Instructor: Kevin Cahill

Phone: 505 205 5448, but please call only after 1 PM.

Due to the pandemic, all classes will be held

online via Zoom from 23 August

except for the week of 25 September.

The Zoom link is https://unm.zoom.us/j/9119790161.

Meeting ID: 911 979 0161

Passcode: class

I plan to use email to send you invitations for each class

but you may find it more convenient just to click on the

Zoom link https://unm.zoom.us/j/9119790161.

466 Classes are at 5:00 on Tuesdays and Thursdays via Zoom.

468 Classes are at 5:30 on Wednesdays via Zoom.

Office Hours: Mondays from 4-6 pm via Zoom,

except today 14 Nov from 1 to 3 because of a medical appointment.

Meeting ID: 911 979 0161

Passcode: class

468 is part of 466; students should attend 468

even if they are not registered for the class.

The plan is to cover the first 10 chapters of the book and also some of the most interesting parts of the second half of the book.

We will be using the second edition of a textbook called

New and used copies of the book are available in the UNM bookstore, but you can also get it from Amazon and eBooks.

Errata for 2d edition

Class notes. This file is frequently updated.

Handwritten class notes from iPad. (Please skip the first blank page.)

Notes on linear algebra, vector calculus, Fourier series, Fourier and Laplace transforms, and infinite series.

Handwritten class notes from iPad. (Please skip the first blank page.)

6.1 Analytic Functions, 6.2 Cauchy-Riemann Conditions, 6.3 Cauchy’s Integral Theorem, 6.4 Cauchy’s Integral Formula, 6.5 Harmonic Functions, 6.6 Taylor Series for Analytic Functions, 6.7 Cauchy’s Inequality, 6.8 Liouville’s Theorem, 6.9 Fundamental Theorem of Algebra, 6.10 Laurent Series, 6.11 Singularities, 6.12 Analytic Continuation, 6.13 Calculus of residues, 6.14 Ghost contours, 6.15 Logarithms and cuts, 6.16 Powers and roots, 6.17 Conformal mapping, 6.18 Cauchy’s principal value, 6.19 Dispersion relations, 6.20 Kramers-Kronig relations, 6.21 Phase and group velocities, 6.22 Method of Steepest Descent, 6.23 Applications to string theory

Handwritten class notes from iPad. (Please skip the first blank page.)

Problems 6.30, 6.31, 7.8; example 7.45, Frobenius's method, singular points, H atom, series via Matlab, falling body, CR circuit, integration of an ode, small oscillations, ode with constant coefficients, Lagrange's equations, singular points, indicial equation, Frobenius's method, quantization, Even and odd operators, Wronki's tricks, boundary conditions, intgerable systems, tunneling, scale factor, making differential operators self adjoint, first-order self-adjoin differential operators, harmonic oscillator, Frobenius's method, completeness of eigenfunctions.

Handwritten notes from iPad from 25 October

How to make the integral equation of a self-adjoint ode. How to use an integral equation to describe the eigenfunctions of an ode. Legendre polynomials -- their definition, their orthognality, their delta-function, their generating function, their main recursion relation. The CMB, Bessel's equation, a Friedmann equation, aborted discussion of current loop, Bessel functions, Bessel's drum.

Handwritten notes from iPad from 9 Nov (Please skip the first blank page.)

Potential of a ring of charge, Bessel delta function, basis vectors, differential forms, general coordinate transformations, upper and lower indexes, invariants, forms, the exterior derivative.

Handwritten notes from iPad from early December

Curvature, invariant volume element, Einstein's equations, negative pressure, time dilation, computation of Christoffel symbols, integration of Friedmann's equations, basis vectors of torus, nonabelian gauge theory.

GRADING — Grading will be based 100 percent on homework.

HOMEWORK — Send to grader via email.

Due Monday 12 September 2022:

do problems 18, 19 28, 32, 33, and 34 of chapter 1. Here problem 34 is a new problem (1.34) described in the online class notes.

Due Monday 19 September:

Do problems 2.1, 2.7, 2.8, 3.19, 3.25.

Due Monday 26 September:

Do problems 4.6, 4.7, 4.18, 5.1, 5.2, 5.12.

Due Monday 3 October:

Do problems 5.19, 6.1, 6.5, 6.7, 6.11.

Due Monday 10 October:

Do problems 6.24, 6.25, 6.26, 6.29, 6.34

Due Monday 17 October:

Do problems 7.9, 7.10, 7.11, 7.13.

Due Monday 24 October:

Do problems 7.14, 7.15, 7.16, 7.17, 7.18 and 7.19.

Due Tuesday 1 November:

Do problems 7.24, 7.43, 7.45, 7.47, 8.3.

Due Monday 7 November:

Do problem 8.4 as listed in the online class notes, and also problems 9.12, 9.14, 9.15, 9.16.

Due Monday 14 November:

Do problems 9.17, 9.19, 9.20, 10.13, and 10.14.

Due Saturday 26 November:

Do problems 10.15, 10.20, and 10.24

Due Wednesday 30 November:

Do problems 13.1, 13.3, 13.4, 13.14.

Due Friday 9 December 13.15, 13.16, 13.17, 13.21, 13.22.

VIDEOS of classes:

Google first displays the videos in SD and then hours later in HD.

So if the images are fuzzy, try them again the next day. If they're still fuzzy, send me email.

23 Aug Tuesday

Arrays, matrices, linear operators, inner products, inequalities, linear independence, dimension, orthonormal vectors, and Dirac notation.

24 Aug Wednesday

Dirac notation, adjoints of operators, hermitian and unitary operators, Hilbert spaces, determinants, jacobians, linear least squares, and Lagrange multipliers.

25 Aug Thursday

Eigenvectors and eigenvalues of square matrices, the characteristic equation, functions of matrices, hermitian and normal matrices, compatible matrices, the singular-value decomposition, pseudoinverses, and entanglement.

30 Aug Tuesday

Singular-value decomposition, derivatives, complex differentiation, gradient, divergence, laplacian, curl, Helmholtz decomposition, Dirac's delta function, covariant and contravariant vectors.

31 Aug Wednesday

Fourier series, the interval, where to put the 2pi's, real Fourier series, stretched intervals, Fourier series for functions of several variables, integration and differentiation of Fourier series, convergence of Fourier series.

1 Sep Thursday

Sections 3.8 How Fourier Series Converge, 3.9 Measure and Lebesgue Integration, 3.10 Quantum-Mechanical Examples, 3.11 Dirac’s Delta Function, 3.12 Nonrelativistic Strings, 3.13 Periodic Boundary Conditions, 4.1 Fourier transforms.

Due to FERPA, I have edited out any segments in which a student's voice (or image) was recorded. To avoid FERPA problems, and since I am hard of hearing, and because some of us have English as a second, third, or fourth language, please use Zoom's chat instead of speaking -- at least until I have stopped recording the lecture.

6 Sep Tuesday

Sections 4.1 Fourier transforms, 4.2 Fourier transforms of real functions, 4.3 Dirac, Parseval, and Poisson, 4.4 Derivatives and integrals of Fourier transforms, 4.5 Fourier transforms of functions of several variables.

7 Sep Wednesday

4.4 Derivatives and integrals of Fourier transforms, 4.5 Fourier transforms of functions of several variables, 4.6 Convolutions, 4.7 Fourier transform of a convolution, 4.8 Fourier transforms and Green’s functions, 4.9 Laplace transforms, 4.10 Inversion of Laplace transforms.

8 Sep Thursday

4.10 Inversion of Laplace transforms, 4.11 Volterra’s Convolution, 4.12 Derivatives and integrals of Laplace transforms, 4.13 Laplace transforms and differential equations, 4.14 Applications to Differential Equations, 5.1 Convergence, 5.2 Tests of convergence, 5.3 Convergent series of functions, 5.4 Power series, 5.5 Factorials and the gamma function

13 Sep Tuesday

5.5 Factorials and the Gamma Function, 5.6 Exponential Integral and Incomplete Gamma Function, 5.7 Euler’s beta function, 5.8 Taylor Series, 5.9 Fourier Series as Power Series, 5.10 Binomial Series, 5.11 Logarithmic Series, 5.12 Dirichlet Series and the Zeta Function, 5.13 Bernoulli Numbers and Polynomials, 5.14 Asymptotic series, 5.15 Fractional and complex derivatives, 5.16 Some electrostatic problems, 5.17 Infinite products, 6.1 Analytic functions, 6.2 Cauchy-Riemann conditions, 6.3 Cauchy’s Integral Theorem, 6.4 Cauchy’s Integral Formula

14 Sep Wednesday

6.1 Analytic Functions, 6.2 Cauchy-Riemann Conditions, 6.3 Cauchy’s Integral Theorem, 6.4 Cauchy’s Integral Formula

15 Sep Thursday

6.5 Harmonic Functions, 6.6 Taylor Series for Analytic Functions, 6.7 Cauchy’s Inequality, 6.8 Liouville’s Theorem, 6.9 Fundamental Theorem of Algebra, 6.10 Laurent Series, 6.11 Singularities, 6.12 Analytic Continuation, 6.13 Calculus of residues

20 Sep Tuesday

6.14 Ghost contours, 6.15 Logarithms and cuts, 6.16 Powers and roots

21 Sep Wednesday

6.17 Conformal mapping, 6.18 Cauchy’s principal value, 6.19 Dispersion relations, 6.20 Kramers-Kronig relations, 6.21 Phase and group velocities, 6.22 Method of steepest descent

22 Sep Thursday

Example of using Mathematica and Matlab to find inverse Laplace transforms, 6.22 Method of Steepest Descent, 7.1 Ordinary linear differential equations, 7.2 Linear partial differential equations, 7.3 Separable partial differential equations, 7.4 First-order differential equations, 7.5 Separable first-order differential equations

4 Oct Tuesday

Matlab tricks. Linear first-order ordinary differential equations, Small oscillations, Systems of ordinary differential equations, Exact higher-order differential equations, Constant-coefficient differential equations, Second-order ordinary differential equations, Frobenius’s solutions of second-order differential equations.

5 Oct Wednesday

Solved examples from chapters 6 & 7, and applications of Matlab.

6 Oct Thursday

Matlab examples of a chaotic autonomous system of ode's. Frobenius’s solutions of second-order differential equations, Fuch’s theorem, Even and odd differential operators, Wronski’s determinant, Second Solutions, Why not three solutions? Boundary conditions.

11 Oct Tuesday

Boundary conditions, WKB Approximation, A variational problem, Self-adjoint differential operators, Self-adjoint differential systems, Making Operators Formally Self Adjoint, Wronskians of Self-Adjoint Operators, First-order self-adjoint differential operators, A constrained variational problem.

12 Oct Wednesday

Examples of self-adjoint differential systems and of the JWKB method.

18 Oct Tuesday

Eigenfunctions and Eigenvalues of Self-Adjoint Systems, Unboundedness of Eigenvalues, Completeness of Eigenfunctions, Inequalities of Bessel and Schwarz, Green’s Functions, Eigenfunctions and Green’s Functions, Green’s functions in One Dimension, Principle of Stationary Action in Field Theory, Symmetries and Conserved Quantities in Field Theory

19 Oct Wednesday

This was a problem session on some of the exercises of chapter 7.

20 Oct Thursday

Principle of Stationary Action in Field Theory, Symmetries and Conserved Quantities in Field Theory, Nonlinear Differential Equations, Hydrodynamics, Nonlinear Differential equations in Cosmology, Nonlinear Differential Equations in Particle Physics, Matlab Solves Differential Equations

25 Oct Tuesday

8 Integral Equations 382 8.1 Differential Equations as Integral Equations 382 8.2 Fredholm Integral Equations 384 8.3 Volterra Integral Equations 385 8.4 Implications of Linearity 386 8.5 Numerical Solutions 387 8.6 Integral Transformations 389 Exercises 391 9 Legendre Polynomials and Spherical Harmonics 393 9.1 Legendre’s Polynomials 393 9.2 The Rodrigues Formula 395 9.3 Generating Function for Legendre Polynomials

26 Oct Wednesday

I did an example of how to construct an integral equation from a differential equation and then discussed Legendre polynomials.

26 Oct Thursday

9.3 Generating Function for Legendre Polynomials 397 9.4 Legendre’s Differential Equation 398 9.5 Recurrence Relations 401 9.6 Special Values of Legendre Polynomials 402 9.7 Schlaefli’s Integral 403 9.8 Orthogonal Polynomials 404 9.9 Azimuthally Symmetric Laplacians 406 9.10 Laplace’s Equation in Two Dimensions 407 9.11 Laplace’s Equation In Three Dimensions 408 9.12 Helmholtz’s Equation in Spherical Coordinates 409 9.13 Associated Legendre Polynomials 409 9.14 Spherical Harmonics 411 9.15 Cosmic Microwave Background Radiation

1 Nov Tuesday

9 Legendre Polynomials and Spherical Harmonics 394 9.1 Legendre Polynomials 394 9.2 The Rodrigues Formula 396 9.3 Generating Function for the Legendre Polynomials 397 9.4 Legendre’s Differential Equation 399 9.5 Recurrence Relations 401 9.6 Special Values of Legendre Polynomials 402 9.7 Schlaefli’s Integral 403 9.8 Orthogonal Polynomials 404 9.9 Azimuthally Symmetric Laplacians 406 9.10 Laplace’s Equation in Two Dimensions 408 9.11 Laplace’s Equation In Three Dimensions 408 9.12 Helmholtz’s Equation in Spherical Coordinates 409 9.13 Associated Legendre Polynomials 409 9.14 Spherical Harmonics 412 9.15 Cosmic Microwave Background Radiation 10.6 Spherical Bessel Functions of the First Kind

2 Nov Wednesday

A discussion of examples involving Legendre and Bessel functions.

3 Nov Thursday

10 Bessel Functions 418 10.1 Bessel Functions of the First Kind 418 10.2 Helmholtz’s Equation 425 10.3 Applications of Bessel Functions of the First Kind 427 10.4 Bessel Functions of the Second Kind 431

8 Nov Tuesday

10.4 Bessel Functions of the Second Kind 432 10.5 Bessel Functions of the Third Kind 434 10.6 Spherical Bessel Functions of the First Kind 436 10.7 Applications of Spherical Bessel Functions of the First Kind 439 10.8 Spherical Bessel Functions of the Second Kind 442 13 General Relativity 534 13.1 Points and their coordinates 534 13.2 Scalars 535 13.3 Contravariant vectors 536 13.4 Covariant vectors 536 13.5 Tensors 537 13.6 Summation convention and contractions 538

9 Nov Wednesday

The potential outside a circle of radius a. Discussion of spherical Bessel functions. Correction of a typo. Solution of problem 13.2. Remarks about basis vectors. Very brief discussion of differential forms.

10 Nov Thursday

Potential inside a hollow sphere, General coordinate transformations, Tensors of various ranks, exterior differentiation, summation convention, tensor equations, inertial frames, the quotient theorem, notations for derivatives.

15 Nov Tuesday

11Tensors 448 11.1 General Coordinate Transformations 448 11.2 Tensors 448 11.3 Invariants, Tangent Vectors and the Metric Tensor 450 11.3.1 Comma Notation 451 11.4 Inverse of Metric Tensor 452 11.4.1 Dual Vectors, Cotangent Vectors

16 Nov Wednesday

11Tensors 448 11.1 General Coordinate Transformations 448 11.2 Tensors 448 11.3 Invariants, Tangent Vectors and the Metric Tensor 450 11.3.1 Comma Notation 451 11.4 Inverse of Metric Tensor 452 11.4.1 Dual Vectors, Cotangent Vectors 11.6 Covariant Derivatives of Contravariant Vectors 454 11.7 Covariant derivatives of covariant vectors 455

17 Nov Thursday

Tensors 448 11.1 General Coordinate Transformations 448 11.2 Tensors 448 11.3 Invariants, Tangent Vectors and the Metric Tensor 450 11.3.1 Comma Notation 451 11.4 Quotient Theorem 453 11.5 Inverse of Metric Tensor 453 11.5.1 Dual Vectors, Cotangent Vectors 454 11.6 Covariant Derivatives of Contravariant Vectors 455 11.7 Covariant Derivatives of Covariant Vectors 456 11.8 Covariant Derivatives of Tensors 456 Covariant curls, Covariant derivatives and Antisymmetry, Parallel Transport, Curvature

22 Nov Tuesday

10.2 Helmholtz’s Equation 425 Tensors 448 11.1 General Coordinate Transformations 448 11.2 Tensors 448 11.3 Invariants, Tangent Vectors and the Metric Tensor 450 11.3.1 Comma Notation 451 11.4 Quotient Theorem 453 11.5 Inverse of Metric Tensor 453 11.5.1 Dual Vectors, Cotangent Vectors 454 11.6 Covariant Derivatives of Contravariant Vectors 455 11.7 Covariant Derivatives of Covariant Vectors 456 11.8 Covariant Derivatives of Tensors 456 Covariant curls, Covariant derivatives and Antisymmetry, Parallel Transport, Curvature

23 Nov Wednesday

Review of the basic mathematics of general relativity: 11 Tensors 449 11.1 General Coordinate Transformations 449 11.2 Tensors 449 11.3 Invariants, Tangent Vectors and the Metric Tensor 451 11.3.1 Comma Notation 452 11.4 Quotient Theorem 454 11.5 Inverse of Metric Tensor 454 11.5.1 Dual Vectors, Cotangent Vectors 455 11.6 Affine Connections and Christoffel Symbols 456 11.7 Covariant Derivatives of Contravariant Vectors 457 11.8 Covariant Derivatives of Covariant Vectors 458 11.9 Covariant Derivatives of Tensors 458 11.10 The covariant derivative of the metric tensor vanishes 459 11.11 Covariant Derivatives and Antisymmetry 459 11.12 Parallel transport 460 11.13 Curvature 462 11.14 Scalar densities and g = | det(g_{ik})| 465 11.15 Divergence of a contravariant vector 466 11.16 Covariant laplacian 468

29 Nov Tuesday

Curvature, invariant 4-volume element, some exercises

30 Nov Wednesday

Curvature, scalar volume element, 465 11.15 Divergence of a contravariant vector 466 11.16 Covariant laplacian 468 11.17 Stationary action and geodesic equation 468 11.18 Weak static gravitational fields 470 11.19 Gravitational time dilation 471 11.20 Einstein’s Equations

1 Dec Thursday

11.12 Parallel transport 461 11.13 Curvature 462 11.14 Scalar densities and g = | det(g_{ik})| 465 11.15 Divergence of a contravariant vector 466 11.16 Covariant laplacian 468 11.17 Stationary action and geodesic equation 468 11.18 Weak static gravitational fields 470 11.19 Gravitational time dilation 471 11.20 Einstein’s Equations 472 11.21 Energy-momentum tensor 474 11.22 Perfect fluids 476 11.23 Gravitational waves 476 11.24 Schwarzschild Metric 477 11.25 Black holes 478 11.26 Kerr metric of a rotating mass 479 11.27 Friedmann-Lemaître-Robinson-Walker Cosmologies 480 11.28 Friedmann Equations 483

6 Dec Tuesday

11.27 Friedmann-Lemaître-Robinson-Walker Cosmologies 480 11.28 Friedmann Equations 483 11.29 Density and pressure 485 11.30 How the scale factor evolves with time 486 11.31 The first hundred thousand years 490 11.32 The next ten billion years 491 11.33 Era of dark energy 493 11.34 Before the Big Bang 493

7 Dec Wednesday

Basis vectors of the torus, how the connection transforms, survey of group theory, the Yang-Mills covariant derivative in nonabelian gauge theory

8 Dec Thursday

20 Path integrals 766 20.1 Path integrals and Richard Feynman 766 20.2 Gaussian integrals and Trotter’s formula 766 20.3 Path integrals in quantum mechanics 767 20.4 Path integrals for quadratic actions 771 20.5 Path integrals in statistical mechanics 776 20.6 Boltzmann path integrals for quadratic actions 781 20.7 Mean values of time-ordered products 784

13 Dec Tuesday

Review of path integrals in quantum mechanics.

All students of physics should read at least the first section of the essay The Trouble with Quantum Mechanics by Steven Weinberg before they graduate.