbers, Roots of Unity, Cauchy-Riemann
equations, analyticity, contour integrals, residues,
Laurent expansion, conformal mapping.
Reading - Boaz: chapters 2 and 14; Arfken and Weber: chapters 6 and 7.
Complex number
Quadratic equation, see also Discriminant in the link
Cubic Equation - solution requires complex numbers even if all roots are real
Fundamental theorem of algebra
Quartic equation
Abel-Ruffini theorem unsolvability of the quintic equation in radicals
Évariste Galois
Quantum Mechanics requires complex numbers
Venn (Euler) diagram for number sets
Euler diagram vs. Venn diagram
Relationships between various basic mathematical structures by Max Tegmark, https://arxiv.org/pdf/gr-qc/9704009.pdf
A Gentle Introduction to Abstract Algebra by B.A. Sethuraman
Complex number
Analytic complex (holomorphic) function
Division algebra
Quaternion
Pauli matrices
Octonion
Sedenion
The Octonions by John C. Baez
Cauchy's integral theorem
Residue
Holomorphic function
Analyticity of holomorphic functions
Real versus complex analytic functions
Do Complex Numbers Exist? - Sabine Hossenfelder - YouTube
Zeros and Poles of complex functions
Cauchy-Riemann equations in polar form
Apr 16 T Lecture notes
Conformal Mapping in electrostatics - my grad E&M notes from PHYS 7311
Complex Analysis and Conformal Mapping notes from Peter Olver
Conformal (angle-preserving) mapping from Wolfram World
Morph Me funhouse mirror distortion effect
Conformal Mapping | Mobius Transformation | Complex Analysis #20 from YouTube
Apr 18 R Lecture notes
Apr 23 T
Differential forms
Gravitation Misner, Thorne, and Wheeler - pages 53-129 (text page numbers, not PDF page numbers)
Differential Forms Crash Course by Jiri Lebl
Lecture Notes on Differential Geometry by Keshav Dasgupta at McGill
Introduction to differential forms by Donu Arapura
A Practical Introduction to Differential Forms by Schulz^2
Differential forms - Wikipedia
Hodge star - Wikipedia
Closed and exact differential forms - Wikipedia
Tangent space - where vectors live
Cotangent space - where 1-forms (covectors) live
Maxwell Equations in terms of differential forms
Apr 25 R
Curved Space - Feynman Lectures
Escher's Angels and Devils moving in the Riemann-Poincaré disc - hyperbolic plane
Gaussian Curvature
Mercedes Gaussian Curvature
Riemann curvature tensor
List of formulas in Riemannian geometry
May 04 Saturday Final Exam 11:30 AM - 2:30 PM online.
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