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. Mathematica notebook for the quantum harmonic oscillator, PDF version 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) 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) 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 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 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.