Mathematical Methods of Physics --- 466.001
       Monday and Wednesday from 5:30 till 6:45 pm in room 184.

       Here is the current version of the class notes.  I often update these notes, so you might want to print out just what you need rather than the whole 560 pages.
       The class notes now have an index to some of the chapters.

       The problem session will be held on Thursdays from 1:30 to 2:20 pm in room 184.

       The grader is Charles Cherqui (

       No textbook is required for this course, but there are some decent ones:
       Mathematical Methods for Physics and Engineering by K. F. Riley, M. P. Hobson, and S. J. Bence (Cambridge Univ. Press),
       A Guided Tour of Mathematical Methods for the Physical Sciences by Roel Snieder
(Cambridge Univ. Press), and
by Arfken; I prefer the later printings of the third edition.

       First homework assignment:  Do the first nine problems at the end of chapter 1 of the on-line notes.  This assignment is due on Wednesday 9 September. 
homework assignment:  Do problems 10-18 at the end of chapter 1 of the on-line notes.  This assignment is due on Wednesday 23 September. 
homework assignment:  Do problems 1-5 at the end of chapter 2 of the on-line notes.
This assignment is due on Monday October 12th.
homework assignment:  Do problem 6 of chapter 2 and all the problems of chapter 3.  This assignment is due on Wednesday October 30th.
homework assignment:  Do all the problems at the end of chapter 4.   This assignment is due on Wednesday November 11th.
homework assignment:  Do the first six problems at the end of chapter 5.   Problems 7 & 8 of that chapter will win extra credit for any eager beavers who do them.   This assignment is due on Wednesday November 25th.
homework assignment:  Do all the problems at the end of chapter 6.   This assignment is due on Wednesday December 9th.

       Videos of lectures (these wmv files are big, so please download each one to your computer before viewing it):
       Due to an unknown glitch, the video of the first lecture is unavailable, so in its place you may view the first lecture of last year's version of this course:
       First lecture of last year: linear algebra up to linear independence.
       Lecture of Wed. August 26th: dimension of a vector space, inner and outer products, and Dirac notation.
       Lecture of Mon. August 31st:
laser demo on polarization vectors, identity operators, vectors and their components, linear operators and their matrix elements, determinants.
       Lecture of Wednesday September 2d: determinants, systems of linear equations, eigenvectors and eigenvalues, the adjoint of a linear operator, hermitian operators, unitary operators, anti-unitary and anti-linear operators, Wigner's theorem on symmetry in quantum mechanics, eigenvalues of a square matrix.
      Lecture of Wednesday September 9th:
  eigenvalues of a square matrix, functions of matrices, hermitian matrices.
      Lecture of Monday September 14th:  hermitian matrices, normal matrices, determinant of a normal matrix, tricks with Dirac notation, compatible normal matrices.
      Lecture of Wednesday September 16th:
  compatible normal matrices, a matrix satisfies its characteristic equation, the singular-value decomposition.
      Lecture of Monday September 21st, part one and part two:  applications of
the singular-value decomposition, LAPACK, the rank of a matrix, the tensor or direct product, density operators.
      Lecture of Wednesday September 23d:  on tensor products, density operators, correlation functions, groups, and complex Fourier series.
      Lecture of Monday September 28th:  on the Fourier series for exp(-m|x|), real Fourier series, the Fourier series for x, complex and real Fourier series for an interval of length L.
      Lecture of Wednesday September 30th:  on Fourier series in several variables, the convergence of Fourier series, and quantum-mechanical examples.
      Lecture of Monday October 5th:  on quantum-mechanical examples, the harmonic oscillator, non-relativistic strings, periodic boundary conditions, and the transition to the Fourier transform.
      Lecture of Wednesday October 7th:  on
the transition to the Fourier transform, the Fourier transform of a gaussian and of a real function, a representation of the delta-function, Parseval's identity, derivatives of a Fourier transform, and momentum and momentum space.
      Lecture of Monday October 12th:  on the uncertainty principle, Fourier transforms in several variables, application to differential equations, Fick's law, and diffusion.
      Lecture of Wednesday October 14th:  on
the wave equation, diffusion, convolutions, Gauss's law, and the magnetic vector potential.
      Lecture of Monday October 19th:  on the Fourier transform of a convolution, finding Green's functions, Laplace transforms, examples of Laplace transforms, how to measure the lifetime of a fluorophore, differentiation and integration of Laplace transforms, inverting Laplace transforms, convergence of infinite series, tests of convergence, series of functions, uniform convergence, the Riemann zeta function, and power series.
     Lecture of Wednesday October 21st:  on power series, the geometric series, the exponential series, factorials and the Gamma function, Taylor series, Fourier series, the binomial theorem, the binomial coefficient, double factorials, the logarithmic series, Bernoulli numbers and polynomials, the Lerch transcendent, asymptotic series, and the exponential integrals.
     Lecture of Monday October 26th:  on dielectrics, analytic functions, Cauchy's integral theorem, Cauchy's integral formula, the Cauchy-Riemann conditions, and harmonic functions.
     Lecture of Wednesday October 28th:  on the Cauchy-Riemann conditions and the contour integral of a general function, harmonics functions, applications to two-dimensional electrostatics, Earnshaw's theorem, Taylor series, and Cauchy's inequality.
     Lecture of Monday 2 November:  on Cauchy's inequality, Liouville's theorem, the fundamental theorem of algebra, Laurent series, poles, essential singularities, and the calculus of residues.
     Lecture of Wednesday 4 November before pizza and after pizza:  on the calculus of residues, ghost contours, third-harmonic microscopy, and several examples of the use of ghost contours.
     Lecture of Monday 9 November:  on logarithms, cuts, roots, contour integrals around cuts, Cauchy's principal value, i-epsilon rules, and application to Feynman's propagator.
     Lecture of Wednesday 11 November:  on dispersion relations, Hilbert transforms, the Kramers-Kronig relations, conformal mapping, the method of steepest descent, phase and group velocities, the group index of refraction, slow light, fast light, and backwards light.
     Lecture of Monday 16 November:  on ordinary linear differential equations, non-linear ODEs, linear partial differential equations, non-linear PDEs, separable partial differential equations, the Helmholtz equation in rectangular, cylindrical, and spherical coordinates,  wave equations, the Klein-Gordon equation, and the field of a spinless boson.
     Lecture of Wednesday 18 November:  on the photon field, the Majorana field, the Dirac field, first-order differential equations, separability, Zipf's law, the logistic equation, hidden separability, exact
first-order differential equations, the criteria of exactness, the condition of integrability, Boyle's law, the ideal -gas law, van der Waals  law, and human population growth.
     Lecture of Monday 23 November:  on the condition of integrability, a method of integration based on exactness, integrating factors, homogeneous functions, the virial theorem, homogeneous first-order ODEs, linear first-order ODEs, and falling bodies.
     Lecture of Wednesday 25 November:  on R-C circuits,  emission from fluorophores,  systems of differential equations,  the Friedmann equations for a homogeneous and isotropic universe,  an open universe of radiation,  a closed universe of matter,  an open universe of matter,  singular points of second-order differential equations,  Frobenius's series solutions of second-order linear ODEs,  the indicial equation,  recurrence relations,  even and odd differential operators,  and parity.
     Lecture of Monday 30 November:  on Fuch's theorem, Wronski's determinant, finding a second solution to a second-order ODE, why not three solutions?, self-adjoint form, self-adjoint or hermitian linear operators, conversion to self-adjoint form, the wronskian and self-adjoint form, and the self-adjoint form of a first-order linear differential operator.
     Lecture of Wednesday 2 December:  on eigenfunctions and eigenvalues, the Bessel and Schwarz inequalities, Green's functions, eigenfunctions and Green's functions, and Green's functions in one dimension.
     Lecture of Monday 7 December:  on Lebesgue's theory of measure and of integration, Fredholm and Volterra integral equations, numerical solutions of linear integral equations, points and their coordinates, contravariant vectors, scalars, covariant vectors, euclidean space, Minkowski space, Lorentz transformations, special relativity, and time dilation and muon decay.
     Lecture of Wednesday 9 December:  on relativistic electrodynamics, Lorentz transformations, tensors, adding tensors, tensor equations, the summation convention, the Kronecker delta, symmetric and anti-symmetric tensors, products of tensors, the quotient rule, the metric tensor, the sphere, a basic axiom, the inverse of the metric tensor, raising and lowering indices, orthogonal coordinates in 3-space, spherical coordinates, the gradient of a scalar field, derivatives and affine connections, Christoffel symbols, and covariant derivatives.
     Lecture of Friday 11 December:  on groups, their representations, quantum-mechanical applications, subgroups, Schur's lemma, tensor products, Lie groups, Lie algebras, generators, structure constants, the Jacobi identity, the adjoint representation, demonstration of the commutation relations of the generators of the group of rotations, SU(n), and Cartan's classification of the compact Lie groups.