# Physics 4321 / 7305 Syllabus - Spring 2023

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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 17 T Lecture notes

Jan 19 R Lecture notes

Jan 24 T Lecture notes

Jan 26 R Lecture notes
Whiteboard photos: 1, 2, 3, 4.

Jan 31 T Lecture notes
Square wave Mathematica example notebook, PDF format;
Triangle wave Mathematica example notebook, PDF format.

Feb 02 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 07 T Lecture notes

Feb 09 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 14 T Lecture notes

Feb 16 R Lecture notes

Feb 21 T  Lecture notes

Wronskian - used to check for linear independence of solutions

Feb 23 R

Feb 28 T Lecture notes, Zoom video, Whiteboard video;

Green functions, Green function for heat equation; Green function example problem; Mathematica notebook.

Mar 02 R

Mar 07 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.
First-order (forward) Euler method for solving differential equations
Runge-Kutta method

Mar 09 R Midterm exam - Open book, open notes, open Mathematica, closed internet; online, during class time.

Green function example problem
Nonlinear Simple Pendulum

Spring Break 13-19 March

Mar 21 T Monte Carlo Methods, video
See also Introduction to Monte Carlo methods by Stefan Weinzierl; Buffon's Needle

Coordinate Systems; Scale functions;
Differential operators: Divergence, Gradient, Curl, Laplacian.

Reading - Boaz: Chapter 6; Arfken and Weber: chapters 1 and 2.

Mar 23 R Lecture notes

Mar 28 T Lecture notes, Lecture notes

Mar 30 R Lecture notes, Lecture notes

Apr 04 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 06 R Lecture notes, Lecture notes

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 11 T Lecture notes

Apr 13 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
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 18 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 20 R Lecture notes

Apr 25 T

Curved Space - Feynman Lectures
Escher's Angels and Devils moving in the Riemann-Poincaré disc - hyperbolic plane
Gaussian Curvature
Mercedes Gaussian Curvature

Apr 27 R

Riemann curvature tensor
List of formulas in Riemannian geometry

May 06 Saturday Final Exam 11:30 AM - 2:30 PM online.

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