Physics 4321 / 7305 Syllabus - Spring 2021


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
   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 26 T Lecture notes, Zoom video;

Jan 28 R Lecture notes, Zoom video;

Feb 02 T Lecture notes, Zoom video;

Feb 04 R Lecture notes, Zoom video;
         Square wave Mathematica example notebook, PDF format;
         Triangle wave Mathematica example notebook, PDF format.

Feb 09 T Lecture notes, Zoom video;

Feb 11 R Lecture notes, Zoom video;

   Gaussian integrals
   Fundamental theorem of calculus

Feb 16 Class cancelled because of the weather.
Feb 18 Class cancelled because of the weather.

Feb 23 T Live Zoom video; Lecture notes, Asynchronous Zoom video;

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 25 R Live Zoom video; Lecture notes 4, Lecture notes 5, Asynchronous Zoom video;

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

Mar 02 Lecture notes, Zoom video;

Mar 04 Lecture notes, Zoom video;

Mar 09 Lecture notes, Zoom video;

   Monte Carlo method

Mar 11 Guest lecture on Monte Carlo methods by Professor Stephen Sekula, Zoom video;

   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

Mar 16 Lecture notes, Zoom video;

Mar 18 Midterm Exam in class

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.

Mar 23 Lecture notes, Zoom video;

Mar 25 Lecture notes, Zoom video;

   Wronskian - used to check for linear independence of solutions

Mar 30 Lecture notes, Zoom video;

Apr 01 Lecture notes, Zoom video;

   Green functions, Green functions (1/2), Green functions (2/2), 
   Green function for heat equation

   "Using a term like nonlinear science is like referring to the bulk of zoology as the study of non-elephant animals." --Stanislaw Ulam

Apr 02 Lecture notes, Zoom video;

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

   Green function example problem
   Nonlinear Simple Pendulum

Apr 06 Zoom video;

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

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

Apr 08 Lecture notes, Zoom video;

Apr 13 Lecture notes, Lecture notes, Zoom video;

Apr 15 Lecture notes, Lecture notes, Zoom video;

    Mixed partial derivatives.

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 20 Lecture notes, Zoom video;

Apr 22 Lecture notes, Zoom video;

   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) J_n (complete, like sin and cos but for cylindrical coord's),
    Weber function, Neumann function, Bessel (second kind) Y_n (infinite at s=0),
    modified Bessel (first kind) I_n (exponential growth for cylindrical coord's),
    modified Bessel (second kind) K_n (exponential decay for cylindrical coord's),
    Legendre Polynomials - P_l(cos θ) (complete in polar angle for spherical coord's)
    Spherical harmonics - Y_lm(θ, φ) (complete in polar and azimuthal angles for spherical coord's)
    Spherical Bessel (first kind) j_n (complete in radius r for spherical coord's)

   Heat equation and wave equation problems/solutions.

Apr 27 Zoom video;

Apr 29 Zoom video;





Group Theory:   Discrete groups: cyclic groups Z_n; dihedral groups D_n; symmetric groups S_n; alternating groups A_n.
                Group multiplication tables. Representations. Sporadic groups.
                Lie groups: SO(2); SO(3); SO(n); SU(n); Sp(2n).

Chaos: Sensitive Dependence on Initial Conditions Nonlinearity, Fractals, Hausdorf Dimension, 
               Lyapunov Exponents, Hennon Map, Logistic Map, 
               Cantor Dust, Sierpinski Gasket, Menger Sponge, Mandelbrot Set, Julia Set,
               Chua's Circuit, Period 3 Implies Chaos
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