Physics 501: Mathematical Physics I
Fall 2018
[Figures following each major topic indicate the approximate number of
lecture hours devoted to it.]
- Introduction and course overview [0.5]
- Partial differential equations [5.5]
(Riley & Hobson Chapters 10, 11)
- some familiar PDEs (10.1, 10.4, 10.5)
- classification of PDEs (10.3.3)
- characteristics and boundary conditions
(10.6)
- method of characteristics
- the Helmholtz equation (11.3.2)
- solution by separation of variables (11.1,
11.2, 11.3)
- cartesian coordinates
- cylindrical polar coordinates: Bessel's equation
- spherical polar coordinates: Legendre equation
- normal modes and general solutions
- Bessel functions; spherical Bessel functions
(9.5, 9.6)
- spherical harmonics (9.1 - 9.3)
- Ordinary differential equations [4.5]
(Riley & Hobson, Chapters 4, 7, 8)
- second-order systems: series solutions; the indicial
equation (7.1 - 7.4)
- second solutions; Wronskian development (7.5)
- Fuch's theorem and properties of first and second solutions
- self-adjoint differential operators; Sturm-Liouville
theory (8.1, 8.2)
- properties of eigenfunctions of Hermitian operators
(8.3)
- orthogonality of eigenfunctions; eigenfunction expansion
- the Euler-Fourier formula (4.1, 4.2)
- completeness and mean-square convergence of Fourier series
- uniform, pointwise, and mean-square convergence
- Bessel's inequality and Parseval's identity
(4.8)
- Fourier series [1.5]
(Riley & Hobson, Chapter 4)
- Euler-Fourier formulae (4.1, 4.2, 4.3)
- the Gibbs phenomenon (4.4, 4.5)
- application to PDEs: spectral methods
- Bessel Functions [1.5]
(Riley & Hobson Sections 9.5, 9.6)
- ODE and series solution
- orthogonality and Bessel series; Fourier-Bessel series
- generating function
- recurrence relations
- normalization
- asymptotic behavior; Hankel functions
- spherical Bessel functions
- modified Bessel functions
^^^^^^^^^^ (Mid-term coverage) ^^^^^^^^^^
- Legendre Functions [3]
(Riley & Hobson Sections 9.1 - 9.3)
- ODE and series solution
- orthogonality and Legendre/Laplace series
- generating function
- recurrence relations
- associated Legendre functions
- spherical harmonics
- multipole expansion
- Fourier series [1]
(Riley & Hobson, Chapter 4)
- Euler-Fourier formulae
- the Gibbs phenomenon
- Integral transforms [4]
(Riley & Hobson, Chapter 5)
- Fourier transforms
- relation to Fourier series
- the Fourier integral and the inverse transform
- transforms of derivatives and integrals
- transforms of convolutions
- applications to linear systems
- Complex analysis [7]
(Riley & Hobson, Chapters 14 and 15}
- functions in the complex plane
- multivalued functions
- analytic functions
- Cauchy's theorem
- Cauchy's integral formula
- Taylor and Laurent series
- zeros and singularities
- the residue theorem
- contour integration
- Greens Functions [5]
(Riley & Hobson Section 11.5)
- definition and properties
- boundary conditions
- fundamental solutions
- method of images
- examples and applications