PHYS 508 :: Physics Illinois :: University of Illinois at Urbana-Champaign
Course Description
Content
The course covers four related areas:
- Calculus of Variations. Equations of mathematical physics as variational problems, conservation laws, Lagrange multipliers, origin of eigenproblems, variational approximation schemes.
- Ordinary differential equations. Linear equations: Solution space, linear independence, Wronskians, normal forms. Eigenvalue problems: importance of boundary conditions, formal and true self-adjointness, completeness of eigenfunctions, Fourier series, continuous spectra and Fourier integrals. Green Functions: Range-nullspace theorem, Fredholm alternative, constructing Green functions via jump conditions.
- Partial Differential equations. Classification of PDE's. Hyperbolic equations: wave equation, method of characteristics, shocks and weak solutions. Heat equation: solution by integral transforms. Elliptic equations: Dirichlet and Neumann problems, Poisson's equation, Legendre functions, spherical harmonics, Bessel and spherical Bessel functions, examples from electrostatics.
- Integral Equations. Type I and type II Fredholm and Volterra equations, solution via Fourier and Laplace transforms, Abel's equation. Separable Kernels: compact and Hilbert-Schmidt operators, Fredholm alternative again. Perturbation methods: Neumann and Fredholm series.
Homework Sets
There will be 10 or 11 weekly homework sets that will count towards the final grade, and one end-of-term set of optional problems. Homeworks will be submitted electronically via Gradescope. Your solutions sets must be submitted before 5pm on the due date, which will usually be Wednesday afternoons.
Exams
The midterm will be in class on ****. The exam will cover everything up to the end of chapter 5.
The final exam will cover everything. Its official date/time is (TBD).
Some old exams for you to review:
Midterm Exam, Fall 2013
Final Exam, Fall 2003
Final Exam, Fall 2014
Textbook and Notes
I recommend (but do not require) that you purchase Mathematics for Physics: A guided tour for graduate students by Michael Stone and Paul Goldbart. (Cambridge University Press 2009). The list price is $90, but Amazon has it for $60 (+shipping). I do not yet know what the UI bookstore is selling it for. (I do not recommend the Kindle version of the book. The equations do not scale correctly.) The book is an expanded version of the lecture notes for both PHYS 508 and PHYS 509. Here is a list of typos and errors that readers have found in the printed text.
A draft version of the book is still available for download, but, now that it is published, the authors are no longer maintaining this version, so typos are not being corrected.
As a cheaper alternative you can buy Mathematics for Physicists by Phillipe Dennery and Andre Krzywicki (Dover Publications, $12.95). This book covers a fair bit of the material in this course, and will be useful for the complex-variable part of PHYS 509.
Here are the brief calculus notes mentioned in the first lecture.
Grades and Gradebook
Students usually access the online gradebook. Some data will still be in that location. However this semester all grades will be in Gradescope.
Your grade in the course will be determined as from your total scores weighted as follows: Homework 50%, Midterm exam 20%, Final Exam 30%.
Late Policy
Extensions on homework and requests for makeup exams will be granted only in the event of conference travel, illness with documentation (call Dial-a-Nurse at McKinley), or family emergency.
AI Policy
No AI is to be used for any course activity. Even though AI can be a valuable tool, it is just simpler if we keep it separate from the learning methods for this course.
Last updated Aug/20/2026